NORTH-HOLLAND
MATHEMATICS STUDIES Notas de Matematica editor: Leopoldo Nochbin
Approximation Theory and Functional Analysis
J. B. PROLLA Editor
'"lRTH-HOLLAND
35
APPROXIMATION THEORY AND FUNCTIONAL ANALYSIS
This Page Intentionally Left Blank
NORTH-HOLLAND MATHEMATICS STUDIES
35
Notas de Matematica (66) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Approximation Theory and Functional Analysis Proceedings of the International Symposium on Approximation Theory, Universidade Estadual de Campinas (UNICAMP) Brazil, August 1-5, 1977
Edited by
Joio B. PROLLA Universidade Estadual de Campinas, Brazil
1979 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM. NEW YORK. OXFORD
© North-Holland Publishing Company, 1979
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0 7204 1964 6
Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK. OXFORD Sole distributors for the U.S.A. and Canada: ELSEVIER NORTH-HOLLAND, INC. 52 V ANDERBILT A VENUE, NEW YORK, N.Y. 10017
Library of Congress Cataloging in Publication Data
International Symposium on Approximation Theory, Universidade Estadual de Campinas, 1977. Approximation theory and functional analysis. (Notas de matematica ; 66) (North-Holland mathematics studies ; 35) Papers in English or French. Includes index. 1. Functional analysis--Congresses. 2. Approximation theo~y--C?ngresses. I. Prolla, Jo~o B.. II. Un~vers~dade Estadual de Campinas. III. T~tle. IV. Series. QAl.N86 no. 66 [QA320] 510' .8s [515' .7] 78-26264 ISBN 0-444-85264-6
PRINTED IN THE NETHERLANDS
FOREWORD
This book contains the Proceedings of the International Symposium on Approximation Theory held at the Universidade
Estadual
de
Campinas (UNICAMP), Brazil, during August 1 -5, 1977.
Besides
the
texts of lectures delivered at the Symposium, it contains some papers by invited lecturers who were unable to attend the meeting. The Symposium was supported by the International Union, by the Funda9ao de Amparo
a
Pesquisa do Estado
Mathematical de
Sao Paulo
(FAPESP), by German and Spanish government agencies, and by
UNICAMP
itself. The organizing committee was constituted by Professors Machado, Leopoldo Nachbin, Joao B. Prolla (chairman),
Silvio
and
Guido
Zapata. We would like to thank Professor Ubiratan D'Ambrosio, director of the Institute of Mathematics of UNICAMP, whose support
made
the
meeting possible. Our special thanks are extended to Miss Elda Morta.ri who typed this volume.
Joao B. Prolla
v
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~ABLE
OF CONTENTS
R. ARON, Po.tynomia.t appJtoximation and a que.6tion
06 G.E. SlUlov.
Ana.tytic. hypoe.t.tiptic.ity 06 opeJtatoJt.6 06 pJt.£nc.ipa.e type. . . . . . . . . . ..........
J. BARROS NETO,
13
H. BAUER, KoJtovkin appJtoximation in 6unc.tion .6pac.e.6 . . .
19
A JtemaJtk on vec.toJt-va.tued appJtoximation
on c.ompac.t .6et.6, appJtoximation on pJtoduc.t .6et.6, and the appJtoximation PJtopeJtty . . . .
K. D. BIERSTEDT,
37
B. BROSOWSKI, The c.omp.tetion 06 paJttia.t.ty oJtdeJted ve.c..tOJt .6pac.e..6
and KoJtovkin'.6 theoJtem . . . . . . . . . . . . . .
63
P. L. BUTZER, R. L. STENS and M. WEHRENS, AppJtoxima.tion bya.t-
gebJtaic. c.onvo.tu.t.£on integJta.t.6 . . . . . . . .
71
J. P. Q. CARNEIRO, Non-aJtc.himedean weighted appJtoximation
J. P. FERRIER, TheoJtie .6pec.tJta.te en une in6inite
. . . 121
de vaJtiabte..6 . . 133
P. M. GAUTHIER, MeJtomoJtphic. uni60Jtm appJtoximation
on
.6ub.6et.6 06 open Riemann .6uJt6ac.e.6 . . . .
•
C. S. GUERREIRO, Whitney'.6 .6pec.tJta.t .6ynthe.6i.6 theoJtem
139
in in• • 159
G. G. LORENTZ and S. D.
RIEr~NSCHNEIDER,
Rec.ent
PJtoge.6.6
in
BiJtkh066 inteJtpo.tation . . . . P. MALLIAVIN, AppJtoximation po.tynomia.te. pondeJtee et
c.anonique..6 . . . . . . . . • . . . . •
vi i
• 187
pJtoduit.6 • • 237
TABLE OF CONTENTS
vi i i
Spac.e-& 06 di 66 elr. entiable nun c.tion-& and the app1toximatio n pita p etr.ty. . . . . . • • 263
R. MEISE,
L. NACHBIN, A
look at appltoximation theolty . . . . . . . . . . . 309
L. NARICI and E. BECKENSTEIN,
Ph. NOVERRAZ,
O. T. W.
Appltoximation
on
Banac.h alg e bltCL6 ovelt va..fued 6ie.fM. . 333 plulti-&ubhaltmonic. 6unc.tion-& . . . . 343
The appltoximation pltopeltty 601t c.elttain -&pac.M 06 holomo~phic. mapping-&. · . 351
PAQUES,
J. B. PROLLA, The appltoximation pltopeltty nolt Nac.hbin -6pac.e-6 . . . 371 I. J.
SCHOENBERG,
M. VALDIVIA, A D. WULBERT,
G. ZAPATA,
On c.altdinal -&pline -&moothing . . .
•
.
383
c.haltac.teltization 06 ec.helon Kothe-Sc.hwMtz -6pac.M . . 409
The Itational appltoximation 06 Iteat nunc.tion-6 . . . . 421 Fundamentat
-6
eminoltm-6 . . . . • . . . . . . . . . . . 429
Index . . . . . . . . . . . . . . . . . . • . . . . . • . . . . . 445
Apppo~imation
TheopY and FunationaZ Analysis J.B. FPoZZa (ed.)
© No#h-HoZZand Publishing Company, 1979
POLYNOMIAL APPROXIMATION AND A QUESTION OF G. E. SHILOV
RICHARD M. ARON Instituto de Matematica Universidade Federal
do Rio de Janeiro
Caixa Postal 1835, zc-OO 20.000 Rio de Janeiro, Brazil
and School of Mathematics Uni vers i ty
of Dub 1 in
39 Trinity
ABSTRACT
Let
space. For
College
2,
Dublin
Ireland
E be an inf ini te dimens ional real or complex
an (E) be the algebra generated
n = 0 ,1,2, ... , 00, let
all continuous polynomials on We discuss the completion of
Banach by
E which are homogeneous of degree < n.
antE) with respect to several
natural
topologies, in the real and complex case. In particular, we prove that when
E is a complex Banach space whose dual has
property, then the
Tw
- completion of
those holomorphic functions
f:E + ~
the
approximation
1
a (E) can be iden ti f ied whose derivative
wi th
df: E +E'is
compact. Let ball
E be a Banach space over
B . For each l
n
continuous polynomials
E
IN, let
JK = lR or
unit
P (nE ,F) be the space of n-hcr.ogeneous
P: E + F, normed by
=
B } (P{oE,F) F). ~(E,F) is the space of m-conI H(E,F) tinuously Frechet differentiable functions from E to F and
sup {lIp{x)1I
: x
E
is the space of holomorphic mappings from
E to
are complex Banach spaces. Throughout, if the
F, where range
E and
F
F
is
space
2
ARON
suppressed, then
F = 1K is understood; thus for example HCE)= H(E,fC).
In this paper, we consider variations on the following problem posed by G. E. Shilov [8 I.
For each
n=O,1,2, ...
,00,
let
antE)
be the algebra generated by the collection of functions P(jE), j E lN, 00 00 n j 2 n; thus a (E) = n~o P( E) = P(E). Then, what is the completion (an(E) ,e)
A
topology
of
antE) with respect to some specified locally
T on
an(E)?
convex
In the real case, this problem has been con-
sidered by many authors. In Section 1, we briefly outline some recent res ul ts in this case. When
E is a complex Banach space,
the
above
problem has apparently not been studied. In Section 2, we discuss me
aleE) and
completion of space
aoo(E) for several common topologies on me
(Related results will also appear in [1 I .) In particu-
H (E).
lar, we characterize the
Tw-completion of
1
a (E) as a space of ana-
lytic functions having weakly uniformly continuous derivatives,
and
in terms of compact holomorphic mappings. Some of the results in this paper were obtained while the
au-
thor was a visitor at the Instituto de Matematica, Universidade
Fe-
deral do Rio de Janeiro, supported in part by the CNPq and FINEP, to which the author expresses his gratitude.
SECTION 1.
Among the most natural, and so far unsolved, versions of
the question of Shilov is the following. Given dimensional Banach space, let
T~
E,
a
real infinite
denote the topology on aoo(E) = peE)
generated by the family of norms P E peE) ->-
where Bm = {x 00
a (E)
1\"['0
E
E:
sup { I p (x) I : x
Ilxll.::.m
00
b of a (E).
E
B }. m
Then, characterize the completion
We recall that to each polynomial P E PinE)
corresponds a unique symmetric continuous n-linear mapping A : E x E x _ Axn.
x E
->-
~,
Thus, since
via the transformation P(x) = A(x, ••• , x)
POLYNOMIAL APPROXIMATION ANO A QUESTION OF SHILOV
Ip(x) - p(y)1
<
3
IA(x, ... ,X) - A(y, ... ,y) I
IA(X, ..• ,X,X) - A(x, ... ,x,y) I
+ IA(x, ... ,x,y) - A(x, ... ,y,y) I + ...
... + /A(x,y, •.. ,y) - A(y, ..• ,y) I
<
for
P E
m Ilx - y II
C
p(nE ), x, y E Bm'
element in
and a constant
m
, we conclude that every
P(E), and hence every element in (P(E) 'T~)~ is uniformly
continuous on bounded subsets of Nemirovski'i and Semenov [6] space
C
E. However, it has been shown
by
that for any infinite dimensional Banach
E, there always exists a uniformly continuous function on
which cannot be approximated uniformly on
Bl
Bl by polynomials. In con-
nection with this, we remark that in many Banach spaces
E, the norm
function (which is obviously uniformly continuous on bounded sets) is not the uniform limit of polynomials on bounded sets. This was served by Kurzweil [4]
who showed that, for example in
ob-
E=C[O,l]
(resp. 91 -
1 :::. p, p not even), the norm is not the uniform limit of P (resp. [p]-) differentiable functions. In particular, as Kurzweil
noted, i f
inf { I p (x) I
II x II =l} = 0 for every
P E p(n E ) and n E lN,
then, the norm cannot be uniformly approximated by polynomials on balls; this condition is closely connected with the uniform convexity of the space [5 ] For arbitrary real Banac h spaces
E, t h e
O I et~on . Tb - comp
of
al(E) was discussed in [2] . We briefly sketch the proof of a generalization of this result. Given a family tion
f : E .... F
P
C
P(E), we say that a flme-
is P - uniformly continuous on bounded subsets of
E
4
ARON
(abbreviated "P-continuous") if for each there is
and bounded set
and a finite subcollection {PI""
0 > 0
x, y E B
if
> 0
£
sati.3fy [Pi(X) - Pity)] < 0
,P }' k
n, then
P E p(nE,F)
pact. Indeed, if is compact in for any
Y
E
p(n E ) C P
> 0, then since
E
F, there are uni t vectors II yll <
+ sup
E
[ 'P. (y) 1.
l
which satisfy ['Pi
lip (x) - P (y) II
0
P
T~
and
-
P (x) - 'PloP (y) I <
The 4pace 06
comp.i'.etion 06
{I}.
A
T~
IThe
ana.f.ogoU4 to the cct.6e
PROOF:
'PI""
I•
,'P k
(i
=
P = E'
we have
P
@
P - continuoU4 6unction4 E, wheJte
A i4 the a.f.gebJLa geneJtated
5 > 0
m
E B
be, P - continuous,
and
{Pl, ... ,P } C P k
<jJ: E .... IRk by
A
hi (y) -< 1
@
F is' P-continuous,
a Tb - complete. > 0
E
To
BeE
and
be selected as in the k
hI""
m for all
xE B,II<jJ(x) -<jJ(xi)1I < 0/2
,hm : IR
k
y E IRk
.... IR such that
,
m
m 1:
comp.t'.ete.t'.y
is given the sup-norm). There exist
IRk
non-negative continuous functions ~
by
<jJ (x) = (PI (x) , •.• 'P (x) ), and
such that for any
i =1, .•. 1m (where
i=l
i4
F = IR) •
above defini tion. Define xl, ... ,x
f: E .... F
;topo.t'.ogy i4 de6bted in a manne!!.
f: E .... F
bounded, and let
is
in(2~
was discussed
It is easy to see that every element of
show density, let
for some
F' sum that
1, ... ,k),
and that the space of P - continuous functions is
choose
in
Therefore, for any x,y E Bl
E
then totally bounded. The case in which
the
for
K:: P(Bl) -P(Bl)
< 2£. The converse implication follows because
PROPOSITION 1:
then
is P-continuous if and only if P is com-
P is compact and
K,
P such that
C
( i = l, ... ,k),
IIf(x) - f(y)1I < £. It is interesting to note that if some
BeE,
1
hi (y)
for
y E
B k (<jJ(x.), 0/2) ,and
U
j=l
i=1 spt h.
1.
C
B
k (<jJ (xi) , a) IR
for
IR
J
i=l, ..• ,m.
POLYNOMIAL APPROXIMATION AND A QUESTION OF SHILOV
Choose polynomials
5
such that for
i =1, ..• ,m,
m
sup { /qi (y) - hi (y) /
y
E
U
j=l
B k (<j>(x.),o) < lR J
E/ (m • max (1, II f (xi) II ) ) • m
Then, the function A ® F
and, for all
g: E
~
F
defined by
g(x)
k q.o<j>(x) • i=l ~
f(x~)
E
•
x E B,
IIg(x) - f(x)1I <
+ II
m k /q. (<j>(x»
i=l
~
- h. (<j>(x» ~
m k hi(<j>(x»(f(x ) i i=l
/ IIf(x.)1I + ~
f(x»1i
< 2E ,
m
k hi(<j>(x» i=l
since
= 1
x E B.
for
Q.E.D.
When E is a separable Hilbert space with {e }, Nemirovski! and Semenov [61 i tion of f
: E
~
p(lE,E) < 0
have proved that the
P(E) contains the regular functions on F, where F is a Banach space, is
any bounded set BeE and
E
orthonormal
E,
basis o Tb - comple-
~(E).
said to be
A
~egu!a~
mapping if
for
> 0, there is a finite set {A1, ... ,A } k
::: [(E,E) and 0> 0 such that if x,y
E
(j=l, ... ,k, i <: :IN), then I/f(x) - f(y)/I
C
B satisfy/(Aj(X - y),e i < E. ~(E,F)
space of regular mappings from E to F, which is a
Fr~chet
)I
denotes the space with
respect to the topology of uniform convergence on bounded subsets of E.
Nemirovskil and Semenov found that every f E
the T~ - completion of
~(E)
is contained
m
P (E) •
Since regular functions are bounded on bounded sets, an algebra. It is not difficult to show that
aleE) C
~(E)
~(E) and
is that
ARON
6
all functions
f
n. dt(E) are in a. (x,e ) ~ ~ i i=l n. > 3 for all i . On the other hand, ~ -
of the form 2
00
~
f(x)
< 00 and ~ !ail i=l a 2 (E) fL dt(E) <x,x} ~ tR (E) . Indeed, suppose since f(x)
provided
let
> 0, and let
y
g: E
AI'··. ,Ak
E
£(E,E)
and
1< Aj (x - y) ,e i ) I < Thus any such
g,
/),
be any function which is unifOl:mly con-
->- JR
tinuous on bounded subsets of
E.
For appropriate
such that i f
> 0
<5
IIx - y II <
then
f E tR (E) ,
x,
and so
E,
> 0,
£
there
Y E Bl
are
satisfy
IIg(x) - g(y)11
<
y.
T~ -limit of polynomials,which
being regular, is a
contradicts the previously mentioned result of Nemirovski'! and Serrenov. It is trivial that left by elements of
tR(E,E) is closed under composition
£(E,E). It is also closed under composition
t(E,E) since, if
the right by elements of respond to a given
f
E
tR(E,E) and
£(E,E) will correspond to space
on the
f
0
£
T, for
>
Al, .•• ,A E t(E,E) k
0, then
Al
0
T, ... ,Ak
on cor-
0
T
E
T E t(E,E). In particular, the
tR(E,E) n £(E,E) is a closed 2-sided ideal in
£(E,E) which con-
tains the finite rank operators. Hence, either dt(E,E)n£(E,E) = £K(E,E) , the compact operators, or lows that for that if then
x, y
> 0,
£ E
Bl
II x - y II <
id E dt(E,E). In the second case,
there are
AI' ... ,A
E £(E,E) and
k
<5
it fol> 0
such
!
satisfy
~
)
But then
£.
! ~ i=l < (
~
i=l However, since
112 is not regular I
id
r;.
dt (E ,E) •
Finally, we briefly review the case of differentiable approxiwith
mation by polynomials. In [71 , the authors examined respect to the topologies
n
TC
for
n = 0 , 1 , ... ,
00.
Here,
Tn c
locally convex topology generated by alL seminorms of the form
is the
POLYNOMIAL APPROXIMATION AND A QUESTION OF SHILOV
7
xEK,yEL},
where
K
and
denotes the
L are compact subsets of
E, j E IN, j ~ n, and
j th - Taylor coefficient of
I t was found that if
E has
the
f
at
x,
djf(x)
an element of P(jE).
approximn tion
property,
then
~ Cn(E) , for n'=I,2, ... ,oo. One notes that in this case, c I n ~ (p(E)'Tn)~. The completion of aleE) with respect (a (E), T ) to c c n n T (resp. T ) was studied in [ 2 1 , where these locally convex to po b u (a l (E) , Tn)
logies are defined by the generating family of seminorms of the form
f E aleE}
where and
X C E
sup {jajf(x)(y)j:x EX, Y E E, lIyll < l } ,
..
varies among the compact (resp. bounded) subsets of
j E IN, j
~
n.
(Of course, since
by the Stone-Weierstrass theorem for
In the case of
T~'
tions
x E E,
j E IN,
f: E .. F
subsets of
(a ~
j
l
E. That is, C
tS > 0
such that if
T~) ~
(E) ,
nL Let
€
wu
(E, F)
n > 1,
E'
'= {f
p(nE).
rI
ap-
has the bounded E
= {f : E .. F:
let
Cn(E} : djf (x) E P f (jE)
on
funcbounded
for all bounded
sets and
k
x, y
'T~) ~
E)
(i) ajf(x} E Pf(J
SECTION 2.
c
Cwu(E,F) be the space of
E
B
satisfy
II f (x) - f (y) II < d. Then, if E' (al(E)
u
C(E)
> 0, there is a finite set {'Pl, ... ,'P } C E'
and all
property,
TO '= TO). For
which are weakly uniformly continuous
BeE
then
=
i t was shown that if
proximation property, then for all
T
(al(E}'T)~
span {'Pn:'PEEI}=(al(E)'T~)~
p(n E ) of
Pf(n E ) :: the closure in
E is real,
{f
E
j'P (x i
E
In this section, E and
C
wu
H(E,F). Too
(i
1, ... ,k),
x
E
E, j E IN,
j
< n,
(E,P(jE»}.
F are complex Banach spaces.
first recall some of the usual topologies on pact-open topology on
y) j < tS
has the bounded approximation
Cn(E} : for all
and (ii) ajf
E
H (E ,F). TO
We
is the
c0m-
is the compact-open topology
of
8
ARON
infinite order, generated by semi norms of the form
f E H(E,F) ... sup {llajf(x) (y)11
where
K
C
E
is compact and
vex topologies
between
1
togolopgy associa ted to
a (E) and
pletion of [1
and
1",
1
1
We will consider locally con-
<5' where
is the bornological
10
in particular, our res ul ts are valid for
;
0
the Nachbin ported topology 1
j E IN.
Y E E, II y II < l},
K,
X E
TW. In this section, we study the
com-
a"'(E) with respect to these topologies.
In
1 , this study is continued for the topology Of course, H(E) = P(E)~
for
1
0
, and hence for all weaker to-
pologies, via the Taylor series expansion. Also, if
E has the
ap-
proximation property, then given a compact set
E,
and
TEE 0 E
f E H(E), we can select sup { I f (x)
f
0
T (x)
I
K } <
X E
such £.
1
we have shown that (a (E) ,TO) (a
1
(E) ,To)
~
=
H(E)
then
~
0
C
E
> 0,
II f - f
that
Then, since
mensional, we can find a complex polynomial
II f iT(E) - PIiT ( K) < <: • Thus, II f - P
K
11K ::
T
0
T(E) is finite di-
P : T(E) ... II:
TIIK < 2<:. Since
P
such 0
T
that aleE),
E
H (E) • The converse implication,
E has the approximation property,
is
if
ap-
parently unknown. 1
To study (a (E), T) {f E H(E) and all
~
, we first remark
: anf(x) E Pf(hE) n E IN} •
for all
1
.cwu(E,F). In [2
~
=
x =0) E ... F
f
Cwu(E,F) defined at the end of
pwu(nE,F) the intersection of
pwu(lE,F) is also denoted
(E) ,1)
(equivalently, for
Now we consider holomorphic mappings
which are also in the space 1. We denote by
x E E
( a
that
Section
p(nE,F) and Cwu(E,F);
1 , the following p=perties
of spaces of functions which are weakly uniformly continuous on
bo'-lI1ded
sets were proved.
PROPOSITION 2:
(a)
16
att bounded
f
E
-6e.t-6
Cwu(E,F), .the.n BeE.
feB) -i-6 c.ompac..t -in
F
60ft
POL. YNOMIAL. APPROXIMATION AND A QUESTION OF SHIL.OV
(b)
Pf(nE ) ® F c pwu(nE,F)
(c)
pwu(nE,F)
(d)
Le~
60fZ. aLe
~~ a Qto¢ed ~ub~paQe 06
E
IN.
p(nE,F).
P E p(nE,F) w~~h a~~oQ~a~ed ~ymme~fZ.~c n-l~neafZ. map~ng Then PEP (~,F) i6 and onty i6 ~he map~ng eEl (E ,P(n-~,F», wu wu
A.
whefZ.e
:= Axyn-l. In thl¢
C (x) (y)
P
wu
ha~
(e) E'
W-6e,
the fZ.ange 06 C
n P ( E,F) K
and evefZ.Y
F
n,
n
P
P ( E) f
such that
181
P (nE , F)
f: E
+
F
to
F,
x = 0) there is a neighborhood
feU) is compact in
F.
Le~
PROPOSITION 3:
60Jt eveJty
E
x E
The
HK (E, F), is the
U of
X
x
E
E
such that
I ,the following is proved.
f E H(E,F). Then
(equ.i.vaientiy,
Let
PROPOSITION 4:
F.
such that for each
(equivalently, for
In [3
consisting
l
E
60fZ. eveny
F = P ( E,F). wu
P (B > is compact in
space of compact holomorphic mappings from set of holomorphic mappings
in
n
denote the closed subspace of
of all those polynomials
.i.¢
(n-1 E F) • '
~he appfZ.o)(ima~ion pfZ.opefZ.~Y ie and only i6
BanaQh ¢paQe
Let
n
9
i6 and onty
f E HK(E,F)
60n. x=O) and
in
n E :N, anf(x) E PK(~,F).
pEP (n E ). Then the 601lowln.g
co
n.ditio H-6
afZ.e
eq uiv al en.t.
(al P E pwu(nE ), (b)dPEP
B = B
l
tion of
l.p.
~
(a) ,
(n-1 EE ,) ' ,
n-l dP E P ( E ,E') • K
(c)
PROOF:
wu
~
(b):
and choose C
(x - y)
wu
I
Let
E >
0 > 0
0 and
and (without loss of generality) let {~l""'~k}
c E'
(E).
By the polarization formula, if
<
(i =1, ... ,k) and if
I')
mapping corresponding to
P,
as in the definix, y E Bl
A is the symmetric
satisfy n -linear
ARON
10
n l IIAX - _ Ayn-l ll = SUp {I Axn-l Z - Ay n-l z I : z E Bl }
sup{ 1_1_ n , 2 n.
£.
~
l: P(£lx+", +£n-I X + £n z ) -P(£lY+'" =±l
+£n-lY+£nzl : ZEBl}
l
n!
n
n
± 1, 1 < i
Since are in II Ax n - l dPEP
x =
~
(£1 x +
Ic,oi Bl and Ayn-l ll < 8
wu
...
(x ;
< n}.
1 + £n-l x + £n z ) and y- =n ( £ 1 Y + ..• +e: n-l Y + £n z )
y)
n
DT
I < IS
(i =l, ... ,k),
aP (x)
Since
= nAx
n-l
we ,
conclude we
that
conclude
that
(n-1 EE ,) ' •
(b)
~
(c):
This implication follows from Proposition 2(a).
(c)
~
(a) :
Assume
X E E ~~ dP(x) n
dP E PK (n-I E ' E') ,
= Ax n - l E E'
mula, i t follows that the
so
that
the
mapping
is compact. By the polarization for -
n-l
linear mapping
is compact. Hence, given £>0, there is a finite set {c,ol, ... ,c,ok} eE' n-l such that for any (xl' ... ,x n _ ) E Bl ' IIA(Xl, ... ,xn _ ) - c,oill <8 for l l some
i =1, ... ,k. Assume now that
x, y E Bl satisfy
(i =1, ••. ,k). Then by the synunetry of
. .• + I Ay
n-l
x - Ay
A,
n-l
YI•
Ic,01 (x - y) I < e:
POLYNOMIAL APPROXIMATION AND A QUESTION OF SHILOV
11
Por an arbitrary term above,
j
lAy x
n-J'-l
J' n-J'-l
x - Ay x
yl
<
< 3E ,
so that
Ip{x) - Ply) I < 3En.
Note that in (c)
~
Q.E.D.
(b) above, it is essential that the polyno-
mial be a derivative since, in general, pwu{nE,F) is properly n
tained in
PK ( E, F). For example, if
ertyand
F =
a:,
E'
pwu{nE ) = Pf{lLE )
then
con-
has the approximation propC
PK{nE) = PInE) , and the
inclusion is proper in general. Combining the above results, we get the following. THEOREM 5:
Le~
alt
and aU,
the
n
E
:IN
app~ox-<-mat~on
f E H{E). Then x
E, dnf{x)
E
p~opeltty,
~6
df E HK (E,E')
then
and only
~6
6o~
Pwu(E). Fu~theltmo~e, -<-6 E'
E
f E
(a
1
(E), T)
~
and
haJ.>
only
-<-6
df E HK (E, E') •
PROOF:
1
We only prove the second part of the theorem. f E (a (E) ,1) -n
n
~
n
x E E, d fix) E P { E) = P (E) f wu by Proposition 2{e). By Proposition 4, dnf (x) E P (nE) if and only wu if and only if for any
E
:IN
and
P (n-1E,E'). Since K follows by Proposition 3. Q.E.D.
if
d{anf{x»
n
E
df
~ n=l
d{dnf (x» n!
'
the
Finally, we remark that in [1 I , we show that for any Banach space
E, given
f
E
H (E), df
locally weakly uniformly continuous.
E
result
complex
H K (E,E') if and only if f is
AAON
12
REFERENCES
(1)
R. M. ARON, Weakly uniformly continuous and weakly sequentially continuous entire functions,
to appear in Proc. Inf.Dim.
Holomorphy 1977, ed. J. A. Barroso, North Holland. (2)
R. M. ARON and J. B. PROLLA, Polynomial approximation of
dif-
ferentiable functions on Banach spaces, to appear. [ 3)
R. M. ARON and R. M. SCHOTTENLOHER, Compact holomorphic mappings on Banach spaces and the approximation property, Journal Functional Anal. 21 (1976), 7 - 30.
[ 4 )
J. KURZWEIL, On approximation in real Banach spaces,
Studia
Math. 14 (1954), 214 - 231. [ 5)
J. KURZWEIL, On approximation in real Banach spaces by analytic
operations, Studia Math. 16(1957), 124 -129. (6)
A.
s.
NEMIROVSKII and S. H. SEMENOV, On polynomial approximation of functions on Hilbert space, Math. USSR
Sbornik
21(1973), 255 - 277. [ 7 )
J. B. PROLLA and C. S. GUERREIRO, An extension
of
Nachbin's
theorem to differentiable functions on Banach spaces with the approximation property, Arkiv for Math.
14(1976),
251 - 258. [ 8)
G. E. SHILOV, Certain solved and unsolved problems in the theory of functions in Hilbert space, Vestnik Moscow Univ.Ser. I, 25 (1970), 66 - 68; Moscov Univ. Math. Bull. 87 - 89.
25 (1972),
Appl'Oximation TheopY and Functional Analysis J.B. P~oZZa (ed.) ©No~th-HoZland PubZishing Company, 1979
ANALYTIC HYPOELLIPTICITY OF OPERATORS OF PRINCIPAL TYPE
J. BARROS NETO Mathematics Department Rutgers university New Brunswick, New Jersey 08903, USA
Let
(1)
P(X,D)
be a differential operator with analytic coefficients in an open set g of lRN. Suppose that
P is of principal type and, in addition, sat-
isfies the hypoellipticity condition: along the null bicharacteristic strip of
Re(P m), the function
It follows from [21
Im(P ) has only zeros of even order. m
that the differential operator
P
is
analytic
hypoelliptic. Indeed, in his paper [21, Treves proves that, for differential operators of principal type, the following properties
are
equivalent: hypoellipticitYi analytic-hypoellipticitYi sub-ellipticity and the above condition on the zeros of the imaginary part of
Pm'
Our aim is to present another proof of the fact that, for
op-
era tors of principal type, the hypoelliptici ty condition
formula
for
pseudodifferential operators, we can replace, modulo analytic regu larizing operators, the differential operator pseudodifferential operator
L
=
D - A(X,t,D ) t x
P
by of
an order
analytic 1,
where
A(x,t,D ) is an analytic pseudodifferential operator of order I, with x respect to the variable x, only, and analytic coefficients depending 13
on
14
(x,
BARROS NETO
t) •
Let, then,
( 2)
L
be a first order analytic pseudodifferential operator defined in an open set in
n l
+
:R
n
which we can assume, without loss of generali ty,
to contain the origin. Let
O(L)
( 3)
A(X,t,~))
(T -
L,
be the symbol of
+
AO(X,t,~)
where the principal symbol
n
analytic function of all its variables on neous of degree 1 with respect to
n
an analytic function in ~.
respect to (O,O,~
write
00
,T )
(~,T),
x
T
-
~n+l
while each
+ .. ,
is an
A(X,t,S) \ { O},
homoge-
A .(x,t,~) -J
x Rn \ { 0 }, homogeneous of degree
=
T
0
A(O,O,~
=
a(x,t,~)
+
o . ) wl.th
ib(x,t,~)
00
(~,T)
is with
-j
We shall reason in a conic neighborhood of the
such that
A(X,t,~)
A_l(x,t,~)
+
'I (0,0).
point If
we
our basic assumption will
be
the following one:
(4)
the 6unction < 2k
b(x,t,~)
hah only zenoh 06 6inite even ondeJ1..
along the iyt;f:egJta.t cunvu. 06 the di66enentia.t
Mj.6tem
dx
dt contained in
Wx
r', wheJte
.6uitabte cone in ~n+l
THEOREM:
r'
i.6 the pnojection
containing
06
a
(~O,TO).
Unden the above a.6.6umptionh, the openaton
L
hah
a .tocat openaton
ANALYTIC HYPOELLIPTICITY OF OPERATORS OF PRINCIPAL TYPE
K : C~(U) -+ V' (U) . -en
whole U i-6 a -6uitable neighboJthood 06 the
I
R n +1 ,
lK (x,t,y,s),
(5)
15
LlK (x, t,y, s)
de6ined
oJtigin U x
.{.vI
u,
o(x-y,t-s) +R(x,t,y,s)
with the 60llowing additional pJtopeJttie-6: a)
lK
i-6 a JtegulaJt
~eJtnel
with Jte-6pect to the vaJtiable-6 (x,t)
and (y,s), i.e., the mapping6
u(y,s) -+ (lK (x,t,y,s) ,u(y,s»
Coo(U) b)
the keJtnel (x,t)
c)
~
COO (U) ;
into
c
and v(x,t) -+(]K (x,t,y,s) ,v(x,t»
i-6 an analytic 6unction
lK (x,t,y,s)
(y,s); tK aJte
the opeJtatoJt K and it6 tJta. n-6 po -6 e local in the 60Uowing -6e.n.6e: let pact open .6ub.6e.t 06 in
d)
PROOF:
then
W,
ill (x,t,y,s)
Ku
U
and
and let
u
w E
ana..tuUc - p-6eudo
be any Jt el at..i.. v ely comE' (U) ; i6
u i-6 ana.R..yUc.
t Ku aJte al60 ana.lytic in
i-6 an analytic 6unc-tion -en
Assuming that the conic neighborhood
never changes sign in b
~
Wx
W x r'
,is
with
Ku(x,t)
connected b(x,t,~)
r'. Moreover, in what follows we are going
O. The case
b
~
0
is treated in a similar way.
Define the Fourier integral operator
(6)
Wj
U x U.
(which is always possible), then condition (4) implies that
to assume that
wheneveJt
16
BARROS NETO
K(X,t,~,T)
(7)
ft
ei~(x,t,t,,~)-i(t-t')Tk(x,t,t' ,~)dt'
-T
T is a small number greater than
where
where the pha.4e 6uYlc.tion
~
0 to be chosen
and the a.mp£Ltude 6unc..tion
later k are
and to be
determined i.n such a way that
LKu
(8)
wi th
u + Ru,
for all
UE
C~(U),
R an analytic regulari zing opera tor.
We choose the phase function
~(X,t,t'/~)
as the solution of
o (9)
with
t
and
t'
belonging to the interval
[-T,T).
Since
A(X,t,t;)
is an analytic function of all its variables, there is a unique lution
~
so-
of (9), analytic with respect to all its variables and haro-
geneous of degree 1 with respect to
~.
As for the amplitude function, write
(10)
~
k(x,t,t',~)
kv(x,t,t',t;)
v=O
kv is homogeneous of degree -v with
as a formal sum where each term respect to
~.
The functions
k v'
v = 0,1,2, •.• , are obtained as
lutions of the following transport equations: n l:
A~ .(x,t,~
j=l
J
(11)
=
i
+
~x)
Dx j kO + CkO
o
s0-
ANALYTIC HYPOELLIPTICITY OF OPERATORS OF PRINCIPAL TYPE
17
and
n D k t v
~
Ac (x,t,~ + ¢x)Dxj k
j'=l c,j
v
+ Ck
v
+
v-I ~ !2.v-v ,kv '
o
v'=O
(12)
o,
with
v
= 1,2, •.•
Setting
J-Tt ei¢(x,t,t' ,~)-i(t-t')Tk.(x,t,t',~)dt' J
(13)
i t can be proved, using sui table estimates for
(14)
k
~
j=O is an analytic symbol [1 J
¢ and
k
j
, that
k. (x,t,t' ,f,;) J
and that
K
defined
by
(6)
is
a
pseudodifferential operator. Finally, one can show that its distri bution kernel
~
satisfies (5) and properties a), b), c) and d)
of
the theorem. The existence of such a kernel implies then the analytic poelliptici ty of
hy-
P.
REFERENCES
[ I l L . BOUTET DE MONVEL, Operateurs pseudodifferentiels analytiques et operateurs d 'ordre infini, Ann. Inst. Fourier 22(1972), 229 - 268. (2 J
F. TREVES, Analytic-hypoelliptic partial differential equations of principal type, Carom. Pure and Appl. Math. 537 - 570.
24 (1971) ,
This Page Intentionally Left Blank
Approximation Theory and Funational Analysis J.B. ?rolla (ed.) ©North-HolZand Publishing Company, 1979
KOROVKIN APPROXIMATION IN FUNCTION SPACES
HEINZ BAUER Mathematisches Institut der Universitat Erlangen-Nlirnberg 0-8520 Erlangen, Bismarckstr. 1 1/2 Federal Republic of Germany
INTRODUCTION The starting point of this survey lecture is Korovkin approximation for a linear space
X of continuous real-valued functions
a compact metrizable space
X where the approximating operators
defined on the total space
C (X)
on
X.
on are
of continuous real-valued functions
This type of setting is called here absolute Korovkin approxi-
mation. Chapter I recalls the main results, in particular the characterization of the Korovkin closure of the given function
space
X.
Motivations, details and references to the relevant literature can be found in the author's survey article [3 I. Chapter II is devoted to the problem of determining the Korovltin closure in cases where it is not all of introduction of the state space of
C (X). The main tools are the
X and the use of convexity argu-
ments. The results of this Chapter arose from discussions Leha. Details will be published
with
G.
elsewhere.
Chapter III studies problems of the so-called theory of
rela-
tive Korovkin approximation. Here the approximating operators are no longer defined on all of subspace
£
of
C (Xl
C(X)
containing
but rather on a fixed closed
linear
X. Most of the results of this Chap-
ter are due to Leha [7 I . 19
BAUER
20
I. ABSOLUTE KOROVKIN APPROXIMATION We shall treat here absolute Korovkin approximation
only for
spaces of continuous functions on a compact, even metrizable though the main results essentially remain true for locally
space compact
spaces [4 1 • Consequently, let C(X)
compact
X be a
met~izabte
space, denote
the linear space of all continuous real-valued functions on
and by Je a nune.t:ion .6pace (on subspace of
C (x)
Le. a point separating,
X),
X
linear
containing the constant function 1. The space usual
will be considered as a normed space equipped with the
11··11
by
C (X)
norm
of uniform convergence (sup-norm). A sequence
(Tn)n
E
IN of positive linear operators
Tn
C (X)
->- C (X)
will be called (Je-) admissible if
lim
n->-oo
A function
f
E
C(X)
11Th - h II n
o
for all
h
E
Je.
satisfying
lim II Tn f n-+ oo
- i II
0
for all admissible sequences will be called a K040vkin 6unction (with respect to Je). The set of all these functions is the Kor(Je)
of
Je. Obviously, it is a linear space satisfying
Je
Je
KMovun dO.6Me
is called a
Ko~ovkin
C
Kor(Je)
.6pace if
C
C(X).
Kor(Je)
C (X).
KOROVKIN APPROXIMATION IN FUNCTION SPACES
21
The Korovkin closure can be characterized by means of the following envelope technique: For an arbi trary
f
E
C (X) twJ envelopes are
defined:
/\
f
inf {h
E
Je
h > f}
f
sup {h
E
Je
h < f}.
-
and
v
-
A
Functions
fEe (X)
for which
f = f (= f) are called Je-a66ine.
v
A
The set ing
Je of these functions is a linear subspace of
C(X} contain-
Je. It turns out to coincide with the Korovkin closure:
Kor (3C )
THEOREM 1:
Another characterization of the Korovkin closure is obtained by means of the representing measures. A positive Radon measure is a
4ep~e~en~ing
mea~u~e
for a point
x E X
(with respect to
for all
hex)
The set of these measures will be denoted by Mx(Je). tains the Dirac measure The set
LEMMA 1:
Fan
Mx(X)
£x
It always
defined by the unit mass in
x
and
].I
E
E X,
M (X)} x
on X
X) if
hEX.
con-
x.
is then described by the following key
f E C(X)
\.l
lemma:
we have
[ f (x)
I
f (x) I .
v
This leads to a new description of the functions in
'" and hence of X,
22
BAUER
the Korovkin closure: A function
f
E
C (X)
is
X-affine if and only
if
J fd\.l for all
x E X
f
(xl
and all representing measures
\.l E
Mx(X).
As a consequence of this and Theorem 1, we obtain a character-
Choque.t: boundaJty
ization of Korovkin spaces. It uses the notion of the 0XX
x
E
of X
X with respect to permitting only
X which by definition is the setof points
Ex
as a representing measure:
{x E X
The. g-<-ve.n 6unct-<-oYl. 6pace. ;}{' -<-6 a KOltovk.-<-n .opace -<-6 and only
THEOREM 2:
'<'6
0xx
=
x.
It is this result which allows in many concrete examples a quick proof of a Korovkin-type theorem. In particular, Korovkin'sclassical result follows almost immediately. It states that, for a compact interval
X = [a,b] on the real line
functions 1, id, id
2
(id
= identity
lR, map
the linear hull of the three x
~ x) is a Korovkin space.
II. DETERMINATION AND GEOMETRICAL INTERPRETATION OF THE KOROVKIN CLOSURE In the existing literature few attention has been given to the determination of space, hence where
Kor (:IC) 0XX
for the case where
* X.
3C
is
not
a
Korovkin
We shall have a closer look at this [a,b 1 where
lem in particular for the case
X
of I, id
u E CeX). A direct application
and a third function
~
X is the linear hull
Theorem 1 in connection with the characterization of
of
X-affine func-
tions by means of representing measures turns out to be difficult, in
KOROVKIN APPROXIMATION IN FUNCTION SPACES
23
general. However, ideas from the theory of integral representation in convex compact sets lead to a satisfactory method. Continuing in the general situation of Chapter I, we denote by the .&:t.a:t.e .&pac.e of
S
S (JC)
'P
JC ~ lR
JC consisting of all positive linear fonos
which are normalized, i.e.
set of the topological dual
E
= JC'
j : X
I,
pp. 79 - 82; [2
ex S
{l
0
is a convex sub-
S
j
a(JC',JC). There is a canoni-
S, namely
~
where
j (x)
x E X. The following properties
the evaluation functional for well-known ([ 1
= 1.
JC'; it is c.ompa.c.:t. me:t.fLiza.b.te in the space
equipped with the weak topology
cal continuous embedding
'P (1)
1,
IS
x
is are
pp. 121 - 125) .
(the set of extreme points) ;
lEE'};
in particular,
S
conv j (X) ,
and h
loj-+ljS
is an order and norm preserving bijection of of
JC onto a dense subspace
A(S), the space of all continuous affine functions
For the case
dim JC < +
this is a bijection of
00
or, more generally, for
JC onto
Y
= j
(X»
JC closed in C (X)
A(S).
Let us consider now a compact set (like
a:S-+lR.
and let us define for
Y
ex
such that
gEe (Y)
the
envelopes inf {a E A(S)
a > g
on
y}
S eye S
"geometrical"
24
BAUER
and
a < g
sup {a E A (S)
By
A(Y,S)
y},
on
we denote the space
A(Y,S)
{g
E
C(Y)
on
As a consequence of the above properties of
y} •
x
j
~
S(X)
we obtain
the canonical isomorphism
" A(j(X),S) -+JC
rP
9 -+ go j
A
of
A(j (X) ,S)
Example:
onto
the state space
X = [a,b]
Consider
the function space u E C([a,b]).
X.
Xu
Since
=
a compact interval in with
lin {l, id, u}
j(X)
= {l} x G
u
where
a
given
u is
ne~the~
convex
no~
and
function
G is the graph of u, u
S can be identified with the closed convex
Suppose that
lR
hull of
concave. Then there
ex-
y E G such that S is the 6ace F y of S generated by u yES. Consequently, for every g E A(Gu'S) the concave function
ists a point
gG
u
- g
-G u
, defined on
S,
Fy = S. This proves that striction of functions of phism between
A(S) and
vanishes at
y
and hence at every point of
is affine on
S.
Hence the
re-
defines a canonical isomor1\
A(Gu'S). Consequently,
Xu
= :K'u'
A
Therefore, we can only expect to have
Je
u
=
Kor (JCu ) '" Jeu
is concave or convex. If follows from Theorem 2 that
" Je
u
=
C(X)
if
U
is
KOROVKIN APPROXIMATION IN FUNCTION SPACES
25
equivalent to the strict concavity or convexity of u. Since
id
JC
E
u
one only has to observe that every representing measure has
x as barycenter. So let us assume that
concave functions in
u is an element of the set
of
K
C([a,b]). In what follows we shall
get
all
addi-
tional information about the behaviour of the map A j{
U
defined on
K. We can introduce a pre-order relation on
K by definX
Then the relation
u
~
v
expresses that
As consequence of the characterization of
[a,b] .
E
v is more concave than
u.
JC-affine functions by means
of representing measures and of Theorem I we obtain the implication
(u,v
K)
E
There are two extreme cases: the affine functions on
[a,b]
are the
minimal, the strictly convex continuous functions on
[a,b]
are the
K.
maximal elements of lin {I, id}
= A(
The
[a,b])
Huch better
and
corresponding
closures
are
C([a,b]), respectively.
results can be obtained by making use of Alfsens
notion of boundary (affine) dependencies [I ] eral framework of this Chapter. For a point all
Korovkin
bounda~y dependencie~
signed Radon measures
v
We return to the genyES
B
the set
y
of
is, by definition, the linear space of all on
S which are supported by the Go-set ex S
and which annihilate all affine continuous functions on
J
IS
adv
o
for
all
S:
a
E
A (S).
BAUER
26
As a consequence of the minimum principle for lower semicontinuous concave functions, a function by its restriction to
g -.. g
I exs
ex-5. Therefore,
defines an order and norm preserving isarorpo.'1ism of A(Y,S)
A(Y,S)
PROPOSITION 1:
.t-ion -in
uniquelydet~ned
is
ex S, hence in particular to
onto a certain linear subspace of quently,
g E A(Y,S)
C(ex:s). This subspace and,conse-
can be described as follows:
A 6tHl.e.tion
q E C (ex S)
i~
lLe~.tIL.ic..tion
.the
-i6 and only i6 .the 6ollow-in9 two
A (Y ,S)
06 a 6une-
c.ond.Ui.o~ aILe ~a.tU-
6-ied: (al
qEA(exS,S);
(b)
f qdv
o
and aU
Condition (b) is still redundant. By using the face generated by a point
PROPOSITION 2:
FOIL
(a)
(b)
eVelLY
6unc..t.i.on
q E C (ex-5)
and
eVelLlj
po-in.t
yES
[a,b J
with
aILe equivalen.t:
o
r qdv J
I
of S,
yES, it can be improved:
eondi.t-ion~
.the 6ollowin9 .two
Fy
o
qdv
V E
U
B
zEF
z Y
We return now to the discussion of the
E x
n
amp l e:
We choose for
u a concave polygon
proper vertices. This means that
where
a l , .•. ,a n + 1
on
u is of the form
are affine functions on
[a,b J
such that a j
~
'i<
KOROVKIN APPROXIMATION IN FUNCTION SPACES
holds only in the trivial case point y in the interior of Furthermore
ex S
j =k
27
(n =1,2, ... ). For an
conv G we have F = S. u Y n + 2 vertices of S. Therefore
S
is the set of the
it follows from the two preceding propositions that and hence
~u
q E A(ex S,S)
arbitrary
A(S} =
A(S,S)
is canonically isomorphic to the linear space of
all
satisfying
o
for all
V E B
Y
A
Furthermore we know that phic to
A(Gu'S)
JC
is (by means of
u
~)
canonically isomor-
and hence to the linear space of all
q
E
A(ex S,S}
satisfying
o
for all
and all
Z E
G \ ex S. u
The latter condition is empty since Indeed, every
z
B = {0 } for all z has a unique representation as barycenter
G u a probability measure on E
ex S
an extreme point. Consequently, A(ex S,S).
tions in
~
n +2
B
y
z is a segment or reduces to isomorphic
y lies in exactly
ex S. These produce
B . Since y
F
is canonically
u
It is easy to check that
with vertices in tors of
since
n - 1
to
n triangles
linear independent vec-
is determined by a system of 3 linear
variables, we obtain
of
equa-
dim By = n - 1. This proves our
final result, namely dim " JC
u
dim
~
u
n + 2.
+ n-l
Formally, this equality also holds for The canonical isomorphism between the same time clear that the elements of
n =0.
'"u
JC
'"JCu
and
A(ex S,S} makes at
are piecewise
affine.
BAUER
28
More precisely: Let
< x
be those points in
n
u has proper vertices. Then C([a,b) where
is
=a
and
n +2
in
[xi,xi+ll, i=O, ... ,n,
xn+l = b. This can be seen also directly by means
of the representing measures. This description of ber
wh<3re
the space of functions
which are affine on every interval Xo
[a,b)
....
JC
u
make s the num-
of its dimension evident.
III. RELATIVE KOROVKIN APPROXIMATION We return now to the situation studied in Chapter I. Hence is a function space on a compact metrizable space of
kel~t~ve
Korovkin approximation if the role of
Korovkin approximation is taken over by a
clo~ed
JC
X. We shall speak C(X) in
absolute
6u~ct~o~ ~pace
t
containing JC as linear subspace:
x
c t
Consequently, a sequence
t c C (X).
(Tn)n
E:IN
of positive linear maps is called
(JC,£)-admissible if
o A function
f E t
spect to JC and
is called a
for all
ketat~ve Kakovk~~
hEX.
6u~ct~o~
(with
re-
£) if
o
holds for all
(X, £) -admissible
is the relative Korovkin closure Kor(JC,£)
sequences. The set of these functions Kor(JC,£)
is a function space satisfying
of JC with respect to £.
KOROVKIN APPROXIMATION IN fUNCTION SPACES
29
JC C Kor (X,£) c £ •
X is called a KOJ!.Ovkin
if
.£
Kor(X,£) = £. As
in the absolute case the main problems are to characterize and to decide whether
Kor(X,£)
X is a Korovkin space with respect to
£.
Let us consider first three
El(ctmpie.
1)
X=[-l,+lj,
and
3 lin {I, id, id 2 , id }. It follows from the considera-
tions in Chapter II that
~j
(-1, -
We shall see that 2)
X=[O,lj,
U
[;
,
11
X.
and
Kor(X,£) = £ .
rl J f(x)dx
£={fEC(X)
f (0) },
o
~
{f E £
(f(
~)
+ fell)
=
f(O)}
. It is easy to check
that 10,1] •
We shall see that 3)
Let
Kor(X,£) = X .
X be the closed unit disk in
of all affine functions on
lR
2
X, and let
,
X the space £
be the space
A (X) of
all functions in
C (X)
Then
is the unit circle, i.e. the topological
QXX = a£x
boundary of
which are harmonic in the open disk.
X. We shall see that
Kor (JC , £)
= £ .
.
Saskin [ 9 I announced a result that - at least for the case of a
finite dimensional space and sufficient for
X - the condition
Kor(X,£)
= £.
QXX
a£ X
is necessary
The first two examples show
that
30
BAUER
this is not true. However, we shall see that Choquet boundaries
and
0JCX
o£X
the
equality
of
the
is sufficient for JC to be a Korovkin
.£ if in addition the common boundary oJCX = a.£ x
space with respect to
is closed. This additional condition is fulfilled in the third
ex-
ample. ~elat~ve
Crucial for the relative theory is the notion of the
Chaquet baunda~y
a;x
which by definition is the set
{x e X
For
.£ = C(X)
aJfx,
hence
this is exactly the definition of the Choquet boundary
oJCX
= a;(X) x.
Immediate consequences of
the
definition
are the following two remarks:
Accordingly, we have in the above Examples:
1)
aJC X = [-1,
; I
U [
~
,
I I ;
I 0,11
2) (since
Mx(JC)
3)
* Mx(.£)
for the origin
x = 0);
topological boundary (since
Mx(X)
Also the notion of a function
-
f E C(X)
* Mx(£)
for all interior points of
Xl.
X-affine functions will be generalized. For
,..
the definition of the envelopes
the one given in Chapter I. A function
f
f
and
f
is
will be called (Je,.£) -aMine
31
KOROVKIN APPROXIMATION IN FUNCTION SPACES
if
f
E
£
and if
A
f (x)
for all
fix)
v
The set
£ = C(X)
for
fi£ c
c
E
a£
x.
£
of these functions is a linear subspace of
JC
Obviously,
X
£ •
xC (X)
we have
"= JC.
The (JC,£)-affine functions do not play the same role as in the absolute theory. The following result generalizes only one
part
of
Theorem 1.
PROPOSITION 3:
Ev~~y
(JC,£)-a66~n~
6unet~on ~~
~eiative
a
Ko~ovkin
6unetion: ,,£ JC
We sketch the proof:
"fix) = fix) a
on the set
" given number
hi,···,h~
and
€
C
Kor (JC, £) .
Let
S = a£
be a function in
f
x.
Compactness of
S
;C£
Then
then leads,
functions
> 0, to the existence of finitely many
hl,···,h~
and
JC
in
h
such
= inf
h < f
<
that
(hl, ••• ,h~)
the
for
functions
two
satisfy
h
and
hex) - !lex)
< £
for all
This implies for an arbitrary (JC,£)-admissible sequence
(Tn)
xES.
that
BAUER
32
(Tnf) converges uniformly on principle follows that Indeed, a function Ig(x) I <
g
S
to
f.
From this
and
(Tnf) converges uniformly to
II gil <
satisfies
£
E
E
the maximum
f
even on
if
and
only
X. if
holds for all
E
We obtain two corollaries:
COROLLARY 1:
Kor (JC, £) = £
This follows by observing Lemma 1 which implies a characterization of the relative Choquet boundary, namely
,.. ()
fE£
where
,.. f}
{f
stands for the set
f},
{£ v
{x
E
X
v
f (x) }.
v
COROLLARY 2:
a£ X
"-
f(x)
Kor (JC,£) = £
M.e eta!.> e.d and eainc.).de..
This follows from the first remark following the definition of the relative boundary. Corollary 2 settles Example 3. It can be seen from Example 1 that one cannot expect to the equali ty
Jf £
= Kor
PC, £) in Proposi tion 3 without additional as-
sumptions. Indeed, since
a£x = X
we have
ample. However, we know from Chapter II that tion
id
3
have
is neither convex nor concave on
Proposition 4 will make clear why
;/i£ '" JC
;/i () JC
in this ex-
£
since the func-
[-1, + 1 J •
Kor (JC,£)=£ and hence
Furthermore "'£
JC
* Kor (X,£) •
The proof of Proposition 3 uses a property of the closure S of the Choquet boundary S
a£x
which holds for much smaller closed
sets
in certain cases. It is this observation which leads from Proposi-
tion 3 to Theorem 3. A set
sex
if a function in
will be called
£-de~e~m..[ning
if it is closed and
£ vanishes identically provided that it vanishes at
KOROVKIN APPROXIMATION IN FUNCTION SPACES
all points of
5. A closed set
tenmil1il1g if for every
will be called
5 C X
there exists a
> 0
£
33
0 >
J...tJtol1gly I- desuch that
0
the
implication
if (x) I holds for all
f
E
xES" II f II <
for all
< 0
I. Obviously, strongly I-determining impliesI-de-
termining. A closed set
5 C X
is strongly I-determining if and only
i S, defined by restricting a function
if the map
PS : I
to the set
S , i s bi j ecti ve and open.
+
£
We have seen that the closure I-determining. If
has finite dimension
set of
of
S is I-determining and if
then, by the open mapping theorem, I
£
5 of cardinality
n
I
f
Xl""Xn E X
always
strongly
: S
is closed in C(5)
S is strongly I-determining.
If
then there exists a strongly I-determining
n. It suffices to choose a base
fl ' ... , fn n
such that
det (f. (x.)) ~
Consequently,
£
is
I. A simple induction argument then yields the existence of
points
E
S = {xl""
,x } n
J
'*
O.
is I-determining and by the preced-
ing argument strongly !-determining. In particular, i f ! .is the set of real polynomials of degree [a,b)
C
~
IR, a'" b, every set of
n
restricted to a compact
n+l
interval
differentpointsxl, ... ,xn+lE[a,b)
is strongly determining. Therefore,in Exarrple 1 the set 5 ={-l, ! is strongly i-determining and contained in aXx.
~, ~, I}
A simple revision of the proof of Proposition 3 now leads
to
the announced improvement:
THEOREM 3: ti~ 11
f E I
Let 5
be a J..tnol1gly
!-detenmil1il1g J..et. Thel1 eveny 6ul1c-
BAUER
34
f(x)
XES
v
i..J i.. n
Kor (:IC, £) . is the intersection of all sets
Since f
E
£, we obtain
COROLLARY: J
with
f
Kor(X,£)
e.t. This corollary settles Example 1. It contains the
corollaries
of Proposition 3 as special cases.
.£ =
For the case of absolute Korovkin approximation, that is for ,,£ C (X), Theorem 1 states that Kor (JC,.£) equals X We have seen
that in the relative theory one cannot expect a similar result out an additional assumption on
£. For
£
= C(X)
the
state
S(C(X)), defined in Chapter II, is the convex compact set all (Radon) probability measures on
X, hence a simplex
withspace
M!(X)
of
(in the sense
of Choquet). It has been proved recently by Leha and Papadopoulou [8] that the corresponding property for general
.£
leads to the complete
generalization of Theorem 1. Continuing the discussion in the general case of the theory,
£
relative
is called Ji..mpii..ci..ai if the state space S (£) is a simplex.
The result then is:
THEOREM 4:
FOIt Ji..mp.U.ci..ai £ a 6uncti.on i..J a Ite.lati..ve. Koltov/Un
6unc-
ti..on i..6 and only i..6 i..t i..J (3C,£)-a66i..ne:
Kor
The proof given in Lazar [6]
[8]
(:IC, £)
=
"'£ 3C •
makes use of the selection theorem of
for (metrizable) simplexes. An immediate consequence
is
KOROVKIN APPROXIMATION IN FUNCTION SPACES
35
then the following result which contains Theorem 2 as a special case:
THEOREM 5:
£
:JC
L~
a KoJtovk-in .6pa.c.e wLth Jte.6pec.t to
a
.6.unpUc.-i.a£ .6pac.e
-i6 and only -i6
For the remaining part of the proof we only have that
il£
to
observe {f'"
=
f}
obtain
a
is contained in the intersection of all the sets
X
with arbitrary
i£ = £.
f E
Since
we
"
partial converse to Corollary 2 of Proposition 3:
COROLLARY:
X -i1> a KoJtovk-in 1>pac.e with
hold1> i6
0xX = o£X
to a .6impl-ic.ial .6 pac.e
Jte1>pec.t
£.
We are now in the position to finish the discussion of Example 2. Here
£
is simplicial since every continuous real function
compact subset of £ (cf. [5
o£X = ]0,1]
1 , p. 169). From
is the restriction of a function
f E £ \ X
for all
according to Lemma 1.
senting measure for definition of
x
cannot be
= 0;
f
fd~ ~
f(O) according to
monic in
U. Again
ary of
U (and
£
as the set of functions
X).
a~cx
C
U*
where
Furthermore
boundary points of the convex set £
X be
n
U c 1R
of an open, convex, relatively compact set and
the
=X
Kor(X,£)
Example 3 can be generalized as follows. Let
X = A (X)
in
X-affine since
however,
X. We thus obtain
a
'" £ = JC '" n £ • [ 0 , 11 it follONs that X
o£ X = X
But a function x E X
on
fEe (X)
, n
~
the closure 2.
which are
Define har-
U* denotes the topolCX]ical Jxnmd-
0XX =
ex X
and
a£ X = U* since all
U are regular (cf. [2
is simplicial since every function
f E C(U*) is the
1 , p. 127). restriction
BAUER
36
of a function in
.£. It follows from the preceding Corollary and Cor-
ollary 2 of Proposition 3, or from Theorem 5, that space with respect to.£ if
if and only if
X
is a Korovkin
ex X = U*, i.e. if and only
U is -6.tJt.i.etiy eonvex.
REFERENCES
[11
E. M. ALFSEN, Compaet eonvex -6et-6 and boundaJty .i.n-tegl!.al-6, Ergebnisse d. Math. 57, Springer-Verlag (1971).
[21
H. BAUER, Silovscher Rand und Dirichletsches Problem, Ann. Inst. Fourier 11 (1961), 89 - 136.
[3]
H. BAUER, Approximation and abstract boundaries, Amer.
Math.
Monthly (to appear). [ 4)
H. BAUER and K. DONNER, Korovkin approximation in Co (X), tiJ.ath. Ann.
[5]
G. CHOQUET, Leetul!.e-6 on a.naiY-6.i.-6, vol. II
w. [6]
(to appear).
A. Benjamin, Inc.
(Repl!.uen:ta...t.i.ontheoI!.Y),
(1969).
A. LAZAR, Spaces of affine continuous functions on
simplexes,
Trans. Amer. Math. Soc. 134 (1968), 503 - 525. [71
G. LEHA, Relative Korovkin-Satze und Rander, Math.
Ann.
229
(1977), 87 - 95. [8]
G. LEHA and S. PAPADOPOULOU, Nachtrag zu "G. Leha: Korovkin-Satze und Rander". Math. Ann.
[ 9]
Y. A. SASKIN, The Milman-Choquet boundary
Relative
233(1978), 273-274.
and
approximation
theory, Funct. Anal. Appl. 1(1967), 170 -171.
Approximation Theory and FUnotional Analysis J.B. Prolla (ed.) ©North-HolZand Publishing Company, 1979
A REMARK ON VECTOR-VALUED APPROXIMATION ON COMPACT SETS, APPROXIMATION ON PRODUCT SETS, AND THE APPROXIMATION PROPERTY
KLAUS - D. BIERSTEDT
FB 17 der GH, Mathematik, D2-228 Warburger Str. 100,
Postfach 1621
D-4790 Paderborn Germany (Fed. Rep.)
INTRODUCTION
After Grothendieck [21), a locally convex (l.c.) said to have the if the identity
app~oximation
i dE
precompact subset of
of
p~ope~ty
space
E
is
(for short, a.p.) if and only
E can be approximated uniformly
on
E by continuous linear operators from
every E
into
E of finite rank (Le. with finite dimensional range).Many "concrete"
l.c. spaces are known to have the a.p., but a
counte~example
06 En6lo
(1972), with subsequent refinements due to Figiel,Davie, and Szankowski, shows that there are even closed subspace of each
p
~
,tP without
a. p.
1, P f 2.
In connection with the a.p., a criterion due
to
L. Schwartz
[26) is very useful: Schwartz introduces for two l.c. spaces E and their
where
for
E-p~oduct
F
by
F ~ is the dual of
on precompact subsets of
F with the topology of uniform convergence F and where the subscript e
37
on the
space
38
BIERSTEDT
.c(F~,E)
F~
of all continuous linear operators from
into E indicates
the topology of uniform convergence on the equicontinuous subsets of F' •
If E and
F are quasi-complete, one can easily show
and the £-product
E £F
of two complete spaces
(cf. [26). Moreover, the
£-ten~on
E ®£ F
F is canplete
of
Grothendieck
E £ F. We can now fonnulate SchwaJt.tz I ~
[21) is a topological subspace of c~ite~ion
pnoduct
E and
E£F~F£E,
60n the a.p. ([26), Proposition 11, cf. also [3), I,
3.9,
and [S) ) :
The
THEOREM (L. Schwartz):
i6 and only i6 i. c.
~paQe
E ® F
i~
den~e
(on, equivalently,
F
and F ane complete l.c. get:
qua~i-complete
~pace~
in
E£F
OM ~uch
v
al~o
Qall, 60n
~hont,
OM
~pace
eaQh
eaQh BanaQh that
E£F = E ®£ F, the completion 06 the
(which we will
i.e.
E
(qua~i-
~pace
on F
the a.p.
ha~
E
) Qomplete
F) . So
ha~
i6
E
the a. p. ,
we
£-ten~on p~oduct
complete
£-ten~o~
E ®£ F
pnoduQt).
In fact, the applications of this theorem, say, in the case of function spaces E derive from the remark that the "abstract"operator space
E £F
can usually be iden ti fied wi th a
F-valued functions "of type
E". And
E ®£ F
"concrete"
is the space of
responding" functions with finite dimensional ranges in proof of the a.p. of
space
"cor-
Hence
F.
of
E is then equivalent to the approximation
a of
certain F-valued functions by functions with values in finite dimensional subspaces of F for every (quasi-) complete l. c. space only for every Banach space
F,
F
or
a result which is of interest in both
directions. In this article, we will give some (rather simple) new exa.mplu of how to apply Schwartz's theorem to function spaces
more
general
than, but essentially similar to the well-known uniform algebras H(K) and A (K) on compact subsets K of a:N (N ~ 1). More precisely, we deal here with spaces of continuous functions on a compact
set
K
which
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
39
either are uniformly approximable by functions belonging, sets
U containing
K,
to a given -6Ub6hea6
F of the sheaf
continuous functions or have restrictions belonging to o terior K of K. The gene4ai
6i~ua~ion
on
F
open
C of all on the in-
is the subject of sections 1 and 2.
In
section 1, the vector-valued case is considered, while section 2 deals with "slice product" - results (on product sets). Finally, in section 3, we look at some of the
mo~iva~ing
exampie6 and survey
the
known
results (and their relations) in this case. So, in a sense, this paper is based on a generalization of the author's old article [21
and motivated, among other things, by
the
more recent article [27] of N. Sibony: We show the connection of of Sibony's results with topological tensor product theory and the a.p. of the spaces of scalar functions in question. The of this paper will be combined with the technique of
&ne
with
results
"localization"
of the a. p. for subspaces of weigh ted Nachbin spaces (cf. [51 and [lol) in a subsequent paper to yield new examples of function spaces
"of
mixed type" with a.p. and to demonstrate applications of the localization procedure in some concrete cases.
ACKNOWLEDGEMENT:
The author gratefully acknowledges
'support
under
the GMD/CNPq agreement during his stay at UN I CAMP July-September 1977 without which it would not have been possible to attend this Conference in Campinas. I would also like to thank oJ. B. Prolla for his
con-
stant interest in my contribution to these Proceedings. As everybody can see immediately, part of the results in this article dates
(at
least) back to the time when the joint publication [10] was prepared. So the author thanks B. Gramsch and R. Meise for many versations and remarks in this connection.
helpful
con-
BIERSTEDT
40
1. THE GENERAL VECTOR-VALUED CASE Let and
X be a completely regular (Hausdorff)
F a c.io-6ed ioc.a.iiy c.onvex (i. c.. J
continuous (real open subset
or complex
U of
X,
F (U)
valued)
1
of the sheaf C
x
functions on
of all
i.e., for each
X,
co. In fact, it would be
plte-6hea.6 only, and we prefer
F to be a
pre sheaf notation throughout this paper. compare [9
space
is a closed topological linear subspace of
C (U) with the compact - open topology ficient to require
~ub-6hea.6
topological
(For some
of our
suf-
to
use
notation
F as above was called "~hea~ 0 6 F-moltpiUc.
and [101. A sheaf
6unc.tio n-6 II in [9 I.) Let E always denote a quasi-complete locally oonvex (Hausdorff) space (over
lR or
We will always assume that
11:).
i.e. that any function f : X ... Y,
f: X
X is a klR-space,
(or, equivalently,
-+ lR
any
function
Y any completely regular space) is continuous if and only
if the restriction of
f
to each compact subset of
X is continuous.
(Each locally compact or metrizable space, and, more generally, each
uc X m -space.) Then each open KlR -space, cf. Blasco [12 I, and hence the sheaves C x k-space is also a
k
p.tete, i.e. the spaces (C(U),co) and
is
again
a
and Fare c.om-
F(U) are complete for each open
U c X.
Under these assumptions, there exists (cf. [10 I, 1.5) the "E-vai.ued
-6hea6
FE 06
F", namely, for any open
E £
U in
X,
F (U)
the space of all continuous E-val ued functions which satisfy
e'o f
wi th the topology subsets of
E
F(U) for each
e'
E
E',
f
on U
endowed
co of uniform convergence on ccmpact
U (cf. [3]
and
[5]),
and the cononical restriction mappings of the sheaf
FE are just the
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
ordinary restrictions of functions. sheaf
C~
41
is a Qto~ed subsheaf of
FE
of all continuous E-valued functions on
X.
Inour definition and in some of our results below, helpful to keep the following motivating examples F-morphic functions in mind (cf. also [9
1. EXAMPLES:
(ii)
X open in
lR
n
(n~l), L
of
it may
sheaves
be
F of
I and [10 I for !lOre examples) :
(i) X = complex manifold or just of holomorphic functions on
the
CJ:N (N > 1), F=O=sheaf
X,
= P(x,D) a (linear) hypoelliptic
differential operator with cOO-coefficients,and F=~ = sheaf of null solutions of L, i.e. NL(U) ={f E Coo(U); (LI U)f:O} for any any open
U in X.
(The closed graph theorem
for
Frechet spaces implies that, on NL(U), the topologies induced by
COO (U) and by
co coincide and hence that NL (U)
is a closed topological linear subspace of (C(U), co).) Especially, the sheaf all assumptions of 1.
JC of harmonic functions on
lR
n
satisfies
(ii) above, and also the "harmonic sheaves" of
abstract potential theory are sheaves of F-morphic functions.
All
the sheaves of example 1. are (FN)-sheaves.
2. DEFINITION: (i)
K of
X,
we define:
C(K,E):= the space of all continuous E - valued on
(ii)
For a compact subset
functions
K with the topology of uniform convergence on
AF(K,E)
K,
E 0
:= {f E C(K,E); f If{ E F (K), i.e.
e'of
10
o
E F(K) for each
e'E E'}, and
K
(iii) HF(K,E)
:= the closure in
C(K,E) of
{f E C(K,E); there exists an open neighbourhood (depending on continuous and
f)
and a function
U of
K
g E FE(U) [i.e. g: U-+-E
e 'og E F(U) for any e'E E'l such that g
iK
= f}.
42
BIERSTEDT
HF (R,E) C AF (R,E) holds, and both are clo.6ed subspaces of e(R,E) which we endow with the topology of uniform convergence on
C(R,E». If
E = 1R or
~,
K (induced
by
we write C(R), AF(R), and HF(R), respec-
lively. Now, of course, if and
E is complete, all the spaces e(R,E), AF (K,E),
HF (K,E) are complete, too. The equation
quasi-complete
E
C (K,E)
= E I: e (K)
for
is well-known (cf. [3]), and, once this equation is
well-understood, the proof of the first part of the following result is clear (see e.g. [31 arbitrary subspace of
or
for a description of
[51
E
E
C(K), from which our result below
F, F is
an easily
derived, too):
(2)
Hence (OIL,
-£6
AF(R,E) = E equ-£vale.ntR.y,
v
AF(K) hold.6 604 all complete
@£
all Banach) .6pace.6
60ft
E
l.c.
-<'6 and only
AF(R) ha.6 the. a.p.
For the second part of 3, Schwartz's criterion for the a.p. the introduction) is needed. In other words, AF(R) has the and only if, for arbitrary Banach space with e' of on
R,
I~
o
E F(R} for any e' E E'
by continuous functions
sional subspaces of
E
9
E,
a.p.
(in if
each function f E e(K,E)
may be approximated, uniformly
on K with values in. finite dimen-
that satisfy
e
0
g
I0
a
E
F (K), too, and
hence
K
have the form
g(x)
n
E:IN
finite (depending on
for all
g),
ei
E
E, and
(Remark that such an approximation with
9i
gi
E
K;
AF(K}, i=l, ... ,n.
only E elK)
possible by the a.p. of C(K) and by the equation for complete l.c. E.)
X E
is
e(K,E) =E
atway.6
®£
C(K)
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
As to the a.p. of the corresponding space there is, in some sense, just the
We a~~ume
4. THEOREM: ha~
K, F(U)
.the a.p. 60IL each that E
aHume ~Mtea.d
(1)
Then
06
HF(K,E)
.e.. c. =
ha~
HF(K)
{f E C(K,E)j
6unct~on
g
E
E 0
E
the a.p.
(OIL eac.h Banac.hl
=
sup xEK
as the
E
OM
each
(e'
and each
=
U(e',E} 06
U
~uch
f) (x) - g(x)
I
in
E
> 0
K an.d
that <
d.
HF(K,E) must be verified for the first as-
p be a continuous seminorm
g E FE ( U) suc h t h at
definition, FE (U) = E on compact subsets of
E
F (U)
on
~~~
E,
E
U~
>
K
and
0
and
a
E B ut, aga1n . by p ( f (x) - g (x) ) <;[.
(with the topology of uniform convergence
U). Without loss of generality, we may assume
and hence the a. p. of
Schwartz's
each
E,
f E HF(K,E). By definition, there exists an open set
Em,
OM
e' E E'
ne~ghbouILhood
0
~o,
and only
~ pace
gte' ,E) E F (U)
I
~o
C(K) is a topological linear subspace of C(K,E) and
° HF(K)
sertion. So let
U
~ub~pace
-tensor product preserves topological linear subspaces, only
density of
f unct i on
al~o
below, we could
E ~~ complete.
.th elL e e x~~ t~ an open
As
[FM (1)
06
HF(K,E), and hence
c.omplete
PROOF:
06 ne~ghbou~hood~
E 0 E HF(K) ~~ a. den~e topol09~cal l~neaJt
Con~eQuenalJ
a
a
the a.p.]
ha.~
hold~ wheneveIL
(2)
Em.
U
HF(K), the situation
oppo~~te:
60~ ~ome ba~~~
that,
43
F (U) or of
E and one
theorem from the introduction imply that
direction
of
E 0 F (U) is dense
44
BIERSTEOT
in FE(U).Therefore we can find h E E 0 F(U) with sup p(g(x)-h(x» < ~2. xEK Now hi K E E ® HF(K) holds and sup p(f(x) - hex»~ < €, which proves xEK
the required density of
E ® HF(K)
in
HF(K,E).
(2) is then clear from Schwartz's criterion because the on the right hand side of the equation is nothing but
HF (K) -
as
true (under the assumption of
4)
a close look will immediately reveal. alway~
In other words, it is that a function
E
space
E
D
f E C(K,E) which can be approximated uniformly on K
by functions extending to elements of
FE
K may also be approximated uniformly on
on open neighbourhoods of
K by functions of the form
n
hex)
L eihi(x) i=l
n E 1N finite (depending on
h),
e
i
for all
E E, and
x E K;
gi E HF(k),i=l, •.• ,n.
But the a.p. of
HF(K) is equivalent to the fact that, for arbitrary
Banach space
each function
given any
E,
e' E E', e' of
that,
f E C(K,E) with the property
K
by
(scalar) functions belonging to F on open sets containing K is
al-
ready an element of
may be approximated uniformly
on
HF(K,E), i.e. can be approximated uniformly
K by E-valued functions belonging to
FE on open sets containing
on K.
Or, to put it this way, HF(K) has the a.p. if and only if, given any Banach space
E and an arbitrary function
exists for any E' , an open set e'
0
go E F (U o )
€ > 0, U
0
uni6o~mly
:> K
for all
and a function
for each
e' E E' 1
The description of
tion in 4. (2) is of course
e' go
in the uni t ball U -.. E a
E' I
of
continuous with
such that
i(e'of)(x)-(e'ogo)(x)i<€ for all
REMARK:
fEe (K, E) as above, there
xEK
and all
e' EEi.
E € HF :K) as the right side of the equa~ndependenz
of the hypothesis on F in 4 ,
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
and so is the inclusion
HF(K,E)
C
description. Hence, as obviously linear subspace of
whenever clearly
45
E e:HF(K) which follows from E ®e: HF(K) is
afway~
this
a topological
HF(K,E), we have
E is complete. So, by Schwartz's theorem, the a.p. of HF(K) implie~
the equality
for complete i.c. spaces Let [or let
E,
even wLthou:t the hypothesis of 4.
E be complete and let the assumption of 4. (1) be satisfied HF(K) have the a.p.]. Then the preceding two theorems imply:
HF(K,E)
C
E e:HF(K)
E e:AF(K)
C
II
II
So we obtain from Schwartz's theorem:
5. COROLLARY:
boufthood<'l 06
Le:t U<'l again aHume :tha:t, K,
F(U)
ha<'l .the a.p.
601t. <'lome ba<'liJ..
60ft each
U
E
Ul
Ul,and fe:t
06 l1ugh-
~(K)
:HF(K)
be vaUd. Then
AF(K)
ha~
= HF(K)
hold<'l 60ft afl comple:te i.
Co
.the a.p. i6 and only i6
(Olt.,
AF(K,E) =HF(K,E)
equivalen:tly, 60ft aU Banach) <'lpacu E.
If, in concrete examples, one examines the methods to a proof of
AF(K)
methods also prove spaces
E.
= HF(K), it turns out very aften AF(K,E) = HF(K,E)
that that
lead these
for, at least, arbitrary Banach
Corollary 5 shows that it suffices to prove
AF(K,E) = HF(K,E) for all Banach spaces
the equality
E to obtain bo.th the a.p. of
46
BIERSTEOT
AF(K) = HF(K) and
AF(K,E) = HF(K,E) even for arbitrary complete t.c.
spaces E. On the other hand, sometimes the methods used AF(K) = HF(K) may also be adapted to yield a of this space, and then
d~~eet
in
proving
proof of the a.p.
AF (K,E) = HF (K,E) holds for all complete Lc.
spaces by Corollary 5, too. In fact, Corollary 5 demonstrates equ~valent.
the two approaches which we have just outlined are
REMARK:
Similarly, if
then the a. p. of ,[11
that
E is a complete l.c. space and i f AF(K) =HF(K),
E or of
AF (K) = HF (K) also implies
AF(K,E) =HF(K,E)
gene~al.
2. APPROXIMATION ON PRODUCT SETS Let us now turn to a description of the £-product plete £-tensor product of two (or more) spaces of type
resp.
com-
AF(K) resp. HF(K).
Such a description follows easily from the (well-known) general".6Uee p~oduct theo~em"
for subspaces of, say, C (K
l
uct theorem was first stated in Eifler (17),
(This slice prod-
x K ).
2
but he points out
the result is already implicitly contained in Grothendieck [21
that
1.
more general slice product theorems, for some ideas connected
For with
the underlying method, and for more applications compare [4 ) and [5].) So let such that
Xl
Xl and x
X 2
X be two completely regular (Hausdorff) spaces 2 ,[.6 a
k
IR
- .6pa.ee. Then both
k IR -spaces, and, on the other hand, Xl x X 2
X are and if at least one of the spaces 2 pact (or i f both resp. F2
Xl and
is
Xl ' X2
X are hem,[eompa.ct 2
denote closed l.c. subsheaves of
know (by applying Blasco's result on the
Xl k
and IR
X must 2
, i f both Xl and
is even locally a:m-
kIR -spaces) . Cx
1
be
resp. C ' X2
Let
Fl
Then we
kIR-property of open
sub-
sets of completely regular kIR -spaces, cf. [12], instead of Arhangel'skiJ: 's proposition on k-spaces in the proof of sheaf
F £F " on 2 l
Xl x X2
exists:
[10 I, 1.10) that the "product
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
Fl
E
F2
is u.n..(qu.ely de.teltm.(n.ed by the following requirements:
For all open subsets
U i
and, for all open
~
Fl
47
E
U i
C
Vi
Xi (i=l,2), (F EF ) (1I x U ) 2 2 I 1 on
Xi (i=1,2),
F2
rUlxU2'VlxV2
F rUV
where
F (U)
denotes the canonical restriction mapping
wi th respect to the sheaf
F and where the E-product
of
F(V)
-+-
continuous
linear mappings is defined in, say, [7 1 • Let us now introduce the following n.o.ta.tion.: cal projection of open subsets
U
Xl x X 2 of
onto
lT
is the canoni-
i
Xi (i =1,2), and,
for
arbitrary
Xl x X2 '
Then we get a gen.eltal description of
Fl E F2
on open sets
UC ~ x~
as follows:
endowed with the topology sets of
co of uniform convergence on compact sub-
U, and the canonical restriction mappings of the sheaf FtF2
are just the ordinary restrictions of functions. Fl
l. c. subsheaf of inheri ted by
6. THEOREM:
have:
Fl
Le.t
CXpX2 = CXl E
F2 ([10],
Ki
E
CX
2 1. 2 c ).
'
Nucleari ty of
be a eompac..t .6ub.6e.t 06
E
F2
is a elo.6ed
Fl
and
Xi (i =1,2).
F 2 is
Then
we
81ERSTEDT
48
{f E C (K 1 x K 2); f ( t, .)
o
I0
and f ( • , x)
E F 2 (K 2)
K2
nOJt
{f
all
E
C(K
(t,x)
l
E
2
,
Kl XK2},
XK2); f(t,')
and
f(',x)
may be app!toximated uni60ltmly on
again with the
(t,x) ~up
F2
on open
~et~
c.ontaining
may be app!toximated uni60!tmly
by 6Unc.UoM belonging to Kl 60!t eac.h
E F 1 (K 1 )
Kl
K2 by 6unc.tion~ belonging to K
0
I0
E
Fl
on open
~et~
on
Kl
c.ontaining
Kl x K }, 2
- nOJtm 06
C (K
l
x K ), 2
and:
( 3)
PROOF:
Parts (1) and (2)
theorem for subspaces of
it suffices to verify
follow immediately from the slice C(K
l
x
K ) 2
quoted above.
HFl (K ) ® HF2 (K 2 ) l
C
HFl
£
product
To prove
F2 (K l x K2 ), which is
HFl E F (K l XK ) C HFl (K ) EHF (K) fol2 l 2 2 from the previous description of the sheaf F1 E F 2 and
immediate, too. The inclusion lows readily
from the description of the E-product on the right hand side, cf. (2).
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
49
Finally
AF (K ) E AF (K ) C AF E F (K l x K ), because l 2 2 1 2 1 2 and hence (by the description of Fl E F ) : 2
for
As 6. (1)
(together with the description of
AF 1 E F2 (K
l
xK )
2
all
at the
shows, AFI (K ) EAF (K ) will in general l 2 2 AFI E F2 (K x K ), and i t is easy to construct 2 l
end of the preceding proof) be -1tft,tctly contained in
example-1 for this phenomenon. However, a simple topological
assump-
tion forces equality here, as part (2) of our next result clerrDnstrates.
7. THEOREM:
(1)
Let, 60te. -1Ome ba-1,t-1
Ui-
l
06 nughbowc.hood6 06 Kl , Fl (Ul )
have the a.p. 6ote. each
U E Uiote. let, 6ote. -1ome ba-1,t-1 112 l l 06 ne,tghboufthood-1 06 K , F2 (U ) have the a.p. 6ote. each 2 2 U E Ui- • [ 1 M tead 06 th,t-1, we could al-1o ftequ,tfte HF (K ) 2
oft
2
1
HF (K 2 ) to have the. a. p. I . 2
HFI E F2 (K l
xK )
2
l
Then
v
hofd-1.
HF (K l ) 0 HF (K 2 ) E 2 1 0
(2)
16
Kl
and K2 aILe "6a:t", J... e. -1a.t,[-16y
K. =K. 1-
1-
(i
1,2) ,
we get:
PROOF:
(1) The remark in brackets is obvious from 6. (3) and Schwartz's
theorem. For the proof of (1)
under the assumption on
Fl resp.
F , 2
50
BIERSTEDT
So let set
> 0
€
U containing
Kl x K2
be given and find
and a function
g E (F
sup jf(x) - g(x) [ < xEK xK l 2 Without loss of generality we may assume
l
an
sF )(U) 2
open
suchthat
s
"2 U.
U
1
E
Ul . 1
(i = 1,2), and hence
by Schwartz's theorem, because
F1(U ) l
or
F (U ) 2 2
has the a.p. Then
h E Fl(U ) @ F (U ) such that l 2 2
there exists
X E
ig(x) - h(x) [ <
sup Kl x K2
~
1<
s.
(2) Notice that, by the identity C(K XK2) =C(Kl'C(K2 »=C(K ,C(K I 2 l
»,
sup [f(x) -h(x) x E Kl x K2
Now
for arbitrary
f E AFI
€
F2 (K l x K2 ), II : t ... f (t,') resp. I 2 : x'" f("x)
yield continuous linear mappings of
Kl
The characterization of
x K ) 2
6 implies
o
AFI
€
F2 (K
l
resp. K2 into C(K ) resp. C(K ). 2 l at the end of the proof of
0
Il(K ) C AF (K ) l 2 2
and
I
2
(K 2 ) C AFl (K l )
and
hence also
and the assertion follows immediately from 6. (l).D
8. COROLLARY:
(1)
16
AF (K.) =H F (K.) i 1 i 1
(i=l,2)
hotd.6 artdi6orte.o6
the.6e. .6pace..6 ha.6 the. a.p., thert we obtairt
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
So we get even .6 et.6 ( 2) HFI
£
Kl F (K I 2
and X
A
HF e:F (K l x K2 ) 1 2 K2 ·
61
F.. (K
Fl e: 2
l
X
K ) 2
6OIl.
6at
K ) ha.6 the a.p. whenevelt both HF (K ) and HF (K ) 2 l 2 1 2
have .the a.p. (3)
16
Kl
and K2
.[6 both
6at and
Me
AF (K )
1 have the a.p.,
l
aVld
ha.6 the a.p., too.
PROOF:
(1)
is clear from Schwartz's theorem, 6. (3), and 7. (2).
and (3)
follow from 7 by aid of the result (Schwartz [26LProposition
11, Corollaire 2) that the e:-product of two compete
~.c.
(2)
spaces with
a.p. also enjoys the a.p. C Induction on 7 and 8. (1)
using, among other (obvious)
things,
that finite e:-products are a.6.6oc..[at.[ve and that e:-products ofccrnplete spaces with a.p. are again spaces with a.p.
9. COROLLARY:
w.[th
Xl x
.6hea.ve.6 06
Let
Xl""
'X
n
yields now:
be c.ompletely lLegulalt (Hau.6Ciolt66) .6pac.e.6
a k lR -.6pac.e, let Fl, ... ,Fn be c.lO.6ed l.c. . .6ubCx , ... , ex ' and Kl , ... ,Kn c.ompac..t .6ub.6eU 06 xl"'" Xn ' 1 n x
xn
lte.6 p ec. t-iv el y •
(1)
Let, nOlL .60me ba.6.[.6 Ul i ha.ve the a. p. 60IL ea.c.h a.t mO.6t one
i)
06 Vle.[ghboulthood.6 06 U
E
i
Ul i
(i = 1, ••• ,n
OIL let a.ll but one
HF (K.) i
Ki'
exc.ept
nOlL
(i =1, . . .
,n)
1.
have the a.p. Then
.[.6 tlLue, and
.[6 aU.
.6ame hold.6 bolt (2)
Let all the .6e.t.6
HF
HF (K.) i
1 e: ...
(i =l, ... ,n)
have thea.p., the
1. E
Kl, ••• ,K
F (K x ••• x K ). n l n
n
be nat. Then
62
BIERSTEDT
hold.6 tllue, and -<'6 all the .6ame hold.6 nOll (3)
Let, OM eac.h
lo
AF
A F . (K ) 1 l. l
E ... E
1 = 1, ••• ,n, K1
have the a.p.,
(i =l, ••• ,n)
F (K
n
1
x . . . x K ).
n
~1 (K ) =H
be 6at and
then all the.6 e .6pac.e.6 (ex:c.ept
1
oak at mO.6t one)
Fi
(K )· 1
have the a.p .•
-<..6 val-<.d. too. For the corresponding spaces of functions w1 th values in a quasi-
.e.. c.
complete
space
10. COROLLARY:
(2)
(1)
Let E
E
(see sect10n 1), we get e. g. :
Let
K , ... ,K
1
be nat. Then
n
be comptete and tet,
, Fi (U ) i i (i =1, ••• ,n). Then
bOUllhood.6 06
HF
l
E ...
F (K e: n 1
oOlt.
have the a.p.
K
uz..l.
nOll .6ome ba.6.t.6
x . . . XKn,E) =E
"
06
eac.h
ne.tgh-
ui
E
uz i
(K.)
OOIl
v
0 c HF (K ) 0 ~ 1 1 e:
-<..6 valU. (3)
Let E be c.omplete, let K. be 6at and AF (K.) i
l.
eaeh 1 =l, ••• ,n. Then -<'6 aU the .6pac.e.6 have the a. p. AF
1 e: ... e:
(i
=1, ... ,n)
1.
= HF i
1.
A . (K ) =H (K.) 1 Fi Fl. l.
,
F (K x ••• x K ,E) 1 n n
hold.6. too.
PROOF:
(1)
is a consequence of 3.(1)
under the hypothesis of (2),
(F
1
and 9.(2). Let us remark that,
e: ... e: Fn) (U)
(as E-product of carplete
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
spaces with a.p.) satisfies the a.p. for each Ul P
: = {U 1 x • •• x Un
of neighbourhoods of 9. (I). Finally (3)
U.
~
Rl x ••• x Rn.
Ul.
E
~
U in the basis
(i=l, ... ,n)}
Hence (2) follows from 4. (1) and
is implied by 9. (2),
very end of section 1.
53
(3)
and by the remark at the
0
3. DISCUSSION OF THE MOTIVATING EXAMPLES In this final section, we will look at some of the known results in the case of our motivating examples of sheaves F (cf. 1 above) and will point out that, between some theorems in the literature, strong relations follow from our previous discussion. here to survey
It is
not
intended
a.ll the relevant articles, but we will rather illus-
trate some of the ideas which might playa role, when one tries
to
apply the results of sections 1 and 2, by specific examples. Perhaps the case most people have been interested in is
F
= 0,
the nuelea.n Frechet sheaf of holomorphic functions on a complex manifold
X. For simplicity, however, we will only deal with holomorphic
functions on of sheaves
X = eN (N .:: 1) here.
It is clear that finite £-products
0 are nothing but the correspending sheaf
uct and that, for any quasi-complete l.c. space sheaf of E-val ued holomorphic functions. When for short, A(R,E), H(R,E) instead of
E, OE
0 on the prodis just
F = 0, we wi 11
the
write,
AF(K,E), HF(K,E), respectively.
F = 0, some of the results in sections 1 and 2 are
appar-
ently part of the "folklore" of the subject, but usually not
easily
For
accessible in the literature: We have already pOinted out in the intraduction that this paper is based on a generalization of the "old" article [2
1. Later on (in [1 ), section 1), o. B. Bekken looked
closed subspaces of
C (K),
K compact, with the so·-called
"~Uee
at
pnopenty"
BIERSTEDT
54
and showed that this property implies the a.p.
After the proper change
of notation and some identifications (using the fact that each Banach space is a closed subspace of
C (K ') for some compact K') his resul ts
there are quite similar to our theorem 3 (for Banach spaces E) . section 3 of [1],
In
(making use of the nuclearity of 0) Bekken obtains
a proposition related to (but somewhat weaker than) our theorem
4.
For a detailed account of the relation of the slice property with the a.p. and the consequences of a theorem of 11ilne in this
connection,
see also [6 ] . we
must
now
2. If
N=1
i. e.
As usual with spaces of holomorphic functions, spli t up our discussion for the cases
N
=1
and
N
>
K is a compact subset of the complex plane, the problem is solved: A(K) and
H(K) have then
alway~
the a.p.
an open p40blem whether even the Banach algebra holomorphic functions on the open unit disk that the a.p. of the disk algebra
A(D)
~letely
(whereas it remains Hoo(D) of all bounded
D enjoys the a.p.
= H(D)
~k
is really quite triv-
ial !). This interesting result is due to the joint effort of several people (and also, unfortunately, not easily accessible in the I i terature in its full generality): Eifler [17], Gamelin-Garnett [19], section H(K), and Davie [15]
6 for
for
A(K)
resul ts). More generally, Gamelin
Vec.tM
(they all use
118], section 12
- valued
has pOinted
out
that the constructive techniques (and the approximation Scheme)
of
Mergelyan and Vi tushkin show that the so-called "T-inva4iant" algebras
= H(K)
in the case of one variable, a nec.u-
have the a.p. As to
A(K)
~a4y
condition (involving
and
~uH.i.c..i.en.t
was given by Vitushkin, see e.g. [19] For
c.ont.i.nuou~
and [29].
N 2.2, there are only pa4tial results. Remark first
by an example of Diederich and Fornaess, there exists compact domain A(K)
~
H(K)
analytic. c.apac.ity)
for
G of holomorphy in K
= G.
leN
with
a
COO -boundary
that,
relatively such that
For a survey of some related recent work
on
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
the question when to Birte 1 [11),
55
A(K) = H(K) in several complex variables,we refer
and for resul ts in "6-t11-t.te S. P -6.
C. mal1-t 6o-td-6"
to
Rossi-Taylor [25]. It is known now that
A(K)
lowing types of compact sets (i)
K
(or H(K»
has the a.p. for the fol-
KeeN:
is the closure of a
p-6cudocol1vex
-6.t~-tc.tiy
~cg-tol1
with
sufficiently smooth (say, C 3 -) boundary, or: (ii)
K
is the closure of a
Both conditions imply
~egulan
WC-tl
polyede~.
K fact (trivially), and
A(K) =H(K)
(in
case (i), this approximation theorem is due to Henkin-Lieb -Kerzman, in case (ii), it is a result of Petrosjan). Bekken [1],
(i) was proved
e.g.
in
section 2, applying a vector-valued version of Henkin's
separation of singularities result. It also follows from Sibony [27l, proposition 4 (in view of our Corollary 5). Sibony [27], p. 173 also remarked that Petrosjan's arguments may be modified A(K,E) = H(K,E) for each Frechet space
E if
to
K is the closure
has yield of a
regular Weil polyeder, and hence (ii) follows again from our Corollary 5.
REMARK:
The method of "locai-tza.t-tol1 06 .the a.p." for certain
tion spaces (cf. [5]
and
the a.p. for compact sets UI1-tOI1I> of sets
[10]) may be used to show that K'
that are "sufficiently well"
func-
A(K') has
d-tl>joJ..n.t
K as above and that some related funct10n spaces have
the a.p., too (cf. [5],
Corollary 15), but we will not go into
de-
tails here. Let us now explicitly state what we get from the preceding results by applying our Corollar1es 9 and 10:
14. THEOREM: (i = 1, ... ,n)
(1)
(l)
H{K) hal> .the a.p.
e.<..then any compac.t l>ub-6e.t 06
a:
16 M
K
56
BIERSTEDT
(ii)
ct04une 06 a
~he
6icientty (2)
in
(1)
®( ... ®(
(3)
H(K) = A(K)
(i) a
ei~hen a~
Le~
then
hotd~ 6a~
¢u6-
wi~h
bounda4Y 04 06 a 4egula4 Weil polyeden.
(i), addiUonaUy,
= A(K ) l
4egion
the a.p. unden the ¢arne conditioM i6 one
ha~
A(K)
~mooth
4t4ic~ty p~eudoconvex
Ki
~o
be
6a~.
And
neQuine~
A(K)
=
A(K ) i~ then tnue. n
604 K =K l x .•. xKn wah Ki (i =l, ... ,n) cornpac~ ¢e~ in ~ wi~h H(K ) = A(K ) 04 i i
in (1) (ii) above. E
be an a4bitnany complete
.t. c.
~pace.
(4)
Unde4 .the
a~wrnpt-i.on~
06 (2),
(5)
Unden .the
a~~umption~
06 (3), we have
A(K,E)
H (K , E),
~oo.
11. (3) is related to a result of Weinstock [301,
p. 8l2,where,
instead of the assumption of a smooth boundary in 11. (1) (ii), he needs only the so-called
"~e9rnen~ p.ILop'ln~y"
of K.
(Weinstock's methods are
quite different, however.) At this point, a few remarks on
Sibony's
paper [271 are also in order (in connection with our preceding results) : Proposition 1 of [271 is, in some sense, easy, if not trivial, as our theorem 4.(1}
(and its simple proof) demonstrates: It is
no~
necessary to invoke Gleason's theorem at this point; the well - known nucleari ty (or even the a. p.) of
0 and simple tensor product argu-
men ts suffice! Corollaire 3 of [271 corresponds with 7. (l) and 10. (2) in this paper. As we have already noted above,
however,
Sibony's
proposition 4 is really a non-tnivial result based on Henkin's method and implies the a.p. of i t is (essentially)
A(K)
equiva.e.en~
in case (i). Hence, by our Corollary 5, to theorem 2.4 of Bekken [1 I. Finally,
Corollaire 8 of (27) corresponds with our theorem 11.(5).
It should
perhaps be pointed out that, whereas part of Sibony's proofs looks as
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
57
though they are based on theorems and methods which are just true in ~p~Qiat
his given
situation, it turns out from our discussion
above
that what is really needed is only a proof of the a.p. of A{K) (=H(K» to make everything work, even in many We turn to sheaves
O~h~4
cases.
F of harmonic functions or, more generally,
of null-solutions of hypoelliptic differential operators with Coo-coef_ ficients now. These are again nuclear Frechet sheaves, and hence our assumption that
F has the a.p. is certainly satisfied.
of the sheaves in axiomatic potential theory, cf.
For nuclearity
Constantinescu-
Cornea [14], § 11. In this case, we will assume for the moment that set
K is the closure of some open subset
U of
U is a
n~gutan
sheaf
F, i.e. for each
f E AF(U)
set for the
with
L: f ~ f1au
g
Vi4iQhl~~ p~obt~m E
C (aU)
compact
(and hence
X
Qompt~~~ty
A very nice phenomenon may occur here which yields a al solution to the question of the a.p. for
the
fat).
ruvi-
AF(K): If we
s~se
that
with respect
to
the
there exists
a
unique
function
f1au = g, the continuous linear restriction mapping
is bijective from
AF(K) onto
C(aU) and hence yields a
topological isomorphism of these Banach spaces. for functions in
AF(K) will imply that
L
(A maximum principle
is even an
i~om~~~y.)
'nlen
AF(K) certainly has the a.p. In fact, i t would be enough for such resul t that ~paQ~ w'('~h
closed set
L: f + f IK' a. p. of
C (K')
AF (K) onto a elo~~d ~ub-
for some closed subset
K'
of
K
(say,
a
K' c aU).
Let for instance tions on
is bijective from
a
F be the sheaf
JC of (real)
ha~moniQ
func-
~n (n ~ 2). We refer e.g. to Ho-Van-Thi-Si [22], p. 617/8,
621/2, 626, 637 for conditions concerning, say, the equalities
= HX(K),
(i)
~(K)
(ii)
~(K)laK=c(aK)
ciple,
L:
and
~(K)
(or, equivalently, by the maximum prin+
C(aK) bijective and isometric) .
BIEASTEOT
58
Let us only note that in general a suitable (outside) c.ol1e d~t~ol1
implies both (i) and (ii) and that, in the case
n =2,
(ii) are valid for a compact set K such that each point
C.011-
(i) and
x E aK
is
a boundary point of a connected component of the complement of K. So A~(K)
then
=
H~(K)
has the a.p.
We also refer to Weinstock [31) for sheaves
F
=
NL (on
for results on
mn) of null solutions of (linear)
partical differential operators
elliptic
L of order m with constant coeffi-
cients in this connection and to Vincent-Smith [28) in the setting of harmonic sheaves
for
F of axiomatic potential theory.
It would lead us too far afield even to give only c.omplete 1te0eJLeJ1c.u for all interesting relevant results in this direction. Another argument then yields the a.p. of
AF(K)
and
even in a much more general setting:
Let a.ga.~11 ~ be the .6hea6
12. THEOREM:
00
ha.ILmol1~c. 6('LI1c.t~011.6
(n > 2) al1d K al1 a.ILb~t.ILa.ILY c.ompac.t .6 ub.6 et 06 ( 1)
Thel1 both
( 2)
Hel1c.e
A~(K,E)
.6pac.e
E, al1d,
Ax(K,E) =
PROOF: A =
a.l1d
Ax(K)
..,
are
.6~mpl-ic.~al
E, Ax(K) =
p. 621, 634 shows, both
spaces, i.e. the null measure
A-max-imal measure (or, equivalently, measure Choquet boundary of
A)
~pUu
A
= H~(K)
is the
concentrated
and
only
in
the
orthogonal to A. This means (cf. Effros-Kazdan
[16), p. 99) that the state space
S = SeA) is a
is order isometric to the Banach subspace tions in
a.lway.6
H~(K)
.
As Ho-Van-Thi-Si [22),
A~(K)
hold.6 6M eac.h complete l. c..
E ®e: Ax(K)
H~(K,E)
]Rn.
alway.6 have the a.p.
H~(K)
60.IL .6uc.h al1
mn
011
.6~mplex
and that
A(S) of all a66il1e
C(S). However, it is well-known that each such
A(S) has the a.p.: In fact, A(S)' is al1 a.b.6tltac.t
(L)
.6~p.e.ex
A
func.6pace
- .6pa.c.e. (This
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
69
argument can be found e.g. in the proof of Corollary 2.6, p. 477 Namioka-Phelps [23].)
~
(2) follows from (1) and 3. (2), 5 above.
For the connection between simplicial spaces and the of "weak ViJtic.hlet pJtoblem,t," see Effros-Kazdan [16]:
of
solution
Ax(K)
(say) is
simplicial if and only if each continuous function defined on a compact subset of the Choquet boundaJty ofAX(K) may be extended to element of
an
Ax(K) of the same norm.
But now we get the a.p. of
AF(K) and
HF(K) for many
F of axiomatic. potential theoJty and all sets K
=
closure of a rela-
U: In fact, under certain
tively compact open set
sheaves
underlying haJtmonic. ,t,pace (X, F), it is known that
axioms
on
the
AF(K) resp. HF(K)
is again '('implic.ial, and then we may proceed as in the proof of theorem 12 to carry the corresponding results over to this
(much
general) setting. For the relevant axioms needed here am the
more
I-
AF (K) resp. H F (K) is a simplicial space, we
refer to Effros - Kaze..,
[16], Cor. 4.3, p. 108 and Cor. 4.2, p. 112.
(For a necessary
sufficient condition for
and
AF(K) = HF(K) in this setting see [16],the-
orem 4.4). In [16], the axioms still excluded geneJtal
open
sets
U
for degeneJtate elliptic equations, but the corresponding problem was solved affirmatively by Bliedtner-Hansen [13],
and we refer to
for the most general results on Simplicial spaces
[13]
AF(K).
In concluding, we should point out that the £-product
Xl £ X 2
of two sheaves of harmonic functions in axiomatic potential
theory
yields nothing but the (multiply resp.) ,t,epaJtately haJtmonic. functions of
Gowrisankaran [20]
resp.
Reay [24]. We leave it to the reader to
combine our preceding remark on the a.p. of
AF(K)
resp.
axiomatic potential theory with the results in section
HF(K) above
to
obtain, say, theorem 11 and lemma 23 of [24] without any effort.
Of
course, we could also immediately state results for holomorphic - harmonic sheaves
0
E:
X
2
in
"mixed"
(say)
etc., but the preceding exarrples
BIERSTEDT
60
and applications may suffice.
REFERENCES
[1]
O. B. BEKKEN, The approximation property for Banach spaces
of
analytic functions, preprint (1974), unpublished. [2]
K.-D. BIERSTEDT, Function algebras and a theorem
Pa.pVt.6
of Mergelyan
.the. Su.mme.Jt Ga.theJting on Fu.nc.tion AlgebJta.6, Aarhus, VariousPub1ifor vector - valued functions,
6Jtom
cation Series 9 (1969), 1 - 10. [3]
K.-D. BIERSTEDT,
Gewichtete
Funktionen und
das
Raume
stetiger
vektorwertiger
injektive Tensorprodukt
I, II,
J.
reine angew. Math. 259 (1973),186-210; 260 (1973), 133-146. [4]
K .-D. BIERSTEDT, Injekti ve Tensorprodukte und
Slice - Produkte
gewichteter Raume stetiger Funktionen, J. reine
angew.
Math, 266 (1974), 121 - 131. [5]
K. -D. BIERSTEDT, The approximation property for weighted
func-
tion spaces; Tensor products of weighted spaces,
Func-
.tion Spa.ce.6 and Ven.6e. AppJtoxima.tion
(Proc.
Conference
Bonn 1974), Bonner. Math. Schriften 81 (1975),3-25; 26-48. [6]
K.-D. BIERSTEDT, Neuere
Ergebnisse
zum Approximationsproblem
von Banach-Grothendieck, JahrbUch Uberb1icke Math. 1976, Bl,45-72. [7]
K. -D. BIERSTEDT and R. MEISE, Lokalkonvexe Untediume logischen Vektorraumen und das
in topo-
£ -
Produkt, Manuscripta
K.-D. BIERSTEDT and R. MEISE, Bemerkungen
fiber die Approxima-
Math. 8 (1973), 143-172.
[8]
tionseigenschaft
1oka1konvexer Funktionenraume,
Ann. 20 9 ( 19 74), 99 - 10 7 .
Math.
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
[9]
61
K.-D. BIERSTEDT, B. GRAMSCH and R. MEISE, LokalkwexeGarben
und
gewichtete induktive Limites F-morpher Funktionen, Func~~on Space~
and
Ven~e Appnox~ma~~on
(Proc.
Conference
Bonn 1974), Bonner Math. Schriften 81 (1975), 59 - 72. [10]
K.-D. BIERSTEDT, B. GRAMSCH and R. schaft, Lifting
und
~ffiISE,
Approximationseigen-
Ko - homologie
bei
lokalkonvexen
Produktgarben, Manuscripta Math. 19 (1976), 319 - 364. [11]
F. T. BIRTEL, Holomorphic approximation to boundary value
al-
gebras, Bull. Amer. Hath. Soc. 84 (1978), 406 - 416.
[12]
J. L. BLASCO, Two problems on klR-spaces, preprint
(1977), to
appear in Acta Math. Sci. Hungar. [13]
J. BLIEDTNER and W. HANSEN, Simplicial cones in potential the-
ory, Inventiones Math. 29 (1975),83-110. [14]
C. CONSTANTINESCU and A. CORNEA, Potential theory
on harmonic
spaces, Springer Grundlehren der Math. Wiss.
Band
158
(1972) • [15]
A. M. DAVIE, The approximation property of A(K) on plane sets, private communication (1969), unpublished.
[16 I E . G. EFFROS and J. L. KAZDAN, Applications of Choquet to elliptic
simplexes
and parabolic boundary value problems,
J.
Diff. Equations 8 (1970), 95 - 134. [17]
L. EIFLER, The slice product of function algebras, Proc. Amer. Math. Soc. 23 (1969), 559 - 564.
[18]
T. W. GAMELIN, Uniform approximation on plane sets, Appltox,[mat~on
Theolty
(Editor: G. G. Lorentz),
Academic
Press,
techniques
in ra-
(1973), 101 - 149.
[19]
T. W. GAMELIN and J. GARNETT, Constructive
tional approximation, Trans. Amer. Math. Soc. 143 (1969), 187 - 200.
62
BIERSTEDT
[20]
K. GOWRISANKARAN, Multiply harmonic functions, Nagoya Math. J. 28 (1966), 27 - 48.
[21]
A. GROTHENDIECK, Produits tensorie1s topo10giques
et espaces
nucleaires, Memoirs Amer. Math. Soc. 16 (1955),
reprint
(1966) • [22
I
HO-VAN-THI-SI, Frontiere de Choquer dans les espaces
de
fonc-
tions et approximation des fonctions harmoniques, Bull. Soc. Roy Sci. Liege 41 (1972),607-639. [23]
I. NAMIOKA and R. R. PHELPS, Tensor products of compact convex sets, Pacific J. Math. 31 (1969), 469 - 480.
[24]
I. REAY, Subdua1s
and tensor products of spaces
of
harmonic
functions, Ann. Inst. Fourier 24 (1974), 119 -144. [25]
H. ROSSI and J. L. TAYLOR, On algebras of ho10norphic functions on finite pseudoconvex manifolds, J. Functional Anal. 24 (1977), 11 - 31.
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valeurs
vectorielles
I, Ann. Inst. Fourier 7 (1957), 1 -142. [27]
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a
valeurs dans un Frechet
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Inst. Fourier
24
(1974),167-179. [28]
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of harmonic func-
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A. G. VITUSHKIN, Uniform approximation by ho10morphic functions, J. Functional Anal. 20
[30 I
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Approximation Theory and FUnctional Analysis J.B. FrolZa (ed.) ©North-HoZZand Publishing Company, 1979
THE COMPLETION OF PARTIALLY ORDERED VECTOR SPACES AND KOROVKIN'S THEOREM
BRUNO BROSOWSKI .Johann wolfgang Goethe Universiti'it Fachbereich Mathematik Robert Mayer-Str. 6-10 D-6000 Frankfurt / Main, Germany
In the present paper we will give a new proof of a generalization of Korovkin' s theorem using the completion of a partially ordered vector space by Dedekind-cuts. The given proof works not only in the case of C[ 0 ,1] but also for certain partially ordered real vector spaces where a mode of convergence is defined, which is compatible with the linear structure and the partial ordering of the considered
linear
space. Let X be a real vector space with a partial ordering defined by a cone space
K,
the set of all posi ti ve elements of
X
including
8 . The
X is called Dedekind-complete if every non-empty subset which
is bounded from above has a supremum and if every non - empty
subset
which is bounded from below has an infimum. In the following we
as-
sume that the partial ordering is Archimedean which is defined by
V
n·v
<
U ~
V <
e
nED
for all elements
uJv E X.
Then we have the following THEOREM:
Let x be a pa.Jr..tia.e..ey oJr.dvl.ed Jr.eat vectoJr. .6pace, which 63
i.6
64
BROSOWSKI
Akchimedean. Then
Theke
i~
we have:
a unique detekmined Vedekind - complete
eX
deked ke.at vee-toll. .6pace
w~th
the 60U.ow-i.ng
pakt~ally
ok-
pf1.opelttie~:
~
Thelte
(i)
eJ(~~t~
~ub~pace X
a
eX .6uch that
06
X
and X ake
i.6omoltph-i.c. (ii)
Evell.y element
x#
inf { u
oX
i~
E
X I x# < u}
called .the Vedekind-comple.tion 06
If in addition the ordering in
x.
X is directed i. e.
] c, dE X
'!I
a, bE X then
~aU.6 6ie.6
oX
E
o ;, a ;, d & c < b < d,
oX is also a lattice. For a proof of the theorem compare LUXEMBURG, ZAANEN [21.
DEFINITION:
A subspace
Vedek~nd-denH
called
X of a partially ordered lR-vector space is
in
Y iff
X eye
oX.
For stating the generalized Korovkin-theorem we have to define the mode of convergence in a partially ordered vector space. We some results of BANASCHEWSKI
[11:
E C K\ {e} defines a convergence generating set in X
A subset
if E satisfies the following conditions: V
3
£,£'EE
I\EE
I.
II.
REMARK:
'!I
3
£EE
oEE
Since we assume
III.
use
inf E
e.
0 < £
2/)
&
1\ < £'
< £
X to be Archimedean we have
COMPLETION OF PARTIALLY ORDERED SPACES AND KOROVKIN'S THEOREM
65
Now we define a mode of convergence as follows: A sequence (x ) C X n
x E X
converges to an element
3 n
o
E
xn
<
<
x +
£
n>n = 0
IN
x
In this case we write
x - (
'tJ
iff
~
n E
x. This mode of convergence has the fol-
lowing properties: (a)
Constant sequences are convergent.
(b)
I f (x)
converges to
converges also to (cl (d)
x
n
~
x
&
E
x n ... x
Yn
=>
~
E
'tJ aEJR
E
Y
=>
-x, then every subsequence of (xn'
x.
xn + ax n
y
~
n E ~
x + Y
ax
E
Further we assume (e)
Let (x ) be a sequence such that n
and such that x
(fl
->-
n E
inf(xn' exists, then inf(xn'.
Let (xn' be a sequence such that
and such that x
sup(xn'
exists, then
n
Now we can state the generalization of Korovkin's theorem:
THEOREl4 1: be
Le;t y
be a paltLially oltdelted
lR-vee;tolt .6paee and let
E
a. eonveftgenc.e genvla.ting ./let in Y • Fufttiteft let X be a.n Aftc.rwnedea.n
BROSOWSKI
66
~pace.,
paJ!..U.a.U.y oftde.fte.d lR-ve.ctOJz. Let (L ) be a n
~uch
06 monoton.ic
~equence
L
n
whIch
.i~
Ve.deHnd-de.n~e.
In
Y.
opeftatoft~
: Y -+- y
that Ln (x)
\I
x
E X
-+- A (x)
E
whefte A : Y -+- Y
I~
~uch
a monotonIc opeftatOJz.
map 06
X
onto
X and
A
lx
~
that the
fte~t!t.ictIon A
06 mOflOtoiUc type (Le..
Ix
.i-6 a b.ijec.tive
A IX(xl)~A IX(x ) ",>x ~x2)' 2 l
Then we have \I yEY
PROOF:
For the proof let
u
E
U
y
.-{uEXly
L
.- {l
Y
and
l E L l
Since
Ln
and
be given. Then define the sets
U
y
For each
y Eye 0 X
E
X I l < y}.
y
we have
< y < u.
A are monotoni c we have
and A(l)
< A(y)
~ A(u).
COMPL.ETION OF PARTIAL.LV ORDERED SPACES AND KOROVKIN'S THEOREM
For abbreviation we set
We now prove:
(Y ) converges to an element Yo. n
Since by assumption
L (U) n
and
->-
E
A (u)
we have
:I
\I
n
EEE
o
< u
\I
E IN
or
n> n
=
A(l)
n
< A(u) + E
0
-
E
< Y
n
< A(u) +
E.
From this we conclude that the elements 8
n : = sup { Y k
k ~ n}
0X
E
and in : = inf {Yk E oX
exist
I
k > n}
and also satisfy the relation \I
n>n
=
0
Consequently we have A(l.)
\I
n>n =
-
E
< i
s < A(u) +
~
E
0
where : = sup {in}
i
This is true for every
( *)
\I
l
E
and
E E
\I
L
Y
uE U Y
s
:=
inf {s n } •
E; thus we have also
AU)
~
i
~ 8
< A(u).
67
68
BROSOWSKI
Now let
u E
i
• Then we have
U
and by
< u
(*)
i A(t)
"LY
l
< i
< u.
E
Since A is of monotonic type and bijective we have 1 1 l = A- A (l) < A- (u)
"Y
tEL
and consequently
1
A-
(u)
From this we conclude
U
E
Y
=u and hence
Now let
A(y}
u E UA(y) • Then we have
Z E" L
and consequently
A-
1
u E U
(u)
U
E
y
•
Using
< AA- 1 (u)
and
we conclude
(*)
= u
UA(y) C U
. Consequently we have
i U
UA(y)
=
i
Similar ly one can prove L
and
= LA (y)
i
this we conclude:
u
l
i.e.
J
i
~
Y
i
and hence
i.e.
E
U
i
=a =
U
-a
A(y).
From the relations
"
e:EE
3
n EJN o
"
n >n =
A (y) 0
-
e: < in ~ an < A (y) + e:
COMPLET~ON
and
OF PARTIALLY ORDERED SPACES AND KOROVKIN'S THEOREM
69
we conclude A (y)
Since
£
E E
-
E: < y n ;;, A (y) +
£ •
was arbitrary we have
Yn ... A(y). E
REMARK:
Let
C [a,
b I be the vector lattice of all real-valued con -
tinuous functions on
f(t)
~
get) for all
[a, b.)
under the ordering defined by .. f ;;, g iff
t E [a,b 1"
Then the set of all polynomials of
degree;;, 2 is Dedekind-dense in C[a,b). Since the set E:={£·lEC[a.bl !E:> O} generates convergence in the sup-norm we can conclude the
classical
Korovkin-theorem from theorem 1.
REFERENCES
[1 l B . BANASCHEWSKI, Uber die Vervolls tandigung geordneter Gruppen ,
Math. Nachr. 16 (1957), 51 - 71. [2
I
W. A. LUXEMBURG and A. C. ZAANEN, R.i.e.&z Apace.&. Vol. 1., North-
-Holland, Amsterdam-London, 1971.
This Page Intentionally Left Blank
Apppoximation Theory and FUnctional Analysis J.B. ppolla (ed.) ©Nopth-HoLland Publishing Company, 1979
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
P.
L. BUTZER, R. L. STENS and M. WEHRENS+) Lehrstuhl A fur Mathematik Aachen University
of Technology
Aachen, Nestern Germany Dedicated to Professor Einar Hille on the occasion of his eightyfifth birthday on 28 June 1979, in friendship and high esteem.
1. INTRODUCTION AND PRELIMINARY RESULTS In 1963 one of the authors [7]
raised the question*)
whether
it is possible to construct a singular convolution integral of a flmC'" tion
f
defined on
degree
n
o(n -a),
0 < a
where
and which approximates
f
uniformly on [- 1 , 1) with order
:= {f E C;
WI (n;fiC)
WI (n;f;C) is the modulus of continuity of
C :: C [ - 1 , 1) functions
f
of
2. 1, provided f belongs to
LiPl (a;C)
(l.l)
[- 1,1] , say, which is an alge!Jraic polynomial
on
f
(see (7.3»
and
is the set of all real (or complex) -valued continuous [- 1 , 1)
endowed with the usual sup-norm II • II C . The
point is that this integral should not be derived under some (trigonorretric)
+)
This author was supported by DFG resear..::h grant Bu 166 /27
which
is gratefully acknowledged.
*)
This problem has since been recalled in some language or other by, for example, G. Freud
[28)
and 71
R. DeVore
[21, 23).
72
BUTZER.STENS and WEHRENS
substitution from a corresponding integral which is a polynomial of degree e.g.
[l4],[9,p.
n
trigonometric
(such as the integral of Fejer-Korovkini see
80]).
This problem was solved in some form or other by Freud (1964), Sallay
[49]
Vertesi-Kis [ 60] (1969), Bojanic Rodina
[46]
(1964), Saxena
(1967), DeVore [5]
[20 I
[14]
(1967), Vertesi [59] (1967),
(1968), Bojanic-DeVore
[18 I
(1969), Chawla
(1973), Freud-Sharma
and Butzer-Stens
[50]
[29]
[ 27]
[6
]
(1970), Mathur [41] (1971) ,
(1974), Szabados [56]
(1976)
(1976). Most of the polynomials construc:ted by
these authors are of degree 4n _. k with k = 2 or 4 I SOMe approximate f uniformly only on
[- 1 +
1 - El for each
E ,
E
> O.
The natural extension of this problem, posed in [8 ] , is whether an algebraic polynomial of degree
n
can be constructed which
uniform approximation to the associate order
o
O(n- l - a ) provided the derivative
f
on the whole fl
gives
[- 1 , 1 ] with
belongs to
LiPl (a i C) ,
< a < 1.
In this respect Bavinck [1, p. 69 -Wehrens
I ,
Lupa~
[38, 39
I and Stens-
[55] considered the integral
(J 2nf) (x)
:=
1
'2
J1
f(u)X2n(x i u)du
-1 (1. 2) X
2n
(Xi U )
3
:=
n
2
+ 3n + 3
lfn k=O
2k +l - - 2 - P (x) P (u) k k
• Il Pk(t) [p~2,O) (t) ]2dt, -1
p~a,(3)
being the Jacobi polynomial. Note that X2n(xiu),;: 0,
L~
X2n(xiu)du
= 2 and, as will be seen below, the kernel can more simply be rewrit-
ten as
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
3
X2n(x;u) n Lupa~,
for
exam~le,
2
T
+ 3n + 3
u
73
[p(2 , O)] 2 (x). n
showed that
(1. 3)
However, (J2nf) (x) is a polynomial of degree
2n I and not
n.
One purpose of this paper is to present a systematic
approach
to these two problems, thus to study direct approximation theorems for algebraic approximation processes that are built up analogously well-known trigonometric convolution integrals. Normally
one
to
would
expect to examine the convergence of convolution integrals like 1
(1. 4)
1-11
"2
to f (x) for
X (x-u)f(u)du n
n'" "". The best known example of such an integral is that
of Landau-SUeltjes (see e.g.
[37,~.
9], [48, p. 147],[ 22, p. 26] /
[40 ]), the kernel of which is given by
X2n (x)
;=
1
Here it is known (see e.g. [22, p. 22l])that f
o
E
Lipl(aiC[-l+£, 1- d),
< a < 1, 0 < £ < 1, implies that
(1. 5)
1 ""2
I.-1l
o (n- a / 2 ) •
X2n (x - u) f(u)du - f(x)IIC[_l + £,1 - d
The integral is again an algebraic polynomial of degree however, difficulties occur at the end points ± 1 [-1, 1] since the classical translation operator (T~g) (x)
:= g(x -
2n.
of the T~
used,
Here, interval namely
u), leads one outside of the interval [-1,1].
74
BUTZER,STENS and WEHRENS
The question now is whether it is possible to employ an alge braic convolution concept (which depends on an associated translation concept) for which these difficulties do not occur and for which there holds some "convolution theorem" in the form that
if
T
is a suit-
able transform such as
(1. 6)
T[
f j(k)
1 [1 f (u)
= 2'
~k
(u)w (u) du
{O,1,2, ••• })
(k E lP
-1
for suitable functions
~k'
orthogonal with respect
to
the
weight
w(x), then (1. 7)
f
*
g
T[f
*
g J (k) = T [ f J (k) T [ g I (k)
being the sui table convolution of
(k E lP) ,
f and g. This would enable
one to use integral transform methods and, as is well - known,
such
methods usually enable one to solve a variety of problems byareduction to a standard procedure (recall the Fourier transform method in the solution of partial differential equations; see e. g. [9, Chap. VII ]). Hereby the aim. is to employ purely algebraic means in the proofs,the only connection with the periodic theory being of structural nature, namely an approach via convolution integrals together with transform methods. Therefore in none of the proofs results of Fourier analysis will be used, as was the case in a few instances in the Chebyshev transform approach of Butzer-Stens [12,13,14,15,16 I, Stens [54 I . The transform we shall apply is the quite well developed Legendre transform. Although Fourier-Legendre series have been known
for
at
least a century, the product formula leading to the translation operator being already known to Gegenbauer [30 I in 1875,
the
Legendre
transform point of view seems to have been first emphasized by Tranter [58] only in 1950 (see also [51, p. 423, 454]
and
cited there). The main results needed here are built
the up
literature in
Stens-
APPROXIMATION BV AL.GEBRAIC CONVOL.UTION INTEGRAL.S
Wehrens
[55J, but let us recall some of the basic concepts.
Letting 1 ~
P
X stand either for the space C [-1,1]
fined on
[- 1,1]
for which the norm :=
II flip
{~ f E X
is finite, the Legendre transform of
Here
or LP(-l,l) =L P ,
00, of all real (or complex)-valued measurable functions f 09-
<
(1. 8)
75
is defined by
1 fl f(u)Pk(ul du := 2" -1
L[f] (k) - r(k)
(k E ]p) •
Pk(x) is the Legendre polynomial of degree _ P
()
k x
(-1' I " k ~
- 2kk!
dk dxk
--(1 -
x
2 k
1
k, namely
(x E
[-1,1); k E]P).
E
[-1,1); kE]P),
In view of the fact that
(1. 9)
1
(x
one has
(1.10)
I L [ f]
(k)
I
< II f II X
(f E X; k E ]P) ,
so that (1.8) defines a bounded linear operator mapping X into (co), the space of all real (or complex)-valued sequences tha t
limk+oo a k
=
,h'
is here replaced by
('hf) (x) := (l/lT)fl f(xh+ u/l-ill-i) Il_i,-l du -1
In contrast to
'h' 'h
ear operator from
such
O.
The classical translation operator
(1.11)
{ak}~=O
defines for each
X into itself with
h
E
(x,h E[-l,l]).
[-1,1] a positive lin-
II 'h II [x,xl = 1
and the usual
76
BUTZER, STENS and WEHRENS
limh~l_ "Thf - f" X
O. The associated convolution product
f
*
g is
defined as (1.12)
If
(f
*
g) (x)
:=
1 '2
Il
(xE[-l,l]).
-1 (Tu f ) (x)g(u)du
f E X, g ELl, the convolution
*
f
g exists as an element of
X
together with ( 1.13)
(1.14)
L [f
*
g
I
L[ f I
(k)
(k)
L [g I
(k E P)
(k)
which is the form taken on by the convolution theorem (1.7)
in the
Legendre case. The derivative also being defined via translation, it is to be expected that the derivative in the Legendre frame will be The strong (Legendre) derivative of
unusual. g
E X
2,3, •••
are
f E X is the function
for which
lim II h+lwe write
Dlf
=
0;
g. Derivatives Dr of higher order
r
=
defined i terati vely. The set of all f E X for which Dr f exists is denoted by Wrx ' Note that the strong derivative Dlf, f E W1X ' coincides with the pointwise derivative d
""dX" (
x 2 -1 2
d ""dX" f (x» •
The counterparts of the modulus of continuity
and
Lipschitz
class here take on the form (1.15)
sup II Thf - f" X
o:s.h:s.l
(-1 < I) < 1)
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
Lip~(a;X)
(1.16)
77
:== {f E X; wi(o;f)
The main purpose of this paper is to give a unified
treatment
of algebraic approximation theory via the Legendre transform method, in particular, to study conditions 1 {Xn}n EP C L (- 1, 1) such that
(1.17)
II f
*
n - f II X
upon the sequence of functions
o
(f E
X),
and to investigate the rate of convergence in (1.17), expressing
it
lim n-+ oo
X
in terms of the modulus of continuity (1.15). In addition
to
such
direct approximation theorems, the matching inverse theorems willalso be considered, emphasis being placed upon the so-called case
of
non-optimal approximation. The case of optimal or saturated approximation is dealt with briefly. The aim will be to employ
elementary
means in establishing the direct theorems (thus without appealing to the general theorems based upon intermediate space methods of Berens [3
1 and Butzer-Scherer
[ 10, 11 ], as was carried
out
Bavinck
by
[1,2]). Concerning the inverse theorems, they will either be dedoced via the associated theorems of best algebraic approximation (as developed in Stens-Wehrens [55]) or from a general result based upon a Bernstein-type inequality. This material is considered in Sec. 2. One major aspect is to study various examples of suitable kernels that can be classified under the Legendre transform
approach.
These are given by various summability methods of the Fourier-Legendre series of
f E X:
namely by the Fejer means of
f
E
X (Sec. 5), by the
Fejer-Korovkin
means (Sec. 3), the Rogosinski means (Sec. 6), certain de La
Vallee
78
8UTZER,STENS and WEHRENS
Poussin sums (Sec. 5), by the Weierstrass and integral means, aswell as by the Landau-Stiel tjes integral in the Legendre frame, all
three
treated in Sec. 4. The Fejer means are defined by
n
(1.19)
k
(anf) (x) : = l:k=O (1 - '""i1+T)(2k + 1) f" (k) P (x) k
(xE[-l,l] :nEP)
which may be rewritten in the form of an algebraic convolution integral
(1. 20)
(a f) (x)
n
:=
r
~
-1
F (u)(T f) (x)du
n
u
where
n
k n+ 1) (2k+l)P k (x)
:= ~k=O (l -
(1.21)
(x e [-1,1]: n ep).
A particular case of the results to be established asserts that
This solves the stated problem in its [ -1,11 of degree
for
0 < ex < 1, anf
original
form
on the
whole
being an algebraic polynomial precisely
n.
For the more difficult case
ex = 1
we proceed as follows.
In
Fourier analysis the Fejer-Korovkin kernel may be defined as that even non-negative trigonometric polynomial
tn of the form
(6 E JR.: n E P)
for which the coefficient (given by
cos (n! (n + 2)
».
a
l
takes on its largest possible
In the corresponding algebraic case
amounts to finding that algebraic polynomial
value this
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
Pn (X)
79
(X E
which is non-negative on [-1, 1 I and for which
b
l
[-1,1); nE lP),
attains its maxi-
mum. The solution of this extremal problem is exactly the
Fejer -
Korovkin kernel for the Legendre case, defined by
(1. 22)
x
~
being the largest root of
notes the largest integer
~
PN(x), and
N
[n/2] + 1
([x] de-
x) .
Again a special case of our results states that
(0 <
a.
~
1) •
This shows that the associated Fejer-Korovkin means
f • K solve our
problem even in its extended form on the whole [-1,1
1,
tually being a
(pure) algebraic polynomial of degree
above results are not only valid in
n
f . Kn
ac-
n. Most of the
C [-1, II but also in LP(-l,l).
As mentioned, two of the authors set up a Chebyshev
transform
method with essentially the same aim in mind, namely to give aunified approach to as many problems as possible on the approximation of functions
f
belonging to
C [-1,
11
or
LP(-l,l), 1 < p <
w
-
00,
polynomials. The Chebyshev method has the advantage that variety of problems can be considered, such as all with moduli of continuity
those
byal.gebJuUc. a
greater connected
of higher order, including the fractional
case. The disadvantage, however, is that it is not as "purely" algebraic as is the Legendre transform approach.
Although the latter is
80
BUTZER, STENS and WEHRENS
more intricate as it is not connected with the periodic Fourier theory, it has the advantage that no "bad-looking" weight factors enter into the picture as is the case with the weight w(x) = (1 - x 2 )-1/2 in the Chebyshev theory. The question of course arises why not treat the matter by the more general Jacobi transform approach. The reason is that
we
first
wanted to present an approach that is not only as uncomplicated
but
also as complete as possible. However, much of the material presented can readily be carried over to the Jacobi frame. As indicated
Bavinck
[ 1, 2] considered more or less some of our results in the latter frane. But it can perhaps be said that his aim was to generalize VUgonomet~~~
approximation theory to the Jacobi frame without being concerned
with the connections to the problems of
algeb~a~~
approximation
in
the classical sense. For a basic unsolved problem in the Legendre approach see [17].
2.
GENERAL THEOREMS ON CONVOLUTION INTEGRALS This section is concerned with theorems on the convergence
of
general convolution integrals
(2.1)
(I f) (x) p
:= (f
*
X ) (x) p
~
I1
(T
-1
u
f) (x}g(u)du
(fEX;XE[-l,l] where
(2.2)
p
{X p
is a kernel, i.e.
}pE A
~
1
f-1
Xp E
Ll(-l, 1) with
=1
X (u)du p
pEA)
(p E
A),
being a parameter ranging over some set A which is either an in-
terval (a,b) with
0
~
a
<
b
~ ~
, or the set
P. Let
Po be
one
of
APPROXIMATION BY AI.GEBRAIC CONVOI.UTION INTEGRAl.S
the points
a,b
or
+
00
In the following
81
•
M denotes a positive constant, the value
of
which may be different at each occurrence. M is always independent of the quantities at the right margin.
PROPOSITION 1:
Let
be a rle)tVle£' buc.h that
{X } p E!A p
(2.3)
(p E 1\) •
II X p III -< M
o
(2.4)
6o~
each
f E
x, i6 and onty i6
(2.5)
lim
(k E IN
1
{l,2,3, .•• }).
P+Po
In this proposition, the proof of which follows by the BanachSteinhaus theorem, it may be difficult to verify condition (2.3) the applications. If the kernel is however positive, i.e. for almost all M=l
u
E
(-1,1), p
E
X (u) p
in > 0
lA, then (2.3) is always satisfied with
in view of (2.2). This leads to the following Bohman-Korovkin-
type result:
PROPOSITION 2:
16 the
ke.lf.net
{X } p E IA p
ib pOb .U:.iv e, eac.h 06 the 6o.e.-
towing abbe.lf..tionb ib equ.ivatent .to (2.4) and (2.5): 1,
(2.6)
<5
(2.7)
lim r. X (u)du p""p -1 P o
o
(-I <
<5
< 1).
82
BUTZER, STENS and WEHRENS
PROOF:
That (2.5) implies (2.6) is obvious. On the other hand,
if
(2.6) holds, then by (2.2) and the positivity of the kernel, 0
1
< (
<
1 Jl (l-u)X (u)du=_2- [l-XA(l)l=o(l) 1-0 -1 p P 1 - 0
J -1
X (u)du < - - - J p
1 - 0
-
This gives (2.7). Concerning (2.7) for
(0
o
~
-1
(l -
u)X (u)du p
(2.4), one uses the fact
that
f E X
(2.8)
II Ipf - f II X
~ ~ <
1(1 II
(T
-1
II f IIx
u
f) (.) -
f (.) II xX (u) du P
a
f-1 X (u)du +
L W
p
which implies (2.4) by standard arguments.
l (o;f)
0
When dealing with positive kernels, there holds the
following
general result on the rate of convergence in (2.4):
PROPOSITION 3:
Let
{xp}p
EfA
be a pO.6).tive l1.eJtne.t, and let
.6tJtic.Uy pO.6itive 6unc.tion de6ined on
fA
.6uc.h that
lim
P-+Po
be
a
'Ii (p) =
o.
'Ii
The 6ollowin9 a.6.6eJttion.6 aJte equivalent: (2.9)
(2.10)
PROOF:
(f E X;
P E fA) •
Let us first mention the following inequalities needed
(see
[55; Sec. 5]),
(2.11)
(g E W1 ; -1 < 0 < 1) x
APPROXIMATION BY AL.GEBRAIC CONVOL.UTION INTEGRAL.S
83
(2.12)
Now if (2.9) holds, then (2.10) follows as in (2.8) by (2.12)
and
(2.2) since
L III f- fII _<12w (1 - '~(p),·f) 21 p .,.. l X
L
12 w (1 -
1
r [1 Ll
+ ~ l-u 1 ()d xp u u
l-XA(l) }_<MWL1(l_
(p E A).
To prove the converse, we apply
(2.13)
for
(g E X;
k = 1 and
Since
P
l
:r1 [1-1 (1
(2.14)
(2.15)
kElP)
(2.9) follows by (2.1l}.0
The particular case
-
(p E A),
p
III f - fllx < 24wLl(XA(l}ifiX} p
gives, noting that
- u)x (u}du > 0
Fait e.a.c.h po!.>LUve. ke.ltne.l
COROLLARY 1:
1;
to deduce
wi,
E
X E [-1,1
p
{Xp}pEIA the.lte. hold!.>
(f E Xi
p E
IA) •
Inequality (2.15) shows that the approximation error dependson the smoothness properties of the functions involved. Therefore
one
might expect that the rate of convergence in (2.4) could bearbitrarily good if
f
is sufficiently smooth. Later on it will be seen
that
84
BUTZER. STENS and WEHRENS
this situation really takes place for at least one convolution
in
tegral. But for a majority of approximation processes there exists a critical order which cannot be transpassed unless
f
is
a constant.
This is the familiar saturation phenomenon. A
handy criterion deciding whether the saturation property holds {Ip} P E/A
for a convolution integral
PROPOSITION 4: ~a:t~~6~e.~
1 - X~ (k)
lim sup P"'P o
06 .the. pltOC.e.H
{Xp}pE/A
c.ondi.tion
(2.16)
whe..I!.e.
a) 16 :the. R.e..I!.ne.t
is given by
i~
a 6unc.tion
>
a~
0
(k E:IN) ,
in P.I!.Op. 3, .the.n
(2.17)
o(
~mpLi.e.-6
.tha.t
f = canst.
(a.e.) *) •
b) 16 mo.l!.e.ove..I!. :the..I!.e.
e.x~~t~ ~ome.
k
o
E
IN
~uc.h
that
1 - X (k ) A
(2.18)
:the.n the..I!.e.
lim sup P"'Po e.xi~:t~
at te.a-6:t
P
0
on~
non-c.on~tant
(non-:t.l!.iviat)
f E
X -6uc.h
.tha.t (2.19)
*)
"(a.e.)" means that an assertion holds for all X=C[-l,ll, and for almost all
x
E
x E [-1,11
if
[-1, II i f X=L P (-l,I), l.::.p
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
PROOF:
85
By (2.17) one has on account of (1.10) and (1.14) that
o ('I'
This means that
lim
II
p->-p 0
(p) )
-XA(k)1 ir(k}II
the
other hand, by (2.16) there exists a subsequence {P.}':" 1 ,lim. p. =p , J J= J-+OO J 0 such that
lim.
J-+03
I XAPj (k)
c
-lll
uniqueness theorem (see [55, Sec.
k
> 0, each
2J) this gives
Concerning part b), one has for
f =P
k
kE IN.
By the
f = const.
(a.e.).
by (2.13) and (2.18) that o O(.p(p»
which completes the proof.D Whereas Cor. 1 gives a direct approximation result, an inverse one is given by
Let {Xn}nE
PROPOSITION 5:
-i..nequal-i..ty 06 ofLdefL (2.20)
16 ~uch
f
f E X
*
Xn E
Y
W~,
JI? be
a kefLnel
~at-i..~6y-i..ng
a
BefLn~te-i..n-type
namely
> 0,
II Dl (f
* Xn) II X
< M nY
II f II X
(f E X; n E JI?) •
can be appfLox-i..mated by .the convolu.tion -i..ntegfLai
that
(2.21)
60fL ~ome
PROOF:
II f
0 <
0,
*
< 1,
Xn - f II X
then
f
o (n-Yo')
(n
-+
cc)
E LiPi(O,;X}.
Let us follow the classical arguments using telescoping sums
(cL [9, p. 1101. Setting n=3,4, ... , then by (2.21)
U2 = f
* X2 2
' Un = f
*
X n 2
f
* X 2n - l
for
86
BUTZER,STENS and WEHRENS
(n
(2.22)
n
lim II ~k=2 Uk -
(2.23)
2,3, ••• ) ,
0,
f II X
n->-oo
Hence for arbitrary
h E (-1,1) and integral
m > 2
(2.24)
Since
Uk belongs to
w~ for every
k > 2, one has by (2.11)
1 < 6 (1 - h) II 0 Uk II X •
(2.25)
Since the convolution product is commutative and associative, one can rewr i te
Uk as
(k = 3,4, ••• ) •
Uk = (f-f * X k-l) *X k - (f-f*X )·X k-l 2 2 2k 2 This implies by (2.25),
(2.20) and (2.21) that
~ M(l - h)2 ky (1-a.),
which is also valid for
(2.26)
k =2. This yields by
(2.24) and (2.22) that
L
(-1<0<1).
wI (0; f)
If one now chooses
m such that for
y m 0 > 1 - 2- , 2 - 1 < (1- o)-lIY :::.~,
then (2.26) gives (as e.g. in [9, p.l0l) This is the desired assertion.
0
that
w~(o;f)
< M(l-o)Ct.
APPROXIMATION BY AL.GEBRAIC CONVOLUTION INTEGRALS
87
Finally a result on best approximation by algebraic polyttmUals will be stated; it may be used for establishing direct approximation theorems for certain approximation processes. way in proving inverse theorems
for
It also gives another
polynomials kernels leading
to
optimal approximation. As usual, let
(n E P)
En (f;X)
be the best approximation of gree
~
n, P n
f E X
be algebraic polynomials of de-
being the set of such polynomials. Concerning the rate
of convergence in
(2.27)
lim En(f;X) n .... oo
o
(f E X)
we have the following counterpart of the classical Jackson and Bernstein theorems (see [55, Sec.
PROPOSITION 6:
a)
Fo~
5]).
eaQh
rEP, the~e hoid~ (f E~; n ElN).
b) The 6oiiowing
(2.28)
(2.29)
~tatement~
a~e
equivaient
6o~
0 < a < 1:
O(n- 2 (r+Ct))
(n ....
L
r
(0 .... 1-).
WI (0;0 fiX)
The purpose of the next sections is to apply results to several concrete approximation processes.
(0),
the
foregoing
88
BUTZER,STENS and WEHRENS
3. THE ALGEBRAIC CONVOLUTION INTEGRAL OF
FEJ~R-KOROVKIN
Concerning Cor. 1 it is of interest to search kernels
{Xn}:=l which belong to
for
Pn such that them:xiulus
positive
wi(~(l);f;X)
is as small as possible. This amounts to determing such kernels which
X~(l)
is asclose to
1 as possible. Note that
X~(1)
for
< 1
in
view of (2.14). This will lead to the Fejer-Korovkin kernel of Sec. 1. Therefore we will now discuss soroe extremal properties of
the
coefficients of the elements of the set
N p+ := {p E P , Pn(x) > 0 n n n
(3.1)
THEOREM 1:
pEN p+ theJte hold.6
FOIL aLe.
a)
x E [-1,1], Pn~(O)=1} (n EIP).
for
n
n
D, and
xm+1
(3.2)
whelLe
m = [(n + 1)/2 b)
fOIL
each even
n
E
P
i.6 the lalLge.6t 1L00t 06
thelLe exi.6t.6 a unique Pn
E
P + . m 1
N P~
.6uch
that P~ (1)
(3.3)
c) FOIL
n
E
IN odd :thelle ex.i.6:t.6 no
d) Fait each
(3.4)
Pn
E
j (j + 1)
NP~
.........l..L.......;.......:.;:"""'2
2 (2n + 1)
and each
< 1 -
P~
j
P E
n
E
IN thelle hold.6
(j) < 72j (j + 1) (1 -
the Itight hand inequality being valid nOlL all inequality only 601!.
PROOF:
N P+ n .6Uch that (3.3) holM.
n E
P~ (1)
,
P, :the le6t
hand
n ~ No = No(j).
First we need the Gauss-Jacobi mechanical quadrature formula
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
(see e.g. [19, p. 741, [57, p. q2k-l E P2k-l'
89
47;(15.3.2}]l. It states that foral!
k E :IN, -there holds
(3.5)
where with
Xj ,k'
1.2. j .2. k, denote the roots of the Legendre polynomial P
-l<xj+l,k<Xj,k
Aj,k
k
the "Legendre
abscissas" 2 J,
2 (1 - x. k)
A.J, k
pEN p+. n n
Now let
(3.6)
Since
k
Choosing
-2
(P ' (x. k»
m
(1.2. j .2. k ) •
J,
[(n+l}/2]
one has by (3.5)
P~ (0)
1
1 Jl
P~(l)=2
(3.7)
-1
up (u)du
-1
n
=..l. ~+l 2 j=l
A
()
j ,m+l Xj ,m+l P n Xj ,m+l •
Pn(Xj,m+l} > 0, Aj,m+l > 0, l.2.j .2.m+l, part a} follows imme-
diately by
1 Ip~ (l) I < "2
max 1 2j.2.m+ l
Ix.J ,m+1 I
~+l
j=l Aj ,m+lPn (x j ,m+l) = xm+l •
To prove parts b} and c} we first assume the existence Pn
E NP+ n
a
satisfying (3.3) . By (3.6) and (3.7) one has
Since all terms in this sum are non-negative and one has by (3.6) (3.8)
of
o
Aj,m+l
(xm + 1 -xj,m+l) ;#0,
90
BUTZER, STENS and WEHRENS
(3.9)
2/ "l,m+l •
Therefore Pn is uniquely determined at the points Since the
X.
],
m+l ,1< -j -<m+l.
xi,m+l' i =2, ... ,m+l are in (-1,l) and Pn is to be posi-
tive, it must have
m double roots. This implies that for even n E F
(3.10)
is the only polynomial For
n odd, i.e.
N p+
in
n
satisfying (3.3).
m = (n + 1)/2, (3.8) states that
sesses (n + 1)/2 double zeros, meaning that
Pn(x)
= O.
Pn
pos --
But this
is
a contradiction to (3.9), proving c). Concerning part d), the right side of (3.4) follows from
72j (j + 1) (1 -
where (2.13), Cor. 1,
(3.11)
1
(0 P.) (x)
J
P~ (1) ) ,
(2.11) and the equality
j (j
+ 1)
- - 2
P j (x)
(x E
[-I,ll: j E P)
(cf. [55, Sec. 3]) were applied. In order to verify the left side of (3.4) fix
(3.12)
j
E IN. For
n
~
j
+ lone has by (3.5) that
91
APPROXIMATION BY AL.GEBRAIC CONVOLUTION INTEGRALS
max \ P (x) \ j x n , n :5. x :5. xl, n
<
the latter equality
max \ P j (x) \ , 0:5. x:5. xl , n
From [57, Sec. 7.211 it now follows that value in
I is even, and Ip.] (x) i attains
being valid as I P. (x) ]
[O,l-nl
at
x=l-n
if
n > 0
x
l,n
=-x • n,n
its maximum
is small enough.
maxO < < I, P. (x) II = I P. (Xl ) I = P. (Xl ) _x xl,n ] ],n],n chosen large enough. So (3.12) yields that
This
if
means that
is
n
< P ]. ( xl ,n )
I ~ ( .) I IP ] I
n
In view of Bruns' inequality (see e.g. [57. (6.21.5)1)
cos (21T/ (2n + 1)
(3.13)
we find with a suitable
1
P. (1)
1 - p,(x
]
Since plete.
P,(X
] ,In
]
lim
l ,n
n-+ oo
I; n ) +
E
< xl,n < cos (1T/ (2n + 1»,
(Xl ,n ' 1)
(1-
xl
,n
that
)P!(I;)
]
n
2
) > !P!(I; )\(1- cos (1Tj(2n +1») > Ip!(I;)/
']
P! (1)
P! (I; ) ]
n
n
J
]
j (j + 1) /2,
the proof
n
(2n + 1)
of
d)
2
is com-
0
The polynomial
Pn of (3.10) satisfies for even
n
E
:IN
the
same extremal property as does the trigonometric Fejer-Korovkin kernel. Therefore one may justly call the kernel (Legendre-) Fejer-Korovkin kernel. If cisely the polynomials
n
Pn of (3.10); if
K
n
of
is even, the n
(1. 22)
Kn
are
the pre-
is odd then Kn(x) =Pn-l(x).
Concerning the approximation behaviour of the associated Fejer -Korovkin convolution integral we have
92
BUTZER,STENS and WEHRENS
by (1. 22) . One ha.t.
*
-
f E X
L fllx .::. 24 wl (xN;f;X)
a)
IIf
b)
f E LiPi(Cli X) -lif
c)
The
X with o~de~
K n
6OIl.
*
(n E P) .
2Cl Kn - f II X = O(n- )
~onvolution integ~a.l
06
Feji~-Ko~ov~in
(n ... 00; O
it.
t.a.tu~a.ted
in
O(n- 2 ).
The proof of part a) follows by Cor. 1. Concerning b), fE LiPi (a;X) - Cl -2Cl ). The converse implies by (3.l3) that wL (xN;f) = O( (1 - x ) ) = O(n N l can be derived either from Prop. 5 since f Kn E Pn C w~ and
*
( 3.14)
(see [55, Sec. 5]), or from Prop. 6. Part c) follows readily by Prop. 4, Thm. ld) and the fact that 1 - K~(l) = 1 - x = O(n- 2 ). N Let us remark that Bavinck [2 ] employs this kernel to establish a Jackson-type theorem (in the Jacobi frame). In [l) the external property but does
he also treats
not distinguish between the cases
n
being even or odd; for odd n his proof is not quite correct since the quadrature formula used is no more exact in this case.Newman-Shapiro [43] and Feinerman-Newman [25, p. 103) handle the extremal
problem
for even n in a somewhat more general frame but do not discuss
re-
sults of the type of Thm. 2 b), c). As an immediate consequence of Thm. ld) one has that for every posi ti ve kernel
{X n }n EP'
Xn E Pn' there holds
(j
E
IN).
An application of Prop. 4 a) then leads to the following algebraic counterpart of a result in
[9, p. 88] which is related to
a
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
93
result of Korovkin [36, p. 1281.
16
COROLLARY 2:
Xn
NP~
E
.then
IIf*X n -fll X 6o~ ~ome
f E X
~mp!~e~
.tha.t
o (n -2 )
(n
f = const.
~
00)
(a.e.).
In other words, one cannot approximate a non-constant function f E X
by positive polynomial convolution integrals of degree n with
O(n-2 ).
an order better than
4. THREE PURTER POSITIVE ALGEBRAIC CONVOLUTION INTEGRALS Whereas the first example below is an algebraic 90lynomial
of
degree 2n, the second and third are non-polynomial.
4.1. THE SINGULAR INTEGRAL OF LANDAU-STIELTJES As remarked in the introduction, the classical Landau-Stieltjes integral approximates functions [-1
+
£,
1 -
£
1
for
E
£
f E C [-I, 11 uniformly
only
on
(0,1). Using the kernel of this integral we
now construct a new singular integral by means of the Legendre convolution which forms an approximation process in in
C [-1, 11 as well as
LP(-l,l). This kernel is defined by
A (xl 2n
:= (I/A
[ 2n ) 1 -
( l-x)2}n 2
(x
E
I -1,1 J; n
(4.1) :=
~
II
[1-
( l ; U )2 Jn du
(n E P) •
-1
The
fl. 2n belong to
N P~n ' and for the
A2n one has
Ao
I,
E
]P),
94
BUTZER, STENS and WEHRENS
2n(2n-2) . . . . . . . 2 (2n + 1) (2n - 1) .•. 3
(n E IN )
< (Tf/(4n +2»1/2
(4.2)
Let us list the properties of
A 2n
(n E P).
needed for the
a~proximation~~-
orem.
LEMMA 1:
a)
F Oft all
n E lP
thefte hold.6
A (l) = 1 - l/(n + 1)A • 2n 2n
(4.3)
(f E Xi n E lP) •
(4.4)
c)
The no.f..f.ow.trtg BeftYl..6te.-LYI.-type -lYl.equa.f.-Lty ho.f.d.6:
(f E Xi
(4.5)
PROOF:
(4.6)
n E lP).
Part a) follows immediately by integrating the identity
(1 - x) A
2n
(2/( n
(x)
' ) + 1) "2n
dd x
[1- ( l ; X )2]n+l.
By applying a) and (3.4) we obtain the right-hand side of inequality (4.4). Concerning the left side, one has in view of the positivity of
A
2n
and (4.6) for each
0 E (0,2), k E IN,
1 - A (k) 2n
1
(1/(n+l)A
2n
)Jr
1-0
1 - Pk (u)
1
u
d
{Tu
[1 _
(~) 2] n+l 2
} du
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
95
1
> (1/(n+l)"2)
inf IPk(u) II {dd [1_(1 ;u)2]n+l }du, U n l-o
Since
= k(k
Pk(l)
+ 1)/2, we may choose some
k
infl_c 0,
0 E (0,2) such that
Furthermore, the integral can
be
estimated
from below by (15/2)2, The desired inequality now follows by (4,2), To prove part c) we obtain from [55, Sec. 3) and (1.13) f
* A2n
E P2n
c
w~
that
and
(f E X; n E lP).
Therefore we have only to show that
(4.7)
-
First let
n
~
"
dx
[x
2
-1
2
2. By the estimate
I...!L [x
A2n . dx
...!L
2
-1
2
d ()]' dx A2n x :
one finds that
n/(n-l)+1<3,
(n E P).
96
BUTZER, STENS and WEHRENS
which yields (4.5) by (4.2), A similar calculation shows that also holds for
n
= 1;
the case
n
=0
is obvious.
(4.7)
0
As an application of Cor. 1, Prop. 5 and Prop. 4 we obtain
Fan the
THEOREM 3. f E
x, thene a) II f
~~ngufan
integnaf 06
Landau-St~eftje~
hofd~
*
A
.s. 24 wLl (1 -
2n - f IIx
[ (n + 1) A2n )
-1
;f;X)
O(n -0./2 )
c) The integ~af 06 Landau-Stieftje& O(n- l / 2 ).
de~
i~
(n E lP).
( n ....
&atu~ated
oo ;
0<0.<1).
in X with
o~
4.2. THE INTEGRAL OF WEIERSTRASS The kernel of Gauss-Weierstrass with respect to the
Legendre
system is given by wt(x) := ~~=O exp {- k(k/1) t} (2k+l)P (x) k
(4.8)
LEMMA 2:
and
The
4ati~6~e~
ke~nef
the
w (x) i~ t
non-negative
60~
(xE[-l,l]; t >0).
all x
E
[-1,1], t > 0,
Be~n~te~n-type ~nequafity
(4.9)
(f E X; 0 < t
.s. 1).
The proof of the first part can be found in Bochner [4, p. 1146) or Hille-Phillips [33, p. 612 I, and (4.9) can be deduced from GOrlich -Nessel-Trebels [31, (3.9)] since the Legendre series of
f E X
is
(C,l)-summable (see Sec. 5 below). An application of Cor. 1, Prop. 5 with
Xn
wl/n ' and Prop. 4
97
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
yields
THEOREM 4. f
*
wt
' f
FOJt the algebJtaic GauJ.>J.>-WeieJtJ.>tJtaJ.>J.> convolution -integJtal
x,
E
theJte holdJ.> (t >
b) f
c)
E
Lipi(a;X) - l i f
The integJtal
f
*
wt - fllx
* wt
0).
(t -+- 0+;
0 < a. < l) •
L6 J.>atuJtated in. x with oltdelt
O(t), t-+-O+.
Concerning the infinitesimal generator of the Weierstrass
as
well as Abel method of summation of the Legendre series see e.g.[33,p.6l0
1.
4.3. THE INTEGRAL MEANS In [55, Sec. 3) there had been introduced the (Legendre-) tegra1 means
in-
Ahf, f E X, h E (-1,1). They are defined as the convo-
lution integral with kernel (l+x)(l-h) -1 log (l - x) (1 + h) log (4.10 )
K
2 (i+h)
-l
(xjh):= {
o
otherwise,
the Legendre coefficients of which are given by
k
1
o
1 - Pk(h) k (k + 1110g (2/ (1 + h) )
Note that
k E IN.
BUTZER, STENS and WEHRENS
98
1 - [K('jh»)"(k)
(4.12)
k (k
1 - h
Since
+ 1)
(k E IN).
8
K(x;h)'::' 0, xE[-l,l) , hE (-1,1) and
limh.... l_[K(·/h),.,(l) =1,
the Ahf define an approximation process on X for
Fo~ th~ integ~at mean~
THEOREM 5.
~f
f
*
K(';h), f
L
E
(h
IIAhf- fllx .::.Mwl(h/f;X)
a)
h .... 1 - .
x,one E
ha~:
(-1,1».
(h .... 1-; 0 < a < 1) •
The
c)
PROOF:
integ~a.t mea.n~
a.~e ~atu.Jta.ted
in X wUh o!tdeJt
o(l-h) , h
Part a) follows by (4.12), Cor. 1 and (2.12), yielding
.... 1-.
also
the direct part of b). Concerning the inverse part in b) ,we make use of the Bernstein inequality (cf. [55, Sec. 3) (f E X; h E
and apply Prop. 5, setting
(-1,1»,
Xn (x) = K(x; 1 - lin). Finally part c) fol-
lows from Prop. 4 by (4.12).0
5.
THE INTEGRALS OF FEJf':R AND DE LA VALLf':E POUSSIN In analogy to the trigonometric theory, the Fejer kernel
with
respect to the Legendre system is defined by
(5.1)
n
k
En D ( ) + 1) (2k + 1) Pk (x) - _1_ n + 1 k=O k x
ex where
Dn
is the (Legendre-)Dirichlet kernel
E [-I,ll; n ElP),
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
n
~k=O
(5.2)
(2k+l)P k (x)
99
E (-l,lJ
(x
The associated algebraic Fejer convolution integral (recall (1.19),
:= f
*
F
n
f EX. The L
1
-
norms
Fn are uniformly bounded (see [32]), i.e. there exists 1< F
II F 111 < F, n E
such that
n
-
1P. Since
k
n+ 1)
with
n
is actually the n-th (C,l)-mean of the prutial
(1.20»
sums of the Fourier-Legendre series (l.18) of of the
(0 f)
n E JP) •
limn+oo F~(k)
(5.3)
*
Fn
< k
k
>
k E IN, one has by Prop.
1,
lim II f n+ oo
o
n
n,
1
o
- f II X
<
(f E X).
Next consider the de La Vallee poussin kernel
(5.4)
v
m,n
(x)
.= __1_ ~n •
m+1
In view of the identity
(5.5)
0
k=n-m k
(F_
l
(x)
(xE[-l,lJ; m, n E F, m,::.n).
0)
vm,n(x)
it follows that
II v
m,n
If m depends on (5.6)
III < (1 + 2 (n - m) ) F m + 1
n
(m,n E lP, m < n).
in such a way that sup nEJN
n men) + 1
<
00
BUTZER,STENS Bnd WEHRENS
100
then II v m,n III -< M, n E JP. In this case Prop. 1 again implies that the associated integral f forms an approximation process on X, v m,n noting that the Legendre coefficients of v are given by m,n
*
o
1
(5.7)
VA
m,n
(k)
(n
+ 1 - k) / (m + 1)
< k < n - m
n - m+ 1
o
<
k < n
k > n.
In contrast to the trigonometric
Fejer
kernel
and
that
(Legendre-) Fejer-Korovkin treated in Sec. 3, the (Legendre-)
of
Fejer
kernel fails to be positive (e.g. Fn(-l) = -1/2 i f n is odd; see also Using (5.5) this enables one to show that the (Legendre-) de
(26).
La Vallee Poussin kernel is not a positive kernel, at tain
lea~t
for cer-
Hence one cannot apply Cor. 1 in order to obtain direct
m,n E JP.
approximation theorems. We proceed in another fashion; in the peri odic case it is due to Steckin [53].
LEMMA 3,
f E X, n,m E JP, m < n
Fait all
(5.8)
IIf*vm,n -fll x < 2(n+l) m+ 1
(5.9)
II f
PROOF: for all
(5.10)
If
*
Fn -
f II X <
8F
n+l
:£n+l
F
th"-It"- hold!.,
En-m (f IX)
(
k=O Ek f; X) •
From (5.7) and the convolution theorem (1.14) it follows that Pn-m E
(p
Pn - m
n-m
p~(f) denotes
* a
v
m,n )
(x)
=p n-m (x)
(xE[-l,l], m,n E JP;
polynomial of best approximation in
m~n).
Pn to
f
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
in
X,
101
then (5.10) yields
II f * v m,n - f II X < II f * v m,n
p*n-m *
v m,n II X + II p* m-n * v m,n - f II X
< "vm,n III II f - p* m-n II X
<
+
" p* m-n - f II X
2(n+l)F m+l
The proof of (5.9) being practically the same as in [53),
we
will only sketch it. One has by (5.l),{5.4) for all k,nElN, 2k - l sn+l<2k
f -
f
*
Fn
k l
+ (n + 1 - 2 - ) (f - f
*
k-l
v
n-2
,n
)}.
One deduces by (5.8)
~
n+l
l=2
k-2
+1
El(fi X )}
8F < n+ 1
which is the desired inequality for
n>l. If
n=O it follows by (5.8)0.
Of course (5.8) gives a usable estimate only in the case
when
102
BUTZER, STENS and WEHRENS
1imn~00(n-m(n»
= +00. Choosing e.g. m
[Sn] for some
0 < S < 1, one
has in view of Prop. 6 and La. 3
THEOREM 6.
Le.t
* v m,n
f
, f e X, be .the a.lgebltaic. de La VaUee PouMin
c.onvolu.tion in.tegte.a.l wi.th in.tegte.a.l One
ha.~
6ote. eac.h
m = [Sn], 0 < S < 1,deMned via. (5.4).
rep
a) IIf*vm,n - fllx .:.Mn
c) The ..integte.a.t
f
*
V
-2r L
w (1-n
-2
1
..i-6 not
m,n
r
II f
*
In the case
~uc.h
f II X
v m,n -
(f E WXi n EIN).
X,..i. e. 6ote. any
~atute.ated..in
Mbi.tte.a.te.y 6unc.tion op a.6 ..in PltOp. 3 thelle (a.e.) (e.g. f any polynomial),
r
iO fiX)
exi~t.6
an
f
X, f tconst.
E
.that
a (IP (n) )
00).
(n -;.
n - m remains bounded, one can proceed simi 1ar1y as
for the Fejer means below, noting that
f
* Fn
f • v
n,n
(see
[34)
for the Gegenbauer case). Concerning the Fejer means
*
f
Fn
we have by (5.9), Prop.
6
and Prop. 4
THEOREM 7:
One
ha.~
60ll
f
*
Fn
If
E
X,
n -;. 00
I
o(n- 2 o.) a)
o(n -llog O(n- 1 )
, 0 < a < 1/2
n)
1/2
, a. > 1/2.
(0 <
CJ.
< 1/2) •
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
Note that Kallaev [35) not only showed that rated in
e (-1, 1] with order
saturation class, IIf
*
i. e.
O(n
-1
the class
103
*
f
is satu-
Fn
) but actually determined
of
functions
l
Fn- file = O(n- ). I t consists of all
f
E
f
for
e[-l, 1]
the which
for
which
G E LiPl (Ii C) , where
G (x)
:=
(
X
1-1
2
(~)1/2 W' (u)du, x- u
w(x)
:=
l
fx
In other words, he studied part b) in the case
f (u) (u - x) -
1/2
duo
u =1/2. For a point-
wise analogue of part a) see Rafal'son [45].
6.
THE ROGOSINSKI SUMMATION METHOD It is well-known that the convolution integral generated by the
Dirichlet kernel X unless
X
Dn of (5.2) fails to be an approximation process in
= LP(-l,l),
4/3 < P < 4 (see [44], [42). Instead of
Dn
we now consider the "shifted" Dirichlet kernel
(6.1)
:=(r
D )(x)
xn n
(n E
lP; X E [-1, 1 1 ) ,
where
{x} is a suitable sequence of reals in (-1,1). This n may by Christoffel's summation formula be rewritten in the
kernel closed
form
(n
+
1)
The construction (6.1) corresponds to the
trigonometric
Rogosinski
104
BUTZER, STENS and WEHRENS
kernel (see [47J and e.g. [9, p. 56, 106J )
n
: = 1 + 2 ~k=l cos k l;n cos k
which forms an approximation process if e.g. l;n = n/(2n). In view of Prop. 1 the
(n E JP;
l; E R)
l;n = n/(2n + 1)
or
I;;
xn in (6.1) have to be chosenin
such a way that
(6.2)
II Rn III < M
(6.3)
1
(n E JP)
(k E IN)
respect.
lim x n-+
oo
n
I,
for in this case
(6.4)
(k E IN).
1
The following lemma gives sufficient conditions for (6.2)
and
(6.3) to hold:
Le.t
LEMMA 4:
{xn}n
(6.5)
wheJte
I
-xn
E]p C
be .6ueh .tha.t
(-1,1)
xn - xn I
(n -+ (0),
i.6 .the laJtge.6.t zeJto 06
Pn(x), Xo
aJtbi.tJtaJt!/. Then
and (6.3) aJte valid.
PROOF:
(6.6)
By Bruns' inequality (recall (3.13»
I l-x n I
<
II-xn'I
+
Ixn
-x
I
nl
we have
(6.2)
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
for
n
-+-
00.
105
This implies (6.3). Applying Abel's transformation twice
we obtain
(6.7)
-
=
llP k (x n )
Here
-2 P k + l (X ) - P (x ) is the first and II P (x ) n k n k n
2P k + 1 (x n ) + P k (x ) n
the second difference of
{P
k
= Pk+2 (x n )-
(xn)}k E lP •
Making use of the well-known representation (see e.g. [57, p. 88])
127f2
Ixl
cos [(k + 1/2) arc cos u 1 du (u - x) 1/2 (1 _ u 2 ) 1/2
(x E
and the generalized mean value theorem, we obtain with
(-1,1)
A(U) E (0,1)
co s [(k + 5/2) arccos u ] - 2ros [ (k+ 3/2) arcros u ] + ros [ (k+l/2) arcros u] du (u - x )1/2(1 _ u 2 )1/2 n
e
12 1f
du.
n
Since
o
<
I~
1 ------~~~--~~~~du
n
(u - x
n
) 1/2 (1 _ u 2 ) 1/2
<
___I_-.;-r.:;- II (l
+ x ) 1/2 n
xn
1
1
(1 + x )1/2
du
0(1)
n
one can estimate the second difference using (6.6) by
(n -+- (0) ,
106
BUTZEFI,STENS and WEHRENS
(n .... 00) •
In a similar way one has for the first difference
< M arccos x
(n .... 00),
n
and using Markov's inequality (e.g. [36, p. 99J) for
IPn(x) -P n
n
-
< MI x - x In n n
(x)1 n
1
x
n -
A
xii P' (x + A (x - x » n n n n n
O(n- l / 2 )
2
(0,1),
E
1
(n .... 00) ,
Applying these three inequalities in (6.7) leads to
which yields (6.2) since
IIFklll
~
F,
kElP, and (see e.g. [32J)
lim n -1/2 II Dn III
n .... "" In view of Prop. 1, La. 4 and that (6.3) implies (6.4),
there
follows
Ix x I ' n-n
COROLLARY 3.
n .... 00, the.n one. ha.6 60ft e.ac.h
(6.8)
REMARK:
lim n-+ oo
II f
o(n -5/2) ,
f E X
*
Rn -
f
II X
°.
Note that the assumptions of La. 4
are
fulfilled
in
the
107
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
particular case
xn
Cor. 3; The kernel
(6.9)
xn • so that one may use
-xn
instead of
x ~ 'xn as
R (x;x ) may also be rewritten for n
in
xn
n
R (x;x ) n n
This form shows that the Fejer-Korovkin kernel actually the square of
R
n
(x;x )
K 2n 2
of (1.22)
is
apart from a normalization factor:
n
K2n - 2 (x)
The analogous result in the trigonometric case, namely
that
kernel is essentially the square of a particular Rogosinski
this
kernel,
was first noticed by Stark [52). Let us finally consider the rate of approximation in (6.8). By the convolution theorem one has for each
(6.10)
(TX
Ixn - -xn I
b) f E
-=
f
*
* Rn
n
Rn - fllx
->-00.
0 l1e h' . a" 1 OO'L
o(n- 2Ct )
f E X
~n ->- 00;
06 LegendlLe-Rogo-,>-<'n-,>Q-<.
-<.-'>
0 < Ct < 1) •
-'>atulLated-<'n
O(n- 2 ).
w-<.th olLdelL Let
O(n- 5 / 2 ) 60 'Lk
LiP~(Ct;X) "* IIf
c) The -<'nteglLal
PROOF:
(xE[-l,l]).
(x)
Let Rn(X;x n ) be de6-<'l1ed a-'> -<'11 (6.1) wheILe {xn}nEpCfl,l)
THEOREM 8.
X
P )
n n
Pn E P n
p~
E Pn
be
a
polynomial of best approximation to f e
x.
108
BUTZER.STENS and WEHRENS
Using Minkowski's inequality and (6.10), one obtains
II f
*
Rn - f II X < II f
*
Rn - p~
* Rn II X +
II p~
*
Rn - f II X
which is the first inequality of part a); the second follows by Prop. 6a), (6.6) and (2.12). Part b) follows by a) and Prop. 6b) or (3.14) and Prop. 5. Finally the saturation result can be derived from Prop. 4 since 1 - R~ (k)
lim
--.!!--
n-+ oo
1 -
xn
1 - P (x ) k n... lim --..:.:...-:.: n-+ co
1 - x
k (k + 1)
(k E IN). 0
2
n
Note that Kallaev [35] states without proof the saturation class of a particular Rogosinski-type integral.
7.
APPROXIMATION RATES EXPRESSED IN TERMS OF CLASSICAL
MODULI
OF
CONTINUITY The reader may object to the fact that the rates of convergence of the various convolution integrals given by parts b) and a) of COr. 1 and Thms. 2-8 are expressed in terms of moduli of continuity Lipschitz classes which are defined via
the
Legendre
or
difference
(Thf) (x) - f(x) and not the classical differences (~f)(X)=f(X+h)-f(X)' 2
(L',hf) (x)
=
f(x+h) +f(x-h) -2f(x). However, the "Legendre" Lipschitz
classes are actually equivalent to certain pointwise Lipsclritz classes
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
i
defined via
(,\f) (x), i =1,2, at least for
shown in [55, Sec. 6
X
C [-1,1
with respect to the differences Lipschi tz classes of order
Ql(11;Xjf;C)
Q2(11i
a >0
:= sup {
i
lihf(x), i =1,2, and the
as
was
for x E [-1,1] associated
are defined by
Ili~f(x)l;
2 jfjC) :=sup{llihf(x)
X
I ,
I.
The pointwise moduli of continuity of f E C [-1,1]
(7.1)
109
I;
hE [-11,111 n[-l-x, l-x]}
hE [-Il,Il]n[-l-x,l-x)n[-l+x,l+x]}
2 P-Lipi(o.;C) :={fEC[-l, 1]; rli(Il;X;f;C) =O«~)a), 11->-0+} 1- x It was indeed shown that
={ P-LiPl (a; C)
(0 < a < 1/2)
Lipi(o.;c)
(7.2)
P-Lip2(a;C)
(0
Using these results we have
The. 6oUowing tU-6eJLtioYlJ.> Me equivalent 60ft f E X and 0 < n < 1:
THEOREM 9:
(il
f E Lipi(n;c) ,
(ii)
f E P- Lip 2(n;C) ,
(iii)
En(f;C}
(iv)
IIf
*
K
(v)
II f
*
A2n - f II C = 0 (n -0./2)
(vi)
IIf
*
wt -
(vii)
IIAhf - file
n
= 0(n- 2a ) -
fll
C
(n ->- 00) ,
= O(n- 2n )
f IIc = 0 (to.)
= 0«1
- h)o.)
(n
-+
00) ,
(n ->- 00) , (t
-+
0+) ,
(h
-+
1-) ,
110
BUTZEFI, STENS and WEHRENS
(viii)
IIf
*
v
[ Snl ,n
-
co i
(n ....
0 < S < 1),
(ix)
(n-+
I Xn -
-X
nI
= 0 (n
-5/2 )
co ) ,
•
Fl.uLtheJtmolte, i6 0 < a < 1/2, the a.Me.!t:tioYl4 (1) - (1x) a.Jte a.iM eqll.[va.-
leYlt to
(n .... <:>0).
If one is only interested in direct ?pproximation theorems, it is also
possibl~
to connect our results on rates of convergence with
the classical moduli of continuity and Lipschitz classes,
even
in
LP - spaces. They are given by
(7.3)
LiPi (al Xl
where
={f
E
X; wi (n I f;X)
O(nCl),n-+o+}
(i=l,2/u>O),
II. "X [a,b] indicates the X-norm taken over the interval [a,b].
By a well known result on best approximation (cf. [24, p. l45)l,rnmely
o(n- r - a )
(7.4)
(n +oo;r EP,a :> 0)
I
we find in view of Prop. 6b)
~
(7.5)
f E Lipl(a;X)
f E LiPt(a/2;X)
(Q < a. .:::. 1)
(7.6)
f E Lip2(aIX) ~ f E Lipt(a/2/X)
(0 < a <: 2).
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
111
The latter three implications will now be used to derive the results. In the case of the integrals of de La Vallee
direct
Poussin
and
Fejer-Korovkin one obtains by (5.8), and Thm. 2b) together with
the
inequality
(fl EX;
(7.7)
1')
>0),
respectively, the following
COROLLARY 4:
one.
ha~
a)
60lt any
Folt ;the.
~ingu,ealt
int:e.gJta£. f
* v[ Bnl , n'
f E X, 0 < B < 1,
r E lP
o(n- r - Cl ) f
*
Kn ,
f E
(n'" "'; 0 < Cl
x, one.
~ 2) •
h4~
(n'" "'; 0 < a. < 1)
(n ... 00; 0 <
(n'" "';
To extend the last implication to the assertion to the case
COROLLARY 5:
Folt
X
=C
*
K
n -
fll C
we have
to
< 2)
0 < a. < 1) •
restrict
[-1,1 I .
fl E LiPl(l;C)
IIf
a =1
Cl
;the.lte.
ho,ed~
(n ... (0) •
112
PROOF:
(7.9)
BUTZEA,STENS and WEHAENS
We need the following implications
L
WI (1 - aifiC) + 0(1 - 8)
fEP-LiP2(1;C)'" wl(a;f;C)
(7.8) follows by (7.7) and the definition of can be found in [ 55, Sec. 6] • I f (7.8) and (7.9) that sertion by Thm. 2b)
L wI (a;f;C)
f'
0(1 - 8) , 8
(7.9)
LiP2' P-LiP2;
LiP1(1;C), then
E
(8 + 1-).
->-
1-,
one
has
by
yielding the as-
.0
Cor. 4b) and Cor. 5 solve the problem posed in the introduction as well as its extension not only in LP(-l,l), 1 ~ p <
00,
C [-1,1] - space
apart form the case
but also
f' E LiPl (liLP). The a l -
gebraic de La Vallee Poussin sums have a much
better
approximation
behaviour. According to Cor 4a) they actually approximate f E X
for
with the same order as do the algebraic polynomials
any given of
best
approximation. Let us recall that the integrals considered in this paper convolution integrals of the form (2.1)
are
(the convolution being under-
stood in the Legendre sense) and not of the form (1.4). These integrals may, however, readily be rewritten in the form
~ Jl-1 f(u)X p (x;u)du
(7.10)
with
II-1
(p E IA)
X (xiu)du=2,x E [-I, I]. For example, the Rogosinski.intep
gral can be wri tten as (7.10) wi·th
(x,u E [-1,1]; n E P).
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
113
Another example of a singular integral written in the form (7.10) was the integral
J
2n
f
of (1.2).
Lupa~'
result (1.3) for this inte-
gral can also be derived from Cor. 1. Indeed, since
n
n 2
+ 3n
(n E P)
I
+ 3n + 3
f E X
one has that for each
3
(7.11)
--:::--....;;;...-- i fiX)
n
2
(n E
JP).
+ 3n + 3
C [-1,1 I, an easy calculation (cf. [55, Sec. 6 I) shows that
X
If
2
(fEC[-l,l]; -1<0<1)
which yields
(f E C [-1,1]; n E ]N).
This is (1.3) apart from a constant. Using our methods it can be shown that
o(n-l-a) rated in holds for
J
2n
provided
f
approximates
2n
in
C [-1, 1] - space with order
f' E LiPl (aiC), 0 < a < 1. Moreover,
X with order J
f
J 2nf
f.
exist a triangular matrix of distinct nodes -
< + 1 k In-
{.,ok,n (x) }~=O
is satu-
0 (n- 2 ). A result of the type of Thm. 2b) also
Let us finally conclude with an unsolved problem.
-1 < x
even
n
and a triangular matrix E
n
{xk,n }k=O '
of
Does n
there E JP
I
fundamental functions
JP, defined on [-1, +1], such that the linear sum-
n mator operators (L f) (x) = L -0 f (x. ).,ok (x), f kn K,n ,n positive algebraic polynomials of degree
nand
E
C [-1, 1),
are
114
BUTZER, STENS and WEHRENS
provided
f
E
Lip2(a;C [-1,1]),0
The point is that the
should be defined as constructive-
ly as the corresponding functions in the case of the Bernstein polynomials defined over
[-1, 1], namely
and these are known to approximate only with order
o(n- a / 2 )
under
the same hypothesis.
REFERENCES
[1]
H. BAVINCK, Jacobi series and approximation. Doctoral Dissertation' Univ. of Amsterdam, 1972.
[2]
H. BAVINCK, Approximation processes for Fourier-Jacobi expansions. Applicable Anal. 5 (1976), 293 - 312.
[3]
H. BERENS, In.te.lLpo.fa.tion.6me.,thode.n ZUIL Be.hand.fung von
ApplLOuma-
tion.6plLoze..6.6e.n au6 BanachlLaume.n. Lecture Notes in Math. 64, Springer Verlag, Ber1in-Heide1berg-New York, 1968. [4]
S. BOCHNER, Posi ti ve zonal functions on spheres. Proc. Nat. Acad. Sci. U.S.A. 40(1954),1141-1147.
[5]
R. BOJANIC, A note on the degree of approximation to continwus functions. Enseignement Math.
(2) 15 (1969), 43 - 51.
[6]
R. BOJANIC and R.DEVORE, A proof of Jackson's theorem. Amer. Math. Soc. 75 (1969), 364 - 367.
[7]
P. L. BUTZER, Unsolved problem. In:
01'1
ApplLoxlma.tlon
Bull.
The.oILY
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
115
(Proc. Conf., Oberwolfach, 1963; Eds. P. L. Butzer J. Korevaar) ISNM Vol. 5,
and
Birkhauser Verlag, Basel-Stuttgart ,
1964, p. 180. [ 8)
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(Proc. Con£.,
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R. DEVORE, On Jackson I s theorem. J. Approximation Theory 1 (1968) , 314 - 318.
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R. DEVORE, An unsolved problem, In: Con6e~ence
on
P~oceeding~
Con~t~uctive Theo~y
06
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the
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Amer.
App~oximation
Theory and Functional Analysis J.B. P~olla (ed.) © No!'th-HoUand Publishing Compa:ny, 1979
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
Jost PAULO Q. CARNEIRO Insti tuto
de Matematica
Universidade Federal do Rio de Janeiro Rio de Janeiro, RJ, Brazil
1. INTRODUCTION Let
F be a non-archimedean, non-trivially valued field, which
is locally compact for its natural topology, Hausdorff space, E over F, and
r
seminorms on and [3
X
a zero - dimensional
a non-archimedean locally convex Hausdorff space
the directed family of all non-archimedean continuous E.
(For the definition of these terms, see [1
I,
[2
I
I .) C(X,E) denotes the vector space, over F, of all continuous func-
tions from
X to
E,
and
An L-weight is a function uppersemicontinuous f E L. If
C(X,E) •
L is a fixed vector subspace of w from
X to
F
such that
p
0
(wf)
pEr
and null at infinity, for every
space. In this case,
sex)
= {sex)
A > 0
E E: s E S}, for each X if for every
WE W
convex
we
set
x E X, and we say that S is
11011-
such that
X
x E X there exists
F O. W is called dilte.r-te.d if, given and
weighted
(L,T ) is called a 11011-altr-hime.de.al1 Nar-hbil1 -6par-e.. W
S is a collection of functions from
val1i-6hil1g on sex)
and
W is a set of L-weights, then the non-archimedean seminorms
f E L ... sup [w(x)f(x»), for p E r , w E W, endow L with a xEX topology TW' under which L is a non-archimedean locally
If
is
E,
s E S
w ,w 2 E W, l
IWil ~ Alwl, for
121
to
such that there exists
i =1,2. The following
CARNEIRO
122
facts are clear:
w
(i)
If
(ii)
If W is directed, then the sets {f E L; for
is non-vanishing on
pEr, w E W,
£
>
X,
then (L, TW) is Hausdorff. ~
p [w(x)f(x)] < d,
0, form a basis of a-neighborhoods
for (L,T ). W (iii) If W is non-vanishing at X, then, for each x E X, ox: L defined by
->
E,
0x(f) = f(x), is continuous and linear.
2. EXAMPLES (i)
If
L = C(X,E) and W is the set of F-characteristic func-
tions of compact subsets of open topology in (ii) If
X then
TW
is the
compact-
C(X,E).
L = Co(X,E), the vector space of all continuous func-
tions from then
TW
X to
E which vanish at infinity, and W = {l},
is the uniform topology in
In particular, if
Co(E,X).
X is compact, we get Co(X,E) =C(X,E).
(iii) If
X is locally compact, L = C (X,E), the vector space b of all bounded continuous functions from X to E, and
W = Co(X,F), then
TW
is called the strict topology
in
Cb(X,E) . We notice that in all these cases
W is non-vanishing
at
X,
and that, in the first two cases, W is directed.
3. THE BOUNDED CASE OF THE NON-ARCHIMEDEAN BERNSTEIN-NACHBIN PROBLEM Given a subalgebra A of
C(X;F) and a Nachbin space
the non-archimedean Bernstein-Nachbin problem consists in describing the closure of a vector subspace that is, such that For ~(x)
=
which is a module over
A,
AM C M.
x, y E X,
~(y),
MeL
for every
we say that ~
x:: y
E A. We denote by
(mod A) X/A
if and
only
the quotient of
if X
NON-AACHIMEDEAN WEIGHTED APPROXIMATION
by this equivalence relation. Each
wi y
of X, one easily sees that
Y EX/ A
is a set of
123
being a closed
L! y - weights, and then
(L I y , TW Iy) is a non-archimedean Nachbin space. M
loeal-tzable undelt A in (L,T W ) if the
is then said to be
TW - closure of
M in
cides with (or, equivalently, contains) the set
£A(M)
that
in
fly The
belongs to the lte~~It-te~ed
T
closure of
Wly
Mly
Lly}.
non-archimedean Bernstein-Nachbin problem
ea~e
con-
holds.
of the non-archimedean Bernstein-Nachbin prob-
lem occurs when the functions in every function in
L coin-
{f E L such
sists in asking for conditions under which localizability The bounded
subset
A are bounded on the
support
of
W.
We have then the following:
THEOREM 1:
In
bounded
~he
loeal-tzable undelt A -tn
PROOF:
ea~e,
evelty
A-module
eon~a-tned
-tn
L
-t~
(L,T ). W
The proof is an adaptation of the one given by Nachbin ([4])
in the real case, and is based on the following
LEMMA:
Le~
A be a
w-t~h ~he ~upltemum
~e~
and
00
un-t~alty e.e.o~ed ~ubalgeblta
noltm. 16, 60lt evelty
X wh-teh -t~ d.t~jo-tn~ 6ltom
~l""'~n E A
(i)
~iIKZ.
06
Cb(X,F)
Y EX/A, Ky
.t~
y, ~hen ~helte ex-t~~
a
equ-tpped eompae~~ub
Zl'."'Zn EX/A
~ueh ~ha~:
i = 1, ... ,n
= 0
~
(ii)
II~ ill
OQ
sup I~· (x) ~ XEX
I
n (iii)
~
~i (x)
= 1,
x E
~ 1,
i
*
1, ••• ,n
x.
i=l PROOF OF THE LEMMA: X
and
S: Cb(X,F)
Let ~
BX
be the Banaschewski compactificationof
C(SX,F) the isometry such that
S(f) = Sf is the
124
CARNEIRO
only continuous extension of of A under
f
SX (see [5 I ). Then
to
S, is a unitary closed subalgebra of
the quotient space
SX/SA
and
n : eX
G
+
SA, the image
C (SX,F). If
G is
the canonical projection,
then G is compact, Hausdorff, and zero-dimensional (see I6 1). Now, for
y n X ,
Y E G, we have that
this case, Y n X E X/A. Then
¢ if and only if
Ky n X
is a compact subset of X, which y E ~ (X)
Y n X. This implies that
is disjoint from
the finite intersection property, there exist
n r:2
that for
n (K y n X) = ¢. Putting i
i-I
= 1, ...
i
(i)
hiln(K
(11)
IIh i II n 2:
00
zi
hl, ••• ,h
o
)
n X
= y.
~
~
1f
(K y n X) =
Yl""'Yn E n(X) and
=
i
~ 1
~
G. By [6),
E C(G,F) such that:
l, ... ,n
1, ... ,n
i
t
hi (t) = 1
n
such
G \n (K . ) , Z
is an open covering of
Lemma 2.1, we can find
(iii)
z.
,n, then
Y E n(X) and,in
G.
E
i=l Letting
~i
element of
= hi
0
~i
n E C(SX,F), then each
is constant
each
in
SX/BA, so that, by the non-archimedean Stone-Weierstrass
Theorem for compact Hausdorff zero-dimensional spaces (see [6 I), each belongs to
SA. Putting
.p
= ~.
~
i
IX '
we get the desired result.
Now we go back to the proof of the Theorem. If algebra of
B is the
C(X,F) generated by A and the constant function 1, then
it is clear that M is an A-module if and only if it is a and that
M is localizable under
calizable under
B in
L.
Suppose first that in and
A in
Thus, we can assume A
C
C (X,F)
b
II E r
such that, for
be given. For each
x E Y
and
B-module,
L if and only if it
is
10-
A to be unitary.
and let A be the closure of A
Cb(X,F). Clearly, X/A = X/ii.. Let then p = II
sub-
fE.cA(M), £>0, wl, ..• ,WmE W
Y E X/A, there exists
j =l, ... ,m,
gy E M
Iwj(x) IlIf(x) - gy(x) II < e.
m U {x E X; Iw. (x) I IIf(x) - gy(x)1i > e} is comj=l ) pact and disjoint from y. By the Lemma, we can find Zl""'Zn EX/A
The set
Ky
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
and gi
,
. ,
"
= gZ· ,
A
E
such that
we get, for
x
E
I Zi
0,
X, i=1, ••• ,ri ,
j
125
1I
Letting
= 1, ... ,m:
~
1
Iw.(x)1
~
]
~
~
as one can see by distinguishing
the
cases
x E K Z.
and
~
It follows that, for
x E X, j
= l"",m,
n
n
~
max
< E
E A, for each i,
i
•
i
such that
k = 1 + max sup IWj (xl I 119 i (x) II. Then i, j xEX n :E [a.(x) -
< ...f. max Iw.(x) I lig. (x)1i
k
]
i,j
~
< e:,
so that
IWj (x)
I
n L ai(x)gi(x) i=l
- f(x)II
n IWj (x) I
As
aig
in
L.
i
E
n
L [ai(x) -
AM
C
M,
this proves that
f
Finally, in the general case when
belongs to the closure of
A
C
C(X,F)
in
X,
,wm
and get
E
W, P
E
r,
we take Y =
Aly C Cb(Y,F). Replacing
m U
j=l
for
and
E > 0, wI""
f(x)11 <e:.
f
E
M
each [A(M) ,
supp w , which is closed j
X, W, L, A, M,
'w
by
126
CARNEIRO
y, Wly' Lly' Aly, Mly, TWl sult, obtaining that in
L Iy.
respectively, we apply the previous re-
y
£Aly(Mly)
is contained in the closure of
Since it is easily seen that
gly E Mly
fly E £AI
,y
p[wj(x) (f(x) - g(x»] <
such that
£,
Mly
(Ml y )' we can find
for x EY, j=l, ..• ,m.
w. (x) = 0 if x E X \ y, this holds for every x E X, which J proves that f belongs to the Tw-closure of M in L. So M is lo-
But since
calizable under
COROLLARY: (i)
In
A in
A L6
(L,T ). W
~epat!.a;U_ng
f E L, then
G.iven
f(x)
eveJty (ii) M X E
.t~
f
beiong~
belong~
to the
to the
eio~uJt.e
c.a~e,
06 M .in
Tw-c.lo~ulLe
06
.in
M(x)
then:
E,
x E x. den~e.tn
L, .in
M(x)
.t~
den~e.tn
L(x),
60Jt eveJty
x.
(i) Since
PROOF:
and we aJte .in the bounded
tons. Given then
A is separating, the elements of > 0, pEr, wI' ... 'W
£
by hypothesis, gEM
which proves that
E W, x E X,
p ( f (x) - 9 (x)]
such that
k = 1 + mjx Iwj(x)l. Then
m
X/A are single-
p[wj(x)(f(x) - g(x»]
<
<
there exists, £
£,
k'
where
for
j=l, •.. ,m,
f E £A(M). The result follows by Theorem 1.
(ii) Follows from (i).
REMARK:
It W is non-vanishing on
X,
then the converses of (il and
(ii) in the Corollary hold, even without the hypotheses on
A.
4. THE SCALAR CASE In the scalar case, E = F absolute value. If the subspace subalgebra A of
is (non-archimedean) normed by L of
C (X,F) contained in
then, as particular case of Theorem 1:
C(X,F) is fixed, then L is an A-module.
We
the every have
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
127
In the bounded c.a.6e, evvl.lj .6ubalgeb!!.a A 06 C(X,F}
THEOREM 2:
c.on-
tained in L i.6 loc.a.lizab.e.e u.nde!!. it.6e.e.6 in (L,T ) ' W
Let A be a .6ubalgeb!ta
THEOREM 3 (Stone-Weierstrass):
c.ontained in the .6ub.6pac.e
L 06
Then, in o!!.de!!. tha.t a given ~n
C(X,F)
Let
t
w-c..e.o.6u!!.e
06
A
L, it i.6 .6u66ic.ient that both (i)
9 (x)
0, 6o!!. eveltlj
(ii)
g(x)
g(y)
9 E A, imp.e.ie.6
6o!!. eveltlj
I
LA (A). Given
9 E A, imp.e.ie.6
Y EX/A,
that there exists constant in
(ii)
implies that
Y,
g E A, such that A
h
then
E
= g(x ) 9
will do}. Then, given
£
>
O.
f(x)
f
E
= f(y). a!!.e
al.6o
belongs to
such that
9 (x)
= 0,
A,
the
9
E
A
'I O.
0,
Since
implies
(i)
9
Y.
is
also
hl y = fly. (If 1.=0, h=O
I
w, Iwj(x}
E
'I
in
Ih(x) -f(x) I ==0<£,
W is non-vanishing on X, and
tw-closure of
for every
is constant
LA(A).
As to the necessity, assume that f
f
Y, if
E
and
A
wl, .. .,wm
y, j == 1, ... , m. So that
X E
Xo
g(x ) o
o
that
0, and
then the.6e conditioY!.6
I
A be this constant value. Fixed
for
=
f(x)
For the sufficiency, it is enough, by Theorem 2, to show that
PROOF: E
C(X,F)
a.nd a.6.6ume the bou.nded c.a.-6e.
be.e.ong.6 to the
f E L
I6 W i.6 non-vani.6hing on X
f
I
06
A in
A, take
Ox
L.
Then if x
as in
§ 1,
E
and
X is
we
get:
{O} ,
f (x)
since
E is Hausdorff, so that
for every f(x)
f (x)
=
O. Similarly, if
g (x) shows
9 E A, the same reasoning with
= 9 (y)
,
that
== f(y).
COROLLARY:
(a)
I6 A i.6 a .6uba.lgeblta 06
C (X,F)
and
f
E
C(X,F),
128
CARNEIRO
the.n f belo rlg.6 -to the. compact-ope.n clO.6ulte. 06 A .[6 and only .[6 (i)
g(x)
(ii) g(x) (b)
g (y) ,
Co(X,F) and
long.6 to the. un.[60ltm clO.6uJte. 06 (i)
0, 60Jt eve.Jty
g(x)
(ii) g(x)
X i..1.> locaU.y compact,
(c) 16 f E
cb (X,F),
then
f (x)
fly) •
f E Co (X,F) , the.n
=
fIx)
b e.-
g E A, .[mpi.[e..6
0
f (x)
A i...6 a .6ubalge.bJta 06
f (y) .
and
C (X,F) b
be.iongl.> to the. I.>tJt.[ct ciol.>uJte 06
f
f
.[6
A .[6 and oniy
g E A, .[mpUel.>
g(y) , 6 OJ!. e.ve.lty
=a
fIx)
g E A, .[mpi.[e.-6
60Jt e.ve.Jty
A .[.6 a -6ubaige.bJta 06
16
g E A, .[mpUe.-6
0, 601t e.ve.Jr.tj
.[6 and
A
a nl!f .[ 6 g(x)
(i)
(ii) g(x)
0,
60Jt e.ve.Jttj
g(y),
g E A, .[mpli..e.1.>
g E A, .[mpl.[e.1.>
60Jt e.ue.Jty
=
f(x)
O.
= f(y)
f(x)
•
5. DENSITY IN TENSOR PRODUCTS If E, then
Sand Tare, respectively, vector subspaces of C (X, F) and S 0 T
the form
x
->
denotes the set of all finite sums of functions sex) t, with
s
E
S, t
E
T. Similarly, if
are zero-dimensional Hausdorff spaces, and tive1y, vector subspaces of denotes the set of all finite
THEOREM 4:
C(X ,F) and 1 sums
of
since
A 0 E
C(X ,F), 2 the
A 0 E
is an A-module, and (A 0 E) (x)
A is non-vanishing at
Corollary.
and
S2
functions
X.
i...6
= E,
and
are,
X 2
respec-
then
16 A i..1.> l.>epaJtat.[ng and non-uani..l.>hi..ng on X,
and i..6 we. Me. i..n the. bounde.d cal.> e., the.n
PROOF:
Sl
Xl
of
of
the form
A 0 EeL,
Tw-de.Me. i..n
for every
L.
x E X,
It suffices then to apply Theorem 1,
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
129
COROLLARY 1: (i)
C ex, F)
® E
i.6 del'!.6 e in
C (X, F),
60ft the compact
-
open
.topology. (E)
16
X i.6 locally compac.t, K (X,F) ® E
60ft .the u.ni60ftm .topology.
(K(X,F)
i.6 den.6e in Co(X,E),
i.6 .the .6e.t 06 aU
con-
tinu.ou..6 .6calaft 6u.nc.tion.6 wi.th compac.t .6UPPOft.t) . (iii) 16
X i.6 locally compac.t, Cb(X,F) ® E )..6 den.6e in
S,(X,E),
60ft .the .6.tftict topology.
COROLLARY 2 (Dieudonne): (i)
(C(Xl,F) ® C(X 2 ,F»
® E
i.6 den.6e
~n
C(X
x X ,E),
l
2
.the compact-open topology. (ii) C(Xl,F) ® C(X 2 ,F)
i.6 den.6e in
C(X
X2 ) ® F.
x
l
6. EXTENSION THEOREMS
THEOREM 5:
r6
E i.6 a non-aftchimedean Fne.che.t .6pace oven
F, and
Y
i.6 a non-emp.ty compac.t .6u.b.6e.t 06 .the zeno-dimen.6ional Hau..6doft66 .6pace X,
then evefty
E -valued continu.ou..6
a bou.nded continu.ou..6 6u.nc.tion on
PROOF:
6unc.tion on
can be extende.d .to
Y
X.
We wi 11 employ a technique due to De La Fuente [ 7 I
linear mapping
Ty: C(X,E) ~ C(Y,E), defined by
Ty(f)
=
fly
S C C(X,E), denote
uni tary subalgebra of
Ty(S)
by
C (y ,F), and
Sly. Then
A = Cb(X,F) Iy
M = C (X, E) I y b
Since the constant functions belong
to
By Theorem 1, Corollary,
is dense in
Assume first that
Cb(X,E) Iy
M, M(x)
X is compact. Then C (X, E)
space, and so is its quotient by the closed subspace
is an
is spaces.
clearly continuous for the compact-open topologies in both For
The
is
a
A - module.
E, for each x E y. C(Y,E). is
a
Frechet Now
130
CARNEIRO
we claim that C(X,E) Iy,
C(X,E)/K
is linearly and topologically isomorphic to
for which it is enough to prove that
homomorphism. Indeed, given
U,
is a topological
a basic neighborhood of 0 in
then
U
{g E C(X,E); p[g(x)]
<
Then
V
{h E C(Y,E); p[h(x)]
< E:; x E y}
of
in
C(Y,E). Since it is evident that
0
Ty
x E X}
E:;
for some
pEr,
is an open Ty(U)
C
C(X,E), E:
> O.
neighborhood
V n [C(X,E) Iy ],
it is enough to prove the reverse inclusion. Let then with
g E C(X,E). Then
joint from
Y.
that
0 on
is
.p
G = {t EX;
By ultra-normality of
is such that
G, 1
fEU
on Y, and
and
Ty(f)
Therefore, C(X,E) Iy = Cb(X,E) Iy
X,
given
g = hlx
THEOREM 6:
16
is complete, and
x, then
tion in
PROOF:
thus
h E Ty(U) .
closed
in
Cb(X,E) Iy = C(Y,E). SFX
the Banaschewski compact-
h E C(SFX,E)
such that
Then,
f =hl y • '!he
is the required extension.
E -il.l a non-altc.himedean Fltec.het I.lpac.e ovelt
il.l a c.!ol.led I.lubl.let I.lpac.e
on X. Then f=.pg E C(X,E)
By the previous result, C(Y,E) = C(SFX,E) I y .
f E C(Y,E), there exists
function
.p EC(X,F) sum
h, which proves that
Now, in the general case, take X.
is compact and dis-
there exists
l.p I 2. 1
C(Y,E). Since it is also dense, we get
ification of
> d
p[g(t)]
06
F, and
Y
the zelto-dimenl.liona! !oc.a!!y c.ompac.t Haul.ldolt66
evelty 6unc.tion in
Co (Y ,E)
c.an be extended to a 6unc.-
Co(X,E).
We omit the proof, which is similar to that of Theorem 5.
REFERENCES
[1]
A. F. MONNA, Ana!Yl.le non-altc.himedienne, Ergebnisse cier Mathematik und ihre Grenzgebiete, Band 56, Springer-Verlag, Berlin, 1970.
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
[21
L. NARICI, E. BECKENSTEIN and G. BACHMAN,
and
Vatuat~on
Theo~y,
131
Fun~t~onat
Anaty~~~
Pure and Applied Mathematics,vol.
5, Marcel Dekker, Inc., New York, 1971. [31
J. P. Q. CARNEIRO, Ap~ox~ma~ao Ponde~ada nao-a~qu~med~ana,{Doc toral Dissertation), Universidade Federal do Rio de Janeiro, 1976; An. Acad. Bras. Ci. 50 (1978), 1 - 34.
[41
L. NACHBIN,
We~ghted App~ox~mat~on
Cont~nuou~
Fun~t~OM:
6M
Reat and
Atgeblta~
and
Set6-Adjo~nt
Modute~
06
CMe~,
Comptex
Annals of Math. 81 (1965), 289 - 302. [51
G. BACHMAN, E. BECKENSTEIN, L. NARICI and S. WARNER,
Rings of
continuous functions with values in a topological field, Trans. Amer. Math. Soc. 204(1975), 91-112. [6
1
J. B. PROLLA, Nonarchimedean function spaces. To appear L~nealt
Spa~e~
App~ox~mat~on
and
in:
(Proc. Conf. ,ObeTh'Olfach,
1977: Eds. P. L. Butzer and B. SZ. - Nagy), ISNM
vol.
40, Birkhauser Verlag, Basel-Stuttgart, 1978.
[71
A. DE LA FUENTE, Atguno~ Ite~uttado~ ~oblte apltox~ma~~on de 6un~~one~
ve~tolt~ate~
t~po
teoltema
We~elt~t~a~~-Stone,
Doc-
toral Dissertation, Madrid, 1973. Etement~
L. NACHBIN,
[91
J. B. PROLLA,
06
Appltox~mat~on
Theolty, D. Van Nostrand Co. Inc., 1967. Reprinted by R. Krieger Co. Inc., 1976.
[81
Appltox~mat~on
06
Ve~tolt
Vatued
Fun~t~oM,
Holland Publishing Co., Amsterdam, 1977.
North-
This Page Intentionally Left Blank
Approximation Theory and Functional Analysis J.B. ProUa (ed.) ©North-Holland Publishing Company, 1979
TH~ORIE
SPECTRALE EN UNE INFINIT~ DE VARIABLES
JEAN-PIERRE FERRIER Institut de Mathematiques Pures Universite de Nancy 1 54037 Nancy Cedex, France
1. L'utilite d'une theorie spectrale et d'un calcul fonctionnel holomorphe en une infinite de variables a ete mise en lumiere par la recherche de conditions d'unicite pour Ie calcul fonctionnel holomorphe d'un nombre fini de variables et des algebres (cf (21).
Disons, de faQOJ1 schematique,
a
spectres non compacts
que l'unicite est
etablie
pour undomaine spectral pseudoconvexe et en particulier polynomialement convexe et que, d'autre part, tout domaine de
~n peut s'inter-
preter comme laprojection d'un domaine polynomialement convexe, mais d'un nombre infini de variables. fa~on
De
a element
classique, etant donnee une algebre
unite (toutes les algebres seront supposees desormais telles),
on se donne des elements ora) de de
([:n
A, commutative et
a
=
a , ... ,an l
de
A et on definit
(al, ... ,a n ) comme l'ensemble des points
tels que l' ideal engendre par
a
l
s
Ie spectre
=
- sl' ... , an - sn
(sl' ... ,sn) soi t pro -
pre, plus precisement comme Ie filtre des complementairesdes parties S, dites spectrales,
sur lesquelles on peut trouver
des
fonctions
2. Pour decrire une situation semblable en dimension infinie, il est nature 1 de remplacer
en
par un espace localement convexe
donnee de
par celIe d'une application lineaire bornee
133
E
et la
a
134
FERRIER
du dual
E'
de
E dans
A.
a
La notion de spectre correspond alors systeme fini
=
(a (
a
semble
de
a
de
u
i
exprimant que
un
dire uneap-
IC , on peut considerer
cr (a ",,),
On s'interesse
plus precisement un en-
a
des familIes
(8
est fixe (ou majore),et les
verifiant des conditions d'uniformite: n fonctions
a
pour
n
E dans
et son spectre
spectral pour
8
E', c'est
(
plication lineaire continue
ce qui suit:
S
independamment
"". Ainsi les ensembles spectraux sont-ils remplaces par des
fa-
~l(S
milles d'ouverts tif.
Avec les notations qui precedent Ie calcul fonctionnel classique est un morphisme
a croissance
lomorphes c'est ou
Os
f
a dire
f[a
-+
soit bornee pour un certain entier
est la distance dans
.A
O(os) des fonctionsho-
polynomiale sur Ie domaine spectral S dans A,
telles que
L' algebre
1 de l'algebre
a;N
au complementaire de
N,
S.
qui intervient en dimension infinie a des elements
de la forme
f
(I)
correspondant f
If'
une famille spectrale (S
oil
L
I 'X
<
00,
ou
E 0(08 ) et verifie dans cette algebre des majorations independan-
tes de
(2)
a
Le calcul fonctionnel s'obtient en posant
f [a J
L'X
f
[a
If'
J.
Plus precisement on Ie definit d'abord pour des sommes telles que (I) n'ayant qU'un nombre fini de termes, et on Ie prolonge au complete, l'algebre
ve ayant
par
passage
ete definie elle-meme de cette ~on.
TH~ORIE SPECTRALE EN UNE INFINIT~ DE VARIABLES
135
3. Un probleme, clef pour l'unicite du cal cuI fonctionnel en un non-
vi came
bre fini de variables, se pose: peut-on considerer l'algebre ~
une algebre de fonctions sur un domaine si
~
de
E? De fa90n
est la partie de la limite projective des
inf Os I{)
.A
on a un morphisme de
eVidente
definie par
(I{)(s}) > 0, I{)
algebre."l.~
sur une
~, dent
de fonctions sur
l'injectivite n'est malheureusement pas claire. 5'il n'y
a
pas
de
probleme dans Ie cas d'un produit, la situation n'est pas dans Ie cas d'un produit fibre sur un domaine de nier est pseudoconvexe (cf [1 1,
[2
~n, sauf si ceder-
1)•
4. De1aissant ici 1e probleme de savoir si 1es fonctions holomorphes sur
du cal cuI fonctionnel sont des fonctions, concentrons-nous
spectre et cherchons si on peut remplacer dans certains cas Ie
nI{)
teme projectif des
par un domaine
a
pouvoir connaitre des familIes (SI{)
n
des parties
de
SI{)
de
~n telles que
SI{)
E. Pour
sys-
il faut
cela
partir de la seu1e donnee de
II est naturel de considerer, pour continues d' applications lineaires
de
1e
n.
n donne, les familIes ~ n E dans IC et 1es familIes contient l'image par
n,
ce qui se traduit plus exactement par 1e fait que
inf U'5 orp}Cs) > 0 10 10
(3)
II faut noter
a
AS.p
s E t2 •
ce sujet que la derniere condition
general impossible pour de
pour tout
A
,
avec
A parcourant
rend
10, 11, Ie
en choix
o
L'ouvert
n
sera spectral si pour tout choix (Stp) con forme
ce qui precede on a
5
avec uniformite par rapport
a
10.
136
FERRIER
Un cas particulierement simple est celui d'une suite bornee (a) n
de.A
et d'une suite bornee (Sn) telle que
par rapport
a
a
n; l'espace
une application a
partie
~
de
du produi t des
Sn
E estl'espace
cr(an) avec uniform:i.te
E
et (a)
.e.oo(ll:)
n
s'identifie
A. Peut-on alors affirmer que la
II (G:!) dans
Sn' def inie par
inf Os (5) > 0, est specn
n
trale pour a ? If faudrai t pour cela que pour un element de E', c'est ait
I{!(n)
a dire
E cr(al{!)'
une suite (X ) de
(4)
~ X
n
a n
de la sphere unite
I Xn I
= 1
on
dire
S
n
cr (~X a ) , n n n
E
et avec uniformite par rapport a (X ). n En effet, s'il existe E > 0 tel que contient la boule ouverte boule ouverte
~
II «[:) telle que
n
c'est
I{!
I{!(~n
B(Zn,E) et
E , alors
=
°
B (~ Xn Zn' E) de sorte que
I{! (n) ••
contient (~X n Z n ) -> E.
5. On peut done se poser de fa90n generale Ie probleme suivant:etant donnee une suite bornee (an) de que
AN et une suite
(Sn) de
telle
G:!N
Sn E cr(an) avec uniformite par rapport a n , est-ce que l'on
la relation (4) pour toute suite (X ) de n avec uniformite par rapport
a
II (G:!) telle que
~IX
n
a
I = 1,
(X n ) ?
Considerons Ie cas particulier d'une algebre
de
Banach.
On
verifie tout d'abord, en prenant des caracteres, l'inclusionsuivante, dans laquelle
sp(a n )
Sn est remplace par l'ensemble
tersection du filtre
(qui est l'in-
cr(an»
~
n
X sp(an):J n
sp(~
n
Xna n ).
Cette meme inclusion montre done que pour tout choix de Sn E cr(ad' on a la relation (4). Cependant il resterait
a etablir
l'uniformite
TH~ORIE SPECTRALE EN UNE INFINITIO DE VARIABLES
137
par rapport au choix d'une suite (An) de la sphere unite de II n' y a pas de difficul te si on remplace la borne sur les coefficients avec
e: >
u
i
a
la distance
par Ie fait que
5 contienne un E-voisinage du
fixe. En effet si
a
A
AE
designe 1 'ensemble des points dent
est strictement inferieure
2:A
n
(sp(a»E
n
On est ainsi conduit
a
spectre
a
E on a
(2: A sp (a » (; .
n
n
etudier la croissance des
coefficients
spectraux en fonction de la distance au spectre. Dans un sens on a l'inegalite:
qui s'etablit facilement en prenant que
IX (u i ) I 2.
t
x(a) E sp(a) et en
sachant
II ui II •
La question fondamentale concerne l' autre sens: peut - on tout
E > 0
trouver une borne des coefficients u i (s) avec qui soi t independante de a, II a II < 1 ?
pour
d(s,sp(a»~E
BIBLIOGRAPHIE;
[1
1
J .-P. FERRIER, Theorie spectrale et approximation par des fone-
tions d'une infinite de variables, Coll. An. Harm. Complexe, La Garde - Freinet 1977. [2 1
K. NI5HIZAWA, A propos de l' unic! te du calcul fonctionnel holomorphe des b-algebres, these, Universite de Nancy, 1977.
[3
1
L. WAELBROEK, Etude spectrale des algebres completes, Acad. Roy. Belg. Cl. 5ci. Mem., 1960.
This Page Intentionally Left Blank
Approximation Theory and FUnctional Analysis J.B. Prolla (ed.) @North-Holland Publishil1{J Company, 1979
MEROl-10RPHIC UNIFORM APPROXIMATION ON CLOSED SUD SETS
OF OPEN RIEMANN SURFACES
P. M. GAUTHIER* Departement de Mathematiques et de Statistique Universite de Montreal, Canada Dedicated in memory
of Alice Roth
1. INTRODUCTION Let
F be a
face R. Denote by
(relatively) closed subset of an open Riemann surH(F) and
M(F) respectively
the
holomorpl1ic
and
meromorphic functions on (a neighbourhood of) F. Let A(F) denote the functions continuous on
F and holomorphic on the interior
F
O
of F.
Recently, the problem of approximating functions in A(F) uniformlyby functions in H (R) has been considered by Scheinberg [17 I . In the present paper, we consider the problem of approximating a given function on
F uniformly by functions in H(R) and obtain, as
a
corollary,
a
result related to Scheinberg's. Our method of approximation is based on the technique of the late Alice Roth [15J. We shall rely on Scheinberg [17 I for some results
on the
to-
pology of surfaces. Without loss of generality, we shall assume that every Riemann surface its closure in of
R if
R is connected. A subset is bounded in
R is compact. A Riemann surface
R'
is an
R
if
ex~en~ion
R is (conformally equivalent to) an open subset of
R'. If
* Research supported by N. R. C. of Canada and Ministere de l' !;ducation du Quebec. 139
140
GAUTHIER
furthermore
R
'I R', R' is an e.6.6ent..i.al
that a closed subset a
exten.6ion of R. We shall say
R is e.6.6 entiatty 06 6inLte 9 enu.6 if F has
F of
covering by a family of :pairwise disjoint open sets, each
nite genus. Denote by morphic on its on
the uniform limits on F of functions r:rero-
M(F)
R with poles outside of
F and by
F of functions holomorphic on
compactification of
of fi-
if (F)
the uniform lim-
R. R* will denote the one point
R.
The central problem in the qualitative theory of approximation is that of approximating a given function on a given set. In thisdirection we state our principal theorem.
(Loc.atiza.tion):
THEOREM 1:
Let F be c.to.6ed and eMentiaUy 06
nite genu.6 in an open Riemann .6ufL6ac.e M(F)
R.
Then, a 6unction
f
i.6
6iin
i6 and onty i6
f I K n F
(1)
60fL evefLy
c.ompac.t .6et K in
E M(K
n F) ,
R.
If we drop the condition that
F be essentially of fini te genus,
then the theorem is no longer true [9 ). dition, for
However, we may drop the con-
R planar, since it is trivially verified by all
F.
In
this situation, Theorem 1 is due to Alice Roth [15). An immediate consequence of Theorem I is the following
Walsh-
type theorem, which was first obtained for planar R by Nersesian [141.
THEOREM 2:
Let F be c.to.6e.d and eMentiaUy 06 6inite genu.6
open Riemann .6ufL6ac.e
R.
A
.6u66ic.ient c.ol1dition 60fL
that
A(F n
V)
in
an
A(F) = M(F)
i.6
141
MEROMORPHIC APPROXIMATION ON CLOSED SlJBSETS OF RIEMANN SlJRFACES
60ft
eveJr.1j bounded open -6et
V -iI'!
R.
By the Bishop-Kodama Localization Theorem [12], the open sets
we may replace
V by parametric discs.
The following is a Runge-type theorem.
THEOREM 3:
Let
F
be c.[o-6ed and e-6-6entiaU.y 06 6inite genu-6
open Riemann -6uJr.6ace
R. Then
H (F)
C
M(F). MOJr.eov eJr. , H (F)
C
in
an
R (F) i6
and onllji6 R*\ F.[.o connected and .f.oc.a.f..f.y connected.
Recently, we proved Theorem 3 for more restricted pairs (F, R) [ 7] •
From Theorem 2, we have a corollary on Walsh-type approximation by holomorphic functions.
THEOREM A:
(ScheinbeJr.g [17]):
Let F
6inite genu-6 in a open Riemann -6uJr.6ace
A(F)
R(F) i-6 that
R* \ F
be c.f..o-6ed and
R.
e-6-6entia.f...f.1j
oS
A -6u66icient conditioI'! naiL
be connec.ted and .f.oca.f..f.1j connected.
Scheinberg actually obtained this result for somewhat nnre general pairs (F,R). For arbitrary pairs (F,R), the condition that R*\F be connected and locally connected is also necessary but
no
longer
sufficient [9]. In fact, Scheinberg has shown that there is no topological characterization of pairs (F,R) for which A(F)
PROOF OF THEOREM A:
Since
= R(F)
[17].
R*\ F
is connected, it follows from the
Bishop-Mergelyan Theorem [2 ] that
F satisfies the hypotheses of The-
orem 2, when the sets f
> 0,
there is a
V are parametric discs. Thus, if
gl E M(R) with
if(z) - gl(z)1
< E/2,
Now by Theorem 3, there is a g E H(R)
z E F.
such that
f E A(F) and
GAUTHIER
142
This completes the proof of the corollary. A closed set F in
R is called a set of Carleman
tion by meromorphic functions, if for each ti ve and continuous on
there is a g E
F,
I fez)
-
< £(z),
g(z)1
f E A(F) and each M (R)
THEOREr14:
£
posi-
with
Z E F •
The next result characterizes such sets completely when result is known for
a~7~)roxima-
F
O
{tf.
This
R planar [14] .
Let F be c.io.6ed w-Lth empty -Lntelt..i.o/t .i.n an open IUemaf'lf'l
6ac.e R. Then F -L.6 a .6et 06 CaILieman appILox.i.mat.i.on
by
6u.Jt-
meILomOlLph1c.
6unc.t.i.on.6 .i.6 and only 16
C(F n K)
601i. eac.h c.ompac.t
.0
et
M(F n K),
K.
2. FUSION LEMr1A Using Behnke-Stein techniques, Gunning and Narasimhan [11] have shown that every open Riemann surface R can be visualized in a very concrete way. Indeed,they showed that fication) above the finite plane
~.
R can be spread (without ramiTo be precise, they proved that
R admits a locally injective holomorphic function
p. Thus
is the spread. We wish to reconstruct the Cauchy kernel of Behnke-Stein on R, something resembling (q - p)
-1
. Conceptually
we prefer to think of p
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
and
q as both lying on
R, however, for proofs, it may be prefera -
ble to think of two copies
z and
the
Rp
p :
R
p
x R
We construct an
r ( .,
_
z) - 1
Rq of
Set on
~
x
z
R spread respectively above
~
;:;
..-+- (z , ;:;) •
cover of
o~en
Dq be discs about
~.
~
q
(p , q)
p
and
;:; planes:
p x
°overand
143
p and
q
R x R. If
(p,q) E R x R,
respectively which lie
U(p,q) = Dp x Dq • Consider the Cousin data U ( p,q. ) S ~nce .
R x R
schlicht
which
is
is Stein, the first Cousin prob-
lem can be solved. Hence there is a meromor"l?hic function whose singularities are on
let
on R x R
the diagonal. In the neighbourhood of
a
diagonal point, we have, in local coordinates (forever more given by p
x
p), that
1
t(l;;,z) -
I; -
is holomorphic. <1>(1; , z) means
z
(p,q), where
pip) = I; and p(q) = z.
We shall persist in this abusive notation, since it is invariant under local change of charts within the atlas given by the function
a Cauchy kernel on
p x p. We call
R since
We shall now extend to surfaces the powerful Fusion
Lemma
of
Alice Roth [15] .
FUSION LEMMA: mann .6uJz.6ac.e
Let
K , K , and 2 l
R, w.i.th
Kl
a.nd
K2
K be c.ompac.t .6ub.6e.t.6 06 an open R-i.ed.i..6jo.i.nt. TheILe '£.6 a. p06.i.tive numbe!t
GAUTHIER
144
a .6uch tha.t.£6 .6a.t.£.6 nljil1g,
m l
a.l1d
m 2
Me. a.111j two me.ftomOftph.£c 6ul1ct.£011.6
011
R
E > 0,
60ft .6ome.
Im l
(1)
- m LK < 2
m, me.ftomoftphic
the.11 the.fte. i.6 a. 6uI1ctiol1
1m -
(2)
E,
R .6uch tha.t 60ft j = 1,2,
011
mj I K UK. < aE J
PROOF:
We may assume
bourhoods and
and U2 of Kl and K2 respectively such that l is precompact. Moreover, we may assume that the
U
R\ U 2
aries of
K2 \ K 'I ¢. Thus, we can construct open neigh-
U
l
curves. Let
and E
U
be the compliment of
U
l
U
U
in
2
(R \ U ) U K2 U K.
~
is uniformly bounded for
z
E
G, where
is
1 on
U
l
and ¢
is
(3) is uniformly bounded, there is a constant
I~
(4 )
¢ (l,;)
G
be a pre-
+ in
is a Cauchy kernel for
We introduce now an auxiliary function ¢
R. Let
then
2
I (z)
in [0,1] such that
bound-
consist of finitely many disjoint smooth Jordan
2
compact neighbourhood of
( 3)
i\nu2 =¢
II
¢
E Cl(R)
0 on a > 2
U
2
R.
with values . Then, since
such that
dn < a - 2 ,
at;
for
z E G.
ml and m2 , we put q =I1J. -~. By (1) we can find a precompact neighbourhood U of K such that Returning now to our meromorphic
Iq (z) I
< E,
Z E
U.
follows. First, set
We replace
q by a function
q 1 constructed
as
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
(5)
q
o
Now set
145
on
elsewhere. Thus,
I
(6)
I ql (Z)
I' < E ,
z E E.
Set
g (Z)
( 7)
From (6) and (4) we have
I g (Z) I
(8)
Since
< (a - 2) • E
g is a Cauchy integral,
Z E
G •
g is holomorphic outside of E.
Consequently,
is holomorphic in
z
E
0(z)ql (z) + g(z),
f (z)
(9)
U2
=
(for
00
z E R,
set
0)
•
For
Ul '
ql (z) + g(z)
fez)
is meromorphic and has the same poles as holomorphic on
U, we invoke the formula
0(z)
- .1. 1T
Hence
ql' To see that
r _Cl_
Jf E
J a~
4> (z, Z;;)
d~
dn
,
z
E
R •
f
is also
GAUTHIER
146
f(z)
For
z E U , ql
Z E
and
q
is holomorphic. Thus morphic on
U
U lJ
R ,
f
is holomorphic in
U U
l 2 Behnke-Stein Theorem [1 I
U
I
and hence
with the same poles as
q.
By
f
is mero-
the Runge-
there is a meromorphic function
m3
on
R
for which
Finally we put
m = m + m3 2
and we have the following estimates:
I
Kl U K
<
I(2l
< £
1m - m2 : <
+
-
+
11
Iq I
(a -
2)£
+ £
If I +
1m3 - f
i m3
I
- f
IgI
+
i
+ 1m3 - f
I
<
a£
<
(2l:
Iq I + Ig I
< £ + (a - 2) £ + £
This completes the proof of the fusion lemma.
a£ •
+
on
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
147
In the fusion lemma, it is clear that if
A(K. U K)
j = 1,2,
]
m. E A(K. J J
we may take
U
K),
= 1,2.
j
Another consequence of the fusion lemma is the follOding BishopKodama
Localization Lemma.
LOCALIZATION LEMMA Md
K
(Kodama (12):
be given
0
n a c.ompac.t -Ilub-
06 an open Riemann -IlUl!·6ac.e, and -Ilupp0-lle that 60Jt eac.h
theJte exi.6t.6 a c.,eo-lled paJtametJtic. di-llc.
Then
f
Let
D
z
with c.enteJt
z
E
K
Z .6uc.h that
f E M(K).
3. PROOFS OF THEOREMS Consider first Theorem 1. The necessity is trivial. To prove the sufficiency, suppose first that which the and that
RI RI
-
closure
F
is open. Let
of {G
R has an essential extension
R'
F is compact. We may assume that n
} be an exhaustion of
R
by
in
R~ F
domains
with
R.
and
In the Fusion Lemma, let
K , K, and l
K2
be the sets
F \ Gn + l , and consider these as compact subsets of the Riemann surface R I . For each ber
n = 1,2,3, •.• , the Fusion Lemma gives us a :9osi ti ve nunr
an' and we may assume that
GAUTHIER
148
If
£
is a given positive number, we select the positive numbers
00
E
and
(10)
n=l By the hypotheses there exist functions
fq n (z)
(11)
-
f(z)1
£n
< ...f... 2
qn E M(R)
such that
£n
<--
2a n
and therefore
Iqn+l (z)
(12)
qn (z)
-
I
Z E F
<
n
n=1,2,3, •••
By the Behnke-Stein Theorem [1], we may assume that by the Fusion Lemma, for each rn
E
M(R') such that, for
n = 1,2.3,. •• , there exists a function
n =1,2,3, ... ,
(13)
<£
n
(14)
The inequalities (13) yield
E : r \) (z) n
Therefore
-
qn E M(R').Thus,
q'J (z)
I
<
E n
Z E G
n
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
q 1 (z) + 1:
m( z )
(r v (z)
-
149
qv ( z) )
1
is meromorphic in
u
R
G
n=l From (11),
1m (z)
From (11),
-
z
(13) and (10), there follows for
f (z) 1 < 1CJ
(13),
n
l
(z)
-
f (z) 1 + [
1r
1
v
(z)
-
q
E Fl '
v
(z)!
<
(14) and (10), we also have n-l
Im(z)
-
£(z) 1 <
l: 1
+ 1:
jr
v (z) - q v +l(z)1 + iq n (z) - f(z)1 +
I r v ( z)
n-l -
qv (z)
<
n
1: 1
n = 2, 3, • . . .
Thus on
f
can be approximated uniformly on
F by functions meromorphic
R, and the proof is complete for the special case that
essential extension in which
R has an
F is bounded.
To prove Theorem 1 in general, we shall invoke the special case just proved to inductively construct a sequence of meromorphic functions which converqes to an approximating function. Suppose
R is an arbitrary open Riemann surface and
closed subset. of
R for which
F has a covering
F
is
a
by pairwise
150
GAUTHIER
disjoint open sets, each of finite genus. We may assume that each V. ]
meets
F, from which it follows that the family
nite. For each
j
we triangulate V.
I
]
{V j }
is locally fi-
and set
where T represents an arbitrary 2-dimensional closed triangle of the triangulation, and
Fj
F n Vj • We call
=
{P.} a !.)olygonal cover and ]
polygonal. It is clear that the segments which make up j are locally finite. Re!.)eating the same argument, we can find, for each
P
j ,
a polygon
Q
j
with
P. C Q. C Q]. C V. •
]
]
]
We may construct an exhaustion
{G } j
of
R by polygonal bound-
ed domains in such a way that
¢
We may also assume that each
k > j
is transversal to each
and k to each aQk' That is, dG j n aP k and dG j n aQk are isolated sets. By a res ul t of Scheinberg [17, Theorem 3.2 1 , each of the Rie dG j
dP
mann surfaces
admits a compact essential extension. Thus, by the special
case
of
Theorem 1, there is a function
If
(z)
-
m (z)
l
There exists a function holomorphic on
G1
U Pl'
Set
I <
PI
E
£
2"
M(R)
such that
ml - PI
is
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
ml - PI
on
G U PI ' l
f - PI
on
F2
{
f2
151
By the special case of Theorem 1, there is a function g2EM(G3UQIUQ2) such that
I f2 (z)
- g2 (z)
I
~,
<
Z
2
E Gl U PI U F 2 •
Set
Im2 (z)
Set
-
f (z)
I
<
E
"2
+
_E_
22
fl == f. Then, we may proceed inductively to construct a sequence
satisfying for
j ==2,3, •..
,
1m. (z) - fez} J
I
j L
<
Z E
n=l
j U
n==l
Fn
and
1m. (z)
J
It is clear that
-
m.
J-
l(Z) I <
Z E
converges to a function
Im(z) - fez)
I
<
E
Z E F •
G.] - 1
m E M(R)
and
152
GAUTHIER
This completes the proof of Theorem 1. Theorem 3 was proved in [7 1 for the special case that R has an essential extension in which F is bounded. Theorem 3 has two
parts,
one on meromorphic approximation and one on holomorphic approxmation. The meromorphic approximation follows from the special case in exactly the same way as the general form of Theorem 1 followed from the special case of Theorem 1. The proof of the holomorphic part of Theorem
3
also
follows
from the holomorphic special case, but we must define the sets Pjl and G
Q j
more carefully so that
j
P.J- 1
U F.
]
is connected and locally connected. First of all the exhaustion
{G } j
is connected, for each
structed in such a way that For each R* \ P j ,
R* \
j , let
K.
J
can be (and usually is) con-
be the set
of
bounded
j.
components
of
Q. , J
and
These are finite in number. Connect each such component to the ideal boundary of R by a simple path which misses F. We may replace this path by a connected polygonal neighbourhood with the same property. Clearly we may assume that the family of all such path neighbourhoods over all
j
is locally finite
and transversal to everything we have
153
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
constructed. Let
P be the union of all these path neighbourhoods. Set
G~ ]
G. \ J
Then by construction,
P
p~
J
p. \ J
R* \
P~
R* \
J
P
Q. \ J
P
OJ ,
R*\ (G~ up' u ... uP~) J I J
and
are all connected. It is easy to see that these sets are also locally connected since the boundaries are locally finite and polygonal. , Q~ have the required J J This completes the proof of Theorem 3.
follows that the covers
G~, p~
J
FO
We now prove Theorem 4. Suppose then, that
C(F (, K)
for each compact set tinuous function on
K.
M(F
Let
n
f E C(F)
properties.
\<1, and that
,
and let
£
be a positive con-
F.
Let {G } be an exhaustion of n
By hypothesis, there is a
I fez)
K)
-
gl (z)
gl E M(R)
I
<
£2 '7
R
It
by polygonal domains. Set
such that
164
Set
GAUTHIER
9 0 = 91 ' Go = ¢, and suppose,
9 ,9 , •.. ,9n -l l 2
to
have been found in
obtain
M(R)
an induction,
with the following
that three
properties:
(15)
19. (z)
-
fez)
J
1< ~
+
u£j+l 2 j +l
2J
Z E F II
(G. \ G.
J-
J
1) ,
z E G.J- 1
(16)
(17)
'
19. (z) J
-
Let us construct continuously to
fez)
1<
£j+l 2j +l
gn' First set
Gn-l
G)
u (F ()
n
f
.
on G Now e;{tend f n-l n = 9n- l n in such a way that f =f on F () ClG n n
and
Since, by assumption,
and since
G is a Lyapunov domain, it follows from Lemma 3 in n
and from the Bishop-Kodama Localization Theorem [12 I that
Hence there is a function
h
n
E M(R)
such that
[10 I
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
Set
gn-l + h n • Then, for
n (G n \ Gn- 1)'
Z E F
/gn(z) -f(z)/ < /g (z) - f (z)/ + :f (z) -f(z)/ < n n n
and so
g
155
£n+l + 2
n satisfies (15). It is easily verified that
g
£n
""2
n also satis-
fies (16) and (17). Thus, we have constructed inductively a sequence gn E M(R) having the properties (15), From (16), we see that
(16), and (17).
gn converges to a function
From (15) and (16), it follows that if
F n (G
Z E
n
g
E
M(R).
\ G 1)' then for n-
m > n,
all
m / f (z) - g
m
(z) I < If (z) - g (z) I + n
+
I:
j=n+l m I:
Ig j (z) -gj-l (z) I <
<
j=n+l
As
m tends to
00,
£
n
we have
I f (z) - g (z) I < £n < £ (z) ,
This completes the proof. If
F
O
= ()
Scheinberg [17]
and
R*\ F
is connected
has shown that
by functions holomorphic on
R
and
locally
connected,
F is a set of Carleman approximation (see also [8 ]). This is implicit in the
proof of Theorem 4. Indeed, we may construct an exhaustion compatible with
F, that is, such that
R* \ (F U
Gj
)
is connected and locally connected for each
j . Thus
we may
choose
156
GAUTHIER
the functions
gn from
H (R) .
4. OPEN PROBLEMS
a) If
R is planar and
f
is given on
Theorem 3 that approximation of on
f
F or by functions meromorphic
F, then it follows fr,om by functions holomorphic on
equivalent. However, the example in [91
R
are
essentially
shows that for
~
closed sets in some Riemann surfaces, there are functions in H(F) which cannot be approximated by functions from The problem of approximation by function in
H(F)
M(R) •
becomes,
then, a separate question which has not been treated on arbitrary open Riemann surfaces. b) If
R is planar, the condition in Theorem 2 is not only suf-
ficient but also necessary (14). It would be of interest to know whether it is also necessary on an open c) There remains the problem
of
Ri~
surface.
considering arbitrary
pairs
(F,R). Only Theorem 4 is complete in this respect. Scheinberg
(17) has shown that there is no topological characterization of pairs (F,R) for which
A(F)
=
H(F). This is not
at
all
obvious, but it is easy to see that there is also no to po logical characterization of pairs
(F ,R) for which A(F) =M(F)
(see [6 ) ) • d) Scheinberg [17] has solved the problem of Carleman approximation by holomorphic functions for the case that (see also ( 8 ) .
In the case where
R is planar,
F
O
=~
necessary
and sufficient conditions are known (necessity [5] , sufficiency [13]) for Carleman approximation, even when What about Riemann surfaces?
FO ~
¢.
MEROMORPHIC I'.PPROXIMI'.TION ON CLOSED SUBSETS OF RIEMI'.NN SURFI'.CES
157
e) There is also the question of uniform approximation on unbounded sets in several complex variables. This is practi cally virgin territory. See, however, [4 I
and [16].
REFERENCES
[11
H. BEHNKE and K. STEIN, Entwecklung Analytischer Funktionenauf Riemannschen Flachen, Math. Ann. 120 (1949), 430 - 461.
[ 2I
E. BISHOP, Subalgebras of Functions on a Riemann Surface,
Pa-
cific J. Math. 8(1958),29-50. [31
S. BOCHNER, Fortsetzung Riemannscher Flachen, Math.
Ann.
98
(1928), 406 - 421. [41
J. E. FORNAESS and E. L. STOUT, Spreading Polydiscs on Complex Manifolds, Amer. J. Math. (to appear).
[51
P. M. GAUTHIER, Tangential Approximation by Entire
Functions
and Functions Holomorphic in a Disc, Izv. Akad. Nauk.
Arm.
SSR 4(1969),319-326. [ 6I
P. M. GAUTHIER, On the Possibility of Rational Approximation, in Pade and Rat~onal App~ox~mat~on, 1977, Academic Press, New York, 261 - 264.
[71
P. M. GAUTHIER, Analytic Approximation on Closed Subsets of Open Riemann Surfaces, P~oe. Con6. on Con~tnuet~ve Funet~on The-
any, Blagoevgrad, Sofia (in print). [81
P. M. GAUTHIER and W. HENGARTNER, Approximation sur les fermes par des fonctions analytiques sur une surface
de
Riemann,
Comptes Rendus de l' Acad. Bulgare des Scienaes(Doklady Bulgar. Akad. Nauk) 26(1973), 731. [ 9I
P. M. GAUTHIER and W. HENGARTNER, Uniform Approximation on Closed Sets by Functions Analytic on a Riemann
Surface,Appna~~on
Theony(Z.Ciesielski and J.Musielak, eds.), Reidel, Hblland, 1975, 63-70.
158
GAUTHIER
[10]
P. M. GAUTHIER and W. HENGARTNER, Complex Approximation andSimultaneous Interpolation on Closed Sets, Can. J. Math. 29 (1977), 701 - 706.
[11]
R. C. GUNNING and R. NARASIMJIAN, Immersion of Open Riemann Surfaces, Math. Ann. 174 (1967), 103 -108.
[12]
L. K. KODAMA, Boundary Measures of Analytic Differentials and Uniform Approximation on a Riemann Surface, Pacific J.Math. lS (196S), 1261 - 1277.
[13]
A. H. NERSESIAN, On the Carleman Sets (Russian), Izv. Akad.NaUk Arm. SSR 6 (1971), 46S - 471.
[14 J
A. H. NERSESIAN, On the Uniform and Tangential Approximation by Meromorphic Functions (Russian), Izv. Akad. Nauk Arm.SSR 7 (1972), 40S - 412.
[lsI
ALICE ROTH, Uniform and Tangential Approximations by Meromorphic Functions on Closed Sets, Can. J. Math.28(1976), 104-111.
[16 I
S. SCHEINBERG, Uniform Approximation by Entire Functions, d'Analyse Math. 29(1976), 16-19.
J.
[171
S. SCHEINBERG, Uniform Approximation by Functions Analytic a Riemann Surface, Ann. Math. (to appear).
on
Appro~imation
Theory and FUnctional Analysis J.B. Prolla (ed.) ©North-Holland PuhlishiYI{J COTTTpany, 1979
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFIlHTE DIMENSIONS
CLAUDIA
s.
GUERREIRO(*)
Instituto de Matematica Universidade Federal do Rio de Janeiro Rio
O.
de Janeiro, Brazil
I;~TRODUCTION
In 1948 H. Whitney [13), based on a conjecture of
L. Schwartz,
proved that, given a non-empty open subset U C IRn, the closure, respect to the compact-open topology of order m, of an ideal
I
with C
8,m(U)
is determined by its set of local ideals. The original proof was simplified in 1966 by B. Malgrange [5).
The main concern of this paper
is to extend Whitney's theorem to open subsets of infinite
dimen -
sional spaces. In finite dimensions there are two equivalent formu lations of this theorem:
THEOREM 1:
16
I C &m(U)
i;., art ideal, thert
I
I, wheJte
n {I + I(a,k); a E U, kEN, k < m}
artd
O,O
I(a,k)
(*) This research was partially supported by
FINEP (Brasil)
through
a grant to the Instituto de Matematica - Universidade Federal do Rio de Janeiro. 159
160
GUERREIRO
~~
THEOREM 2:
"I
an
n {I + I (a,k,£); a
~deaf,
E
V, k E
m i {f E S, (V); II 0 f (a) II <
I (a,k,£)
-I
~hen
£,
V
I,
m, k
0 <
whelle
and
< m, £ > O}
< k}.
i
In infinite dimensions, Whitney's theorem is false in formulation 1, even in the case
V
H, a real separable Hilbert space, and
m=l. We present an example of this in section 2. In formulation
2
it is true, with respect to the usual compact-open topology, for the case
m=l
with some restrictions. The case
m > 2
remains an open
problem and our guess is that the theorem is false in this context. Two other directions arise naturally in infinite dimensions:the first one is to consider subspaces of dimensions, with the whole space new topology in
S,m(V)
S,m(U) which coincide, in finite
S,m(V); the second is to look for a
which coincides, in finite dimensions,
with
the usual one. In section 2 we consider the concept of differentiability which gives us a unified way to deal simultaneously subspaces of
~,
several
with
S,m(V).
In [12] Restrepo studied the closure of the algebra of nomials of finite
~
poly-
in a Banach space of a certain kind, for the to-
pology of the uniform convergence of the function and its derivative on bounded subsets. In [1]
Aron and Prolla extended this result
to
a more general class of Banach spaces, considering the case m ~ 2 and polynomial algebras of vector functions weakly uniformly
continuous
on bounded subsets. In section 3 we study ideals of functions weakly uniformly continuous on bounded sets, with respect to the topology of the uniform convergence of order m on bounded sets. In section 4, we consider the topology in [101 •
T
C
introduced by Prolla
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
161
Finally, in section 5, we use the results of section 4
to es-
tablish some facts about modules. The results of this paper are taken from the author's Doctoral Dissertation at the Universidade Federal do Rio de Janeiro,
written
under the guidance of Professor J. B. Prolla.
1. PRELIMINARIES In the sequel {O,l,2, ... }, elements of Let cal duals E' ® F
stands
:IN
for
the
set of
m stands for an element of
:IN U
natural
{co}
and
integers i, j,k
for
:IN.
E t- 0 E'
and
and F I
be real normed linear spaces with topologi-
F
respectively,
the linear subspace of
U
[(E;F)
C
E
a non-empty open subset,
spanned by the
applications
rp ® v : x E E + rp (xl v E F, rp EE', V E F.
For
X a real Hausdorff locally convex space, a function
is called b-di66elten.tiabie i f there is unique)
such that, for
x
Df(x)y
E
lim
U, "
U -+ [(E; X)
f(x + AY) - f(x) A
y on each bounded subset of
In the same way, we define c-di66elten.tiabiii.ty b
by
c
(necessarily
JR,
E
A+O
uniformly with respect to
Df:
f: U-+X
and bounded by compact. We observe that if
space, b-differentiabili ty is Frechet
E
E. by
replacing
is
a
normed
differentiability and c - dif-
ferentiability is Hadamard differentiability (Nashed (91). Let gy
Tb
[b(E;X) denote the space
[(E;X) endowed with the topolo-
of uniform convergence on bounded subsets of
denote the space
l(E;X) endowed with the topology
convergence on compact subsets of [ b ( 0 E IF)
E.
By induction
E and T
C
we
of may
[c (E; X) uniform define
= F and, for k .:: 1, [ b ( k E I F) = [ b (E;[ b ( k-l E IF)). In the same
162
GUERREIRO
way, replacing
b
by
c,
we have
.cc(kEiF). Furthermore, let
denote the vector space of all continuous functions from endowed with the compact-open topology The space
U
C(U;X) to
X,
TO
/i,bm(U;F) and its topology
T
bm
will
be
defined
inductively as follows: For if
m=O,
/l,bo(U;F)
C(U;F) ,
T
bo
TO
and we denote Oaf = f,
f E C(U;F). For
m = 1, define
/I,
bl
(U;F) as the vector space ofal! f
which are b-differentiable and such that bl T
pology
E
C(U;F)
Df E C(U;.cb(E;F)). The to-
is defined as the topology for which the isomorphism
f E /l,bl(U;F)
~ (f,Df)
E C(U;F)
x C(U;.cb(E;F))
is a homeomorphism. For uniformity of notation, olf = Of. Suppose we had already defined
/l,b(k-l} (UiF), Tb(k-l}
ok-I: /l,b(k-l) (U;F) ~ C(U;.cb(k-l EiF )), for some Define such that
k > 2.
/l,bk(UiF) as the vector space of all
ok-If
Ok: /l,bk(U;F) ~ C(U;.cb(k EiF )) by
po logy
bk T
f
E
&b(k-l) (U;F)
D(Ok-l f ) E C(U;.cb(k EiF }).
is b-differentiable and
Oefine
and
Dkf = O(Ok-l f )
and the to-
as being the only one for which the isomorphism
f E /l,bk(U,"F) ~ (Oi f )
k
o
E
n
C(U;.cb(i E1F })
i=O
is a homeomorphism.
n /i,bk(U;F) and consider as kElN the topology for which the isomorphism Finally, define
f Eli.
is a
boo
(UiF)
homeomorphism.
/i,boo(U;F} =
~
i
(0 f)~ EN E
...
n
i E IN
b i C(U;.c (EiF)}
T
boo
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
For the case The space
F = JR,
we will write
e.cm(U;F) and its topology
163
[&bm(U;F) = [&bm(U), cm T
ti vely in the same way, by jus t replacing b by
is defined c
induc-
in the above defi-
nition. There is a natural identification between £(k EiF ) and £(kE;F), the vector space of continuous k-linear maps from Ek there is a homeomorphism between and
£b(k EiF ) (respectively
with the topology
Tb
£b(k E1F )
(respectively
T
F.
(respectively
£c(k EiF )), the space C
to
£c(k EiF ))
£(kE;F)
endowed
).
k
On the other hand, the natural isomorphism between
£s( E;F),
the vector space of continuous symmetric k-linear maps from F, and
k
P (EiF), the space of continuous k-homogeneous
from E in to
F,
In fact,
Ek
to
polynomials
is, actually, a homeomorphism, if we endow both spaces
with the topology
Tb
Moreover, given we may associate
C
or both with the topology f
belonging to
T
•
[&bm(UiF)or [&ern(UiF) , xE U, k~m,
Dkf(x) with an element
dkf(x) of
£s(kE;F)
which
may be identified with a polynomial akf (x) of pk (ElF) • bm In that case, the T topology may be defined in by the family of seminorms of the form
PK,k(f)
K C U
sup {lIaif(x)lIi x E K, 0 < i < k},
a compact subset, k < m. The topology
Tern
may be defined in
[&cm(UiF) by the
family
of seminorms:
PK,L,k (f)
K C U, LeE
sup {lIdif(x)yll;
compact subsets,
k < m.
For details, see Nachbin [8 I.
(x,y) E K
x
L, 0 < i < k},
164
GUERREIRO
2. IDEALS AND DIFFERENTIABILITY TYPES The concept of holomorphy type for complex functions is already well known (Nachbin [7 I ). The same definition may be applied to real spaces (Aron and Prolla [1 I ).
A di6 6eJl.e.n;tia.bLt,Lty ;type nJt.om
DEFINITION 2.1: of Banach spaces by
P
P
Sk
(E; F), k
E
E ;to
F is a sequence
the norm on each being
IN,
denoted
II P 118 ' which satisfies the following conditions:
-+-
i)
pSo (E; F) E
is the normed space of all constant functions fran
to F, identified with
F;
k P (E;F);
p8k(E;F) is a vector subspace of
ii) each
iii) there is a real number
> 1
(J
j, k E IN , j
such that
and
p E pSk(E;F) imply
DEFINITION 2.2:
Let
S be a differentiability type from E to
all functions
&sm(V;F) as the vector subspace of
f
such that, for x
and the mapping We endow
E
U
-+-
akf(x)
x
E
V,
k,
ajp(x) E pSj(E;F) and
x E E
define the space
~
~
k
F.We
&bm(V;F) ~J<
m, we have d'f(x)
E
of 8k P (E;F)
pSk(E;F) is continuous.
E
&sm(V;F) with the topology
T
8m
defined by the fam-
ily of seminorms;
~i
PK,k (f)
where
K
C
V
sup{l!d f(x)II
; x
E
is a compact subset and
In the case
F = 1R we will write
We remark that the space definition.
S
K, 0 < i < k},
k E IN, k
~
m.
&em(U;F) = &sm(U) .
&bm(V;F) is a particular case of this
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
DEFINITION 2.3 (Aron and Prolla [11): from
E
to
165
A differentiability type
8
is called compact if it satisfies the following condi-
F
tions for each
k E IN:
k
i)
Pf(E;F), the vector space of continuous k-homogeneous polynomials of finite type, is densely contained in
ii)
for each
v E F
~ ~ ~k ® v
the map
p8k(E;F);
is continuous
k (E',II'II) to (Pf(E/F), 11,11 )/ 8 iii) i f PEE' ® E, then Q oP E p8k(E;F)
for all
Q
from
8k E p (E;F)
and II Q 0 pil 8 < II Q II 8 II P II k.
EXAMPLES 2.4:
P~(EiF)
in
For each
k E lN,
let
pck(E;F)
pk(EiF) for the usual norm. Then
ferentiability type called
k E lli,
II· liN' and if
of
8 = c is a compact dif-
pNk(E;F), the Banach
of all nuclear continuous polynomials from nuclear norm
the closure
compact type.
cu~~ent
If we consider, for each
be
E
to
space
F, endowed wi th the
E has the approximation property,
then
is a compact differentiability type called nuc.e.eM type (see [ 2 J ) •
8 = N
Let F be a Banach '-'pace and 8 a
PROPOSITION 2.5:
ty type
16
6~om
E
to
di66e~entiabLU.-
F.
p8k(E;F) = pk(E;F), k E lli, k ~ m, then
&bm(U;F) =&8m(U;F)
topologically.
PROOF: map
As we have (P
8k
p8k (E iF) c pk (E; F)
equivalent norms.
COROLLARY 2.6:
(E;F), II· 11
8
) a Banach space and the inclusion
is continuous, then
II' II
and
II . 118
are
0
Let E be a 6-i.nite dimeMio It
compact di6 6e~entiability type
6~om
E
to
F.
no~med
'-'pace and
8
a
166
GUERREIRO
~(E;F)
PROOF:
pk (E i F),
k E :IN.
0
DEFINITION 2.7: Let 8 be a differentiability type from E to F and 8m A c a (U;F) a non-empty subset. We define: (\ {A + I (a, k); a
A
where
where
a
n fA + I (a, k ,e:); a
8m
(U;F)
Fix
r6 e
If
i
.s.
k}
and:
e: > a}
f
1..6 a. di 66 elLenUabULty type 6J!.om
a E U, k < m
A
(') {A
i-6
E to F a.nd
Tern -c..to-6ed.
and consider
+ I (a, k, e);
¢ B(a,k) there is
e
>
0
IS > O}.
such that
p(f - g)
>
for
IS,
g E A, where
p(h)
Consider -
:IN}
U, k .:: rn, k E IN,
a. non-empty -6u.b-6'!..t, then
B(a,k)
T
E
.s.
E
I (a,k,e)
PROOF:
am
m1 k
0, 0
PROPOSITION 2.8:
every
.s.
U, k
I(a,k)
A
A c
E
-1 sup {lid h(a)lIai 0 < 1 < k},
V = {h E ,am(U;F); p(f - h)
neighborhood of
If there exists
< e / 2},
which
is
a
f.
h E V (\ B(a,k)
I
we have
p(h - g) < e/2
for
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
some
g E A. Then:
p (f - g)
B (a,k)
is closed.
DEFINITION 2.9:
+ p (h - g) <
< p (f - h)
which is a contradiction. So
V n B(a,k)
9m
proves
that
0
Let
B be a differentiability type from
E
to
F
GeE' ® E
and
(U;F) non-empty subsets.
We say that and
E,
which
satisfying condition (iii) of Definition 2.3, A c a
167
V c U
Ao (gIV)
(A,G)
~ax~~6~e~
eond~x~on
(L) if, given g(V) c u,
a non-empty open subset such that
g
we
E
G
have
(AIV), the closure being considered in (&Bm(V;F),1 8m ).
C
Similar conditions have been used by Lesmes [3] , Prolla [10], Llavona [ 4 ] •
EXAMPLE 2.10: and P n
Let E be a space with a Schauder basis {eo,e , ... ,en""} l
the projection of
E on the vector
subspace
spanned
by
{e ,el, .•. , e } .
o
n
~ A e be a let gn (x) = An' Let 8 nElN n n is compact differentiability type from E to lR such that &8m(E)
Gi ven
x
E, x =
E
an algebra and consider quence
{gn,
i
i
E
I c &Bm(E), the ideal generated by a subse-
IN}. If
~
G
=
{P
n
;
n
E
IN}
then (I,G)
satisfies
condition (L). This example may be extended to a space with a generalized basis. We remark that for the differentiability types introduced 2.4,
aBm(u) is an algebra. More generally, if
bili ty type from E to we have from
PQ E P
p6i(E)
x
6 (k+ ')
J
lR
6 is a differentia-
such that given
(E) and the mapping
p9j(E) to
in
p6 (i+j) (E), then
(P,Q)
+
PQ
is continuous
&8m(U) is an algebra.
168
GUERREIRO
DEFINITION 2.11: quence
{P
ii)
We say that
; n E IN} eE'
n
I{J 0
P n ...
I{J,
I{J
@
E has PJtope.Jtty (B) if there is a
E
such that
EE'.
This definition was used by Restrepo [12] condi tion that the
Let a
THEOREM 2.12: lR
~uch that
Pn
se-
with the
additional
are projections.
be a compact di66eJtentiabLU_ty type oJtom
gam(U) i~ an algebJta and let
Suppo-:le that theJte. .[-:1 a -:lequence
G
be an ideal.
I c gam(U)
=
{P
n
;
to
E
n E IN}
E'
C
@
E
-:luch that: i)
ha~
E
ii)
(I ,G) y
Then I
pJtopeJtty
(B)
condition
~ati~ 6ie~
i-:l the
T
8m
Jte~pect
with
clo-:luJte
-
to
G;
(L).
00
I
in
g
8m
(U).
For the proof we need several lemmas.
Let
LEMMA 2.13:
that
8
be a di66eJtentiabLLUy type 6Jtom
to
~uch
lR
&am(U) i-:l an algebJta and (E'lu) C &8m(U).
Let
El
C
E
dimen~ional
be a 6inLte
a non-empty open ~ub~et and con~ideJt
16
&bm(U ). MOJteoveJt, i6 l
to the Tbm-clo~uJte 06
PROOF:
R: g
vectoJt E
~ub~pace,
&8m(U) ... g:U
l
I E g8m(U) i-:l an ideal then the Tbm-clo-:luJte. 06
ideal 06
R
E
Let
R(I) in
f E &8m(U), f E ~, then
U l E
C
El nu
&bm(U ). l
R(I) .[-:1 an
be.long¢
Rf
gbm(U ). l
A = R(g8m(U», which is a subalgebra of
gbm(Ul)because
is an algebra homomorphism. Now
cause
1
E
A satisfies the hypotheses of Nachbin I s theorem A
and (E' I U)
It is clear that
C
g
8m
[6
I be-
bm _hm (U). Therefore A is T -dense in lO (UJ!.
R(I) is a vector subspace of
&bm(U ). On the l
169
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INF INITE DIMENSIONS
other hand, if
Rf E R(I)
implies R(I) • A
C
and
Rg E A
we have
R(fg) E R(I),
R(I). By continuity of multiplication, R(I)
and we conclude that
R(I)·
A
C
which
·AC
R(I)·A
R(I), which completes the proof that
R(I) is an ideal.
'" a E f E I, Ul '
Let now
Defini tion 2.1, there is
g E I
-i d g(a)Ii
II aif(a)
k < m, -
(
a > 1
If
is given
by
such that
k
S
o.
>
< (/a ,
0 < i
-<
k.
Then
-i d g(a)Ii
So, we have orem, Rf
E
R(I).
LEr.1MA 2.14:
:to {P ;(;0
n
;
n
Le;(; E
< k.
E
be
a .6pac.e .6aA:i.6 eying pltopeft;(;y (B) wi.th S
be. ct
Q E K
E
c.ornpctc..t .6ub.6e..t.6, 12. i 2. k.
Ki C pSi(E;F)
and nair. all
fte.6pe.c..t
c.ompctc..t dio 6e.lr.e.n.tictbili.ty .type. oltom
i
,
1 < i
< k.
Le.;(; E be a .6pctc.e .6a.ti.6eyin9 plr.opeft.ty (B) wi.th
LEMMA 2.15: {P
< i
See Aron and Prolla [1 1 •
PROOF:
.to
o
o
0
lN }, le..t
n > n
< (,
Rf E (R(I»v and, by the classical Whitney's the-
( > 0, k E lN,
F,
S
n
; n
E
lN }, te..t
e
be. ct
c.ompa.c..t
di~6e.lr.e.n.t..i.a.bLe.i.ty
fte.6pe.c..t
.type. Oltom
E
170
GUERREIRO
to
<
TheJte a.Jte
and
no E :IN
m, K
C
a. c.ompac..t .6u.b.6e..t,
V
E: >
o.
a non-empty open .6u.b.6et,
V C V,
.6u.c.h
that: i)
K C V
n > n
-
a
PROOF: Let M > 1 be such that be such that x
o
< i
E
K, Y
E
lip nil':: M, n E :IN, and let 0 < 15 < dist(K,E\ V)
v,lIx-yll< 8
imply IIdif(x) -dif(y)lI
< E:/2Mk,
.:: k. By (B) and Lemma 2.14, there is
IIX - P xII {
Let r
< 15/2,
110 - On'Pnll"
= 8/2M and,
x
such that, for
,/2.
xl, ••• ,x
K C U {B (xi' r);
s
Q E aif(K). 1
E
K
1 < i
Consider the non-empty open subset a E V, lIa-xili
.:: IIPna -
Pnxill
< r
for some
2 1'
a
k.
= {t E vll1t -
xII < r}.
< s}.
V
=U
lIa - xiII
{B
(xi' r) 1 1 < i.::. s}.
n ~ no'
+
~
II xl."
< 8,
Pn(V) C U. Finally observe that for
-
such that
i . Then, i f
+ IIPnx i - xiII < IIPnll
n >n :
K
E
for each x E K, B(x,r)
By compactness, there are
If
e
x E K, 1 < i
< k,
n > no
-
P n all -<
and we have
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
LEMMA 2.16:
L e:t
PEE'S E,
El
e
be a compac:t di6 6eJten:tiabLU:ty :type 6tr.om E :to F,
= P(E).
The ~pace SIHlac.h
PROOF:
& pace
171
endowed w.Uh :the
60ft each
k
E
nOJtm
/I Q /I P
a
]N •
See Aron and Prolla [1 I .
v
PROOF OF THEOREM 2.12:
I
C
It is clear that
I C I. By Proposition 2.8,
I. Let
...
f E I, K
C
Lenuna 2.15, there are such that
Fix
U
a compact subset, k E
no E
K C V, Pn(V) C U
n > n
o
and
]N
C
U
I
k
~
a non-empty
m,
P
open
then norm
/I Q /l p
then (pi(E ), /I ./l p ) is a Banach space by Lenuna 2.16, and l
Hence the topology
,bm
By
subset
PtE) ,
pi (E ), 1 < i < k 1
equivalent to the usual norm
> O.
E
and
and consider
If we define in
V
]N
/I.
in
IIQ
0
/I • /l p
P/le'
is
i
P (E l ).
may be defined in
by
the
family of seminorms:
L C U a compact subset, j E 1
]N,
j
< m.
By using notation and results from Lenuna 2.13, there is such that
Then:
g E I
172
GUERREIRO
< (£/3) + II 21 i (Rf) (Px)
for all
0
x E K,
So we have
~
(L),
PK,k(g
(hIV»
(plv) -
1~
- Q.e.o~
0
Pile
is
possible
< 2£/3
to
and as (I,G)
h E I
find
< £/3. We conclude that
PK,k(f - h)
~ame
is any compact differentiability type from
E
the I
=
that
< £
and
0
~6
ed, then
satis-
such
2. 12,
COROLLARY 2.17: 8m
(Rg) (PX)
k.
it
the proof is complete.
T
21 i
P -
PK,k«flv) - go (plv))
fies condition 0
~
i
0
-
I
=
Qo~d~t~on~
06
Theo~em
I
'"
I .
COROLLARY 2. 18 : ~eqtlenc.e
The~
G = {P ha~
i)
E
11)
(I,G)
I
PROOF:
' n E
:IN} C E'
p~ope~ty ~a.ti~6~e~
E'
e
~tlc.h
0 E
~e~pect
(B) wLth
c.ond~t~on
~~ the Tbl-Q.e.ow~e 06
If
we have
n
I
~n
that: to
G.
(L).
&bl(U).
to
lR
= pl(E) = p8l(E) = pl(E). f
By proposition 2.5,
EXAMPLE 2.19:
If
&8l(U)
I e &em(U)
= &bl(U)
topologically. c
is an ideal and
sional, i t is not always true that
i
E
is closed.
is infinite dimenHence
Whitney's
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
theorem is false in the formulation example for
I
173
I. We will give
m = 1, remarking once more that
&81(U)
logically for any compact differentiability type
a counter-
= &bl(U)
topo-
8.
Let H be a real separable Hilbert space of infinite dimension and let
{e
S C H
Denote by rei
S
i E :IN}. Then Consider
gi(x)
= <x,e
be an orthonormal basis for
i E IN}
i
H.
the vector subspace algebraically spanned by = H.
I C &bl (H)
the ideal generated by {gi ; i E :IN} where
) , x E H. Explicitly:
i
n
{~f. g.
I
~
i=O
If
P n : H ... H
space spanned by respect to G = {P
{e n
fi E &bl (H),
n E IN} •
1.
denotes the projection of
o
i
~ n}
< i
in E IN}
then
H
H
on the vector sub-
has property
and (Example 2.10)
with
(B)
(I,G) satisfies con-
di tion (L). Consider the closed ideal let us characterize For a fixed I + I(a,l)
o}
{f E &bl(H); f(O)
Io
and
I. a E H,
= {f E &bl(H)13g E I, g(a) =f(a), dg(a) =df(a)}.
If
g E l , then
g(O)
as
gi E I, i E lN,
= 0
and
d9i(0)
=
e
dg(O) E S i
,
given
and, on the other u E S
there is
hand,
g E l such
that dg(O) =u. Then, i f a=O, I+I(a,l) = {fE&bl(H); f(O) =0, M(O)ES}. a
If
t-
f
E
df(a)
O. Consider
Then
t I
I + I(a,l)
0, we claim that
= A,
f(a) a
'I
= gi
= &bl(H). In fact, let f E &bl(H),
such that gi(a) F 0, because 1 Ae i h E &bl(H), heal =_A__ and dh(a) =g. (a)(v-g.(a»' gi (a) v. There is
i E lN
1.
h E I, tea)
=
A
and
dt(a)
= v,
which
proves
1.
that
+ I(a,!). We conclude that
Tbl-closed. In fact:
I
I + I(O,l) and we claim that it is
not
174
GUERREIRO
the function
I + I
(0,1) ~ 1
fix)
(x,y)
0
,
because for
belongs to
v E H, v
¢
S,
but does not belong
10
to
1+1(0,1).
On the other hand, if {si; fi(x) and
i
such
IN } C S,
E
g E 10' dg(O) that
v, there is a
sequence
as
Consider
1+1(0,1) g(x) - (x,v - si) , x E H. Each fi belongs to bl 1+ 1(0,1) =1 , fi ~ g, for the topology 1 as i .. 00. Then 0
which establishes the claim. I f
Furthermore does not belong to
i. In fact:
I =
(X,X)
i, but
belongs to
I.
By Corollary 2.18, I plies
fix)
v
I. Horeover:
I
C
v
I
I
::: (I)
C 10
im-
I0
Then:
c
I
'*
I
t "I
I
I
t
a
gbl(H) .
3. IDEALS OF FUNCTIONS WEAKLY UNIFORHLY m-CONTINUOUSLY
DIFFERENT~LE
ON BOUNDED SETS
is called we.ak..e.y uni601tm.e.y
c.on-
tinuou4 on bounded 4et4 (WUC.b4) if given BeE a bounded subset
and
DEFINITION 3.1:
£
>
0,
x,y E B,
A function
there are
/) > 0
l
DEFINITION 3.2:
Let
a
1
<
f: E .. F
and
c ,
1 < i .::. k, imply
is wucb s ,
k E IN,
k < m}.
that
/If(x) - f(y)/I < e.
be a differentiability type from
We define:
such
E
to
F.
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
We endow
&!m(E;F) with the
8m
TW -topology,
175
defined by the fam-
ily of seminorms: -1 sup { II d f (x) 118 ; x E B, 0 < i
PB,k (f)
BeE
a bounded subset, We remark that
differentiabili ty
k E lN, k .::. m.
&!m(E;F)
8, whenever
= &bm(E;F) E
for any compact
type
8
a compact differentiability type from
Pf(E;F) c &!m(E;F)
(see Aron and Prolla [1
Let
8
of
is a finite dimensional space. No-
tice also that for
DEFINITION 3.3:
< k},
E
to F,
1) •
be a differentiability type from
E
to F and
A c &!m(E;F) a non-empty subset. We define:
..,
n {A + I(a,k,E); a E U, k < m,
A
£
> O}
where i {f E &8m(E'F)' w ' , IId f(a)1I
I(a,k,£)
We may define
8
< E, 0 < i
< k}.
A in a similar way and, by introducting the na-
tural modifications in Example 2.19, we see that
-
A
is not always a
8m-closed subset of TW
8m (E;F). On the other hand, A " &w
for any
a non-empty subset. The proof of this fact
A c &!m(E;F)
is
T 8m-closed
w
is
similar to 2.8.
PROPOSITION 3.4: ~ati~6ying
aLI'.
06
8
i~
a di66e4entiability type 640m
E
to
F
Ve6inition 2.3, then
PEE' ® E.
PROOF: then
(iii)
16
Let
a
k (f
0
f E &8m(E;F) and
PEE' ® E. If
w
P) (x)
= akf (Px)
0
P.
k E lN,
k < m, x E E,
176
GUERREIRO
Let
BeE
be a bounded subset and
/) > 0
bounded subset, there are
I~.
x,y E B,
~
(Px) - ~. (Py) ~
I
£
and
< /), 1
<
O. As
>
~l""
'~s E
P(B) C E
E'
is a
such that
i < s, imply
Then:
x
which proves that
DEFINITION 3.5:
Let
E
E
->-
ak(f oP) (x) E p8k(E;F) is wucbs.
8 be a differentiability type from
0
E
to
F
satisfying (iii) of Definition 2.3, and let GeE' ® E and A c &!m(E;F) be non-empty subsets. We say that (A, G) &ati&6ie& condition (L) if given have
A 0 g C
A,
g E G
we
the closure being considered with respect to the to-
pology
DEFINITION 3.6: quence
{P
n
; n E IN}
THEOREM 3.7: lR
& uch
We say that
Let
e
C
E' ® E
E
ha& PJt0PVtty (B*) if there is a such that
~
oP
n
...
be a compact di66eJtentiabi!ity type 6Jtom I C &8m(E)
that
w
Suppo&e theJte i& a .6equence
ii)
E
ha& pJtopeJtty (B*) with Jte.6pect to
(I,G) &ati&6ie& condition (L).
Th en v I .-<.& th e
8m TW
D -c,,-O.6uJte
0
6 ' I-<.n
For the proof we need the following lemmas:
G;
~
EE'.
E
to
be an idea!.
G = {P n ; n E IN } C E' ® E
that: i)
for all
se-
&uch
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
LEMMA 3.8:
Let
be a d.-L66elLent.-Lab.U..,uy type 61L0m
&6m(E) .-L-6 an alg ebJta and w
that
E
Let
R:g
6
E
l &6m(E) w
C
E
to
E
be a O.-Ln.-Lte d.-Lmen-6.-Lonal -6ub-6paee
"* glEl
E
-6uch
]R
E' c &8m(E) w • and
eOn-6.-LdelL
&bm(E ). l
I C &bm(E) .-L-6 an .-Ldeal then the Tbm -elo-6UlLe 06 w
16
&bm(E )· MOJteovelL, .-LO f E &!m(E), and f l &bm(E ) . belong-6 to the Tbm -elo-6uJte 06 R(I) .-Ln l
.-Ldeal 00
PROOF:
177
v
E
I,
R(I) .-L-6 an then
Rf
Analogous to 2.13.
LEMMA 3.9: to {Pn;n
Let E E
be a -6pac.e -6at.-L-66y.-Lng plLopeJtty (B*) w.-Lth
IN}; 6 a c.ompact d.-L66eJtent--Lab.-Llity type 61L0m
f E &6m(E'F) w ' , k E IN, k _< m, BeE
a bounded -6ub-6et,
e:
Jte-6pect
E
to
F,
> O.
PB , k (f - foP n) < e:, n > no
PROOF:
See Aron and Prolla [1 J •
PROOF OF THEOREM 3.7: Conversely, let
e: > 0
It is clear that v
f E I, BeE
v
I C I.
a bounded subset,
k < m,
and
be given. By Lemma 3.9, there is
no E N
such that
n > n Fix
n > n
o
and let
P
and results from Lemma 3.8, Rf in
= Pn
If
o
El = P (E), by using notation
belongs to the
Tbm-closure of
R(I)
&bm(E ). Furthermore, P(B) C El is a bounded subset, then a relal tively compact subset, and the topology Tbm may be defined in &bm(E ) l by the family of seminorms:
178
GUERREIRO
Pp , L , J. (h)
L C El
a compact subset, j E IN, So, there is
g E I
·i
~
< m.
such that:
lid (Rf) (Px) 0 P -
X E B, 0 ~ i
j
·i d
(Rg) (Px) 0 pli
e
< E/3,
k,
and using the fact that (I,G)
satisfies condition (L), there is h EI
such that PB,k(g oP - h)
< E/3.
Then:
.i ·i lid (g oP) (x) - d h(x)ll
X
E B, 0 < i
4. IDEALS OF
e
< E/3 + E/3 + E/3
< k, which concludes the proof.
E,
0
&cm(U)
DEFINITION 4.1:
For
A
C
&cm(U;F) a non - empty
subset
we
define
"
A = n {A+I(a,k,L,E); a E U, k < m, LeE compact, E > O} where
I(a,k,L,E)
{f E Il,cm(U;F);
Iidi f(a)vll
< E, vEL, 0 < i
< k}.
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
179
The definition of
A
extends naturally and obvious modifica-
tions in 2.19 show that
A
may be fail to be Tcm-closed. By contrast,
v A
is always
Tcm-closed.
The definition of condition (L) for a pair (A,G),
GeE' 0 E
a non-empty subset, is naturally extended too.
Let
THEOREM 4. 2:
I c &cm (U) bean .i.deal and .6UPPOH :theJLe..u., GeE' 0 E
.6 uch that i)
the .tdenLUy
iE'
06
E, beiong-6 to the cio-6ulte 06
.tn
G
tC(E;E); ii)
(I,G) .6at.t.6 6.te-6 cond.tt.ton (L). y
Then I
LEMMA 4.3:
cm .t.6 the T - ciO-6 ulte 06 I.tn
Let
cm & (U).
I c &cm(U) be an .tdeai, El c E
a 6.tn.tte
ciimen.6.tona..t
vectolt -6u.b.6pace, Ul c U () El a non-empty open .6ub.6et. 16 we con-6.tdelt R : g E S,cm(U) ->- glu l E S,cmCU1):then:the Tcm-c.tO-6UJte
06 then
&cm(U ) . Molteovelt, .t6 f l cm Rf beiong-6 to the T -c.e.O-6uJte 06 R(I) in
R(I) .t.6 an .tdeai 06
E
&cm(U),
v
f
E
I,
£
>
O.
&cm(U ) • l
PROOF: Analogous to 2.13. We just remark that pologically because El is a finite dimension vector space.
PROOF OF THEOREM 4.2:
It is clear that
I c "I.
y
Let
f E I, K C U and
LeE
compact subsets, k
By Lemma 3.1, Prolla and Guerreiro [11], there are
<
u E G
m,
and V C U
a non-empty open subset such that
PK,L,k«fiV) - f
Consider
El =u(E), U =E l l
II
0
(uiV»
U, Kl
< £/3.
U(K)
and
L~
u(L) .
By
180
GUERREIRO
using notation and results from Lemma 4.3, there is
PK
L
g E r such that
keRf - Rg) < E/3.
l' l'
On the other hand (r,G) satisfies condition (L) h E I
so
there
is
such that
PK,L,k(g
0
(uIV) -
(hlv» < E/3.
Then:
-i -i Id g(u(x»u(v) - d h(x)vl <
PK,L,k (g
(X,V) E K
L, 0 < i
x
This shows that
5.
we &cm(U;F) an
all
fEr.
Le.t
IP E F'
I
F
(uIV) -
(h IV»
<
E,
< k. 0
&cm(U) - SUBMODULES OF
THEOREM 5.1:
0
&cm(U;F)
be. a lIpace. wLth the. appJtoxi.mati.on p!Wpe.Jtt!J
&cm(U)-.6ubmodu.ee .6ati.6o!Jing:
(IP oW)
@ V
V E F.
Suppo.6e theJte i.6
GeE' @ E
.6ueh that:
belong.6 to the clO.6uJte. 06 G in
i)
iE
1i)
(W,G)
.6ati.66ie..6 condition
(L).
.cc(E;E);
c
W,
and
OM
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
The proof of 5.1 uses the following W c &cm(U;F) is an
LEMMA 5.2:
(I{) 0
I{)
oW =
lemmas.
In
both,
~ E F',
&cm(U)-submodule and
The vec.tOlt J.>ubJ.>pa.c.e
&cm(U). Mofteoveft,
two
181
{I{)
og; g E w}
W,G) J.>a.t,[J.> 6'[eJ.> c.ond,[t,[on (L),
'[6
'[J.> a.n '[deal 06 (W,G)
J.>a.t'[J.>-
6,[eJ.> c.ondit,[on (L).
PROOF: h
If
h E &cm(U) and
a g) E
(I{)
I{)
g E W, then
oW. Therefore
I{)
aW
hg E Wand, so
I{) 0
(gh)
is an ideal.
Suppose now that (W,G) satisfies (L) and let
g E G and
V c U
be a non-empty open subset such that
g(V) C U. If we consider K C V
and
~
LeE
compact subsets, fEW, k
m,
£
> 0,
there
hEW
is
such that
(x,y) E K
x
L,
0 < i
< k.
Then:
(x,y) E K
This proves that
LEMMA 5.3: 60ft J.>ome
16
(I{)
SuppoJ.>e tha.t
x
L,
oW) a (gIV)
iE
0
< i
<
k.
(l{)oWIV).
C
0
belongJ.> to the c.lo).,ufte 06 G
in
tC(E/E),
GeE' ® E, a.nd tha.t (W,G) J.>a.t,[J.>6,[eJ.> c.ond,[t,[on (L). fEW, then
I{)
of
belongJ.> to the Tcm-c.loJ.>uJte
06
I{)
oW
,[n
&cm(U).
.., PROOF:
Consider
fEW, a E U, k < m,
£
> 0
and
LeE
a
compact
GUERREIRO
182
subset. There is
Y
E
L,
0 < i
g E W such that
< k.
Then:
Y E L, 0 .2. i .2. k, which proves that
<{i 0
f E
oWr . Since Lemma 5.2
(<{i
enables us to apply Theorem 4.2, we conclude that the Tcm-closure of
<{i 0
W in
&cm (U) •
I{! 0
f
belongs
to
0
v
PROOF OF THEOREM 5.1: sets, k .2. m,
Then
£
fEW, let
and define for
> 0
A =
Let
{Ai; 0 .2. i .2. k}
U
approximation property, there are
K
0 < i
U, LeE
C
be compact sub-
< k the set
is a compact subset of n E N,
<{i
0
E
J
F
I,
F.
By the such
that: n ~
Ily -
<{iJo (y)vJoll < £/3,
yEA.
j=l By Lemma 5.3, each
<{ij
0
f
belongs to
<{ij oW,
so
there are
w.
Consider
gj E W such that
n where
El
E/3 (1 +
~
j=l
n Let
h =
t E W such that
~
j=l
(<{i
0
J
0
go) J
II Vj II).
o
Vj
.
By hypothesis,
h E
WHITNEV'S SPECTRAL SVNTHESIS THEOREM IN INFINITE DIMENSIONS
183
PK,L,k(h - t) < £/3.
Then:
,
n
II a~ ( L (V', j=l ]
0
f) ® v],}(x) v - aih (x)vll + II aih (x) v -
ait (x)vll <
n
i E/3 + j:l I a (IP j
f - V'j 0 gj) (x)v I IIvjll + PK;L,k (h - t) <
0
n
£/3 + j:l £lllvj II + £/3 < E,
(X,V) E K
This proves that
x
L,
0 < i < k.
PK,L,k(f - t) < £, and so
fEW, as desired. D
In order to drop the approximation property of the space F, we will introduce a new topology.
DEFINITION 5.4:
We will denote by
Tcm_w the topology
defined
in
&cm(U;F) by the family of seminorms:
f
where
K
U, LeE
C
->-
PK, L ,k (V'
above definition that
for each
f) ,
are compact subse.ts, and
Notice that for any subset
A if, and only if,
0
IP
A
C
IP E F', k 2. m, k E IN.
&Cffi(U;F) it follows from
f E &cm(U;F) belongs to the 0
f
,cm_ w closure of
belongs to the ,cm-closure of IP 0 A in &Cffi(U) ,
EFt.
DEFINITION 5.5:
Let
A
the
C
&cm(U;F) be a non-empty subset.
184
GUERREIRO
~ve
A*
define:
n {A+I(a,k,L,
E
U, k < m, LeE compact, II'
E
F', £ >
a},
where I(a,k,L,
If
{f E 8.
cm
·i
(UIF);!d (
A c &cm(U;F) is a non-empty subset, an argument similar to
2.8 shows that
DEFINITION 5.6:
A*
is
T
cm
_* closed.
A c &cm(U;F) and
Let
GeE' ® E
be non-empty sub-
sets. ~a~i~6ie~
We say that (A,G) V c U
(L*) if given g E G
a non-empty open subset such that g(V) c u, we have Ao(g!V)
the closure considered in
THEOREM 5.7: ~he~e i~
Let
we
(W,G)
8.
C
and (AIV),
(&cm(V;F), T cm _*).
cm (U;F) be an
~uch ~ha~
GeE' ® E
tC(E;E) and
iE
&cm(U)-~ubmodule.
belong~
~o
the
clo~u~e
Suppo~e
06
G in
~ati~6ie~ condition (L*).
Then w* i~ ~he
PROOF:
condi~ion
T
cm_* clo~u~e 06
w.
Apply Lemma 5.2 and Theorem 4.2.
REFERENCES
[ 11
R. ARON and J. B. PROLLA, Polynomial approximation of
differ-
entiable functions on Banach spaces (to appear).
[ 21
S. DINEEN, Holomorphy types on a Banach space, Studia Math. 39 (1971), 241 - 288.
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
[ 3I
185
J. LESMES, On the approximation of continuously differentiable functions in Hilbert spaces, Rev. Colornbiana de Matem.8 (1974), 217 - 223.
[41
J. L. G. LLAVONA, ApptLOx-Lmad.on de. 6unc.-Lone.'!'
d-L6eJLe.nuabte..6, Doc-
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B. MALGRANGE, Ide.al~
06
d-L66e.~e.n~-Lable. 6unc.~-Lon~, Tata Insti-
tute of Fundamental Research, Bombay, 1966. [61
L. NACHBIN, Sur les algebres denses de fonctions differentiables sur une variete,C. R. Acad. Sci. Paris 288 (1949), 1549 - 1551.
[71
L. NACHBIN, Topology
Olt
6JOac.e.~
06 holomotc.ph-<-c. ma.pp-<-ng.~,Springer
Verla.g, 1969. [81
L. NACHBIN, On continuously differentiable
mappings
between
locally convex spaces (to appear). [ 9I
M. Z. NASHED, Differentiability and related properties of nonlinear operators: some aspects of the role entials
Func.~-Lona.l
AnatY6-L6 and Appt-Lcat-Lon6 (ed.
Academic Press, [101
differNonl-<-n~
L. B. Rall),
(1971), pp. 103-309.
J. B. PROLLA, On polynomial algebras of continuously differentiable functions, Rendiconti dell'Accademia Nazionale dei Lincei, Serie 8, vol. 57 (1974),
[111
of
in nonlinear functional analysis, in
481 - 486.
J. B. PROLLA and C. S. GUERREIRO, An extension
of
Nachbin's
theorem to differentiable functions on Banach spaces with approximation property, Arkiv for Mathematik 14 (1976), 251 - 258. [12]
G. RESTREPO, An infinite dimensional version of a theorem
of
Bernstein, Proc. Arner. Math. Soc. 23(1969),193-198. [13]
H. WHITNEY, On ideals of differentiable functions, Arner. J. of Math.
70 (1948), 635 - 658.
This Page Intentionally Left Blank
App~oximation Theo~
and Funotiona~ Ana~ysis (ed.) €)No~th-Holland Pub2ishing Company, 1979
J.B.
P~o~Za
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
G. G. LORENTZ
t
Department of Mathematics The University of Texas Austin, Texas, U.S.A. S. D. RIEMENSCHNEIDER* Department of Mathematics University of Alberta Edmonton, Alberta, Canada
§l. INTRODUCTION The first paper [3 J on Birkhoff interpolation is due to G. D. Birkhoff himself, which he presented to the American Mathematical Society when he was only 19 years old. Its style is old-fashioned; the
main interest is in identities, remainder formulas, theorems. Birkhoff was interested in the sign of
and
mean value
the kernels
appear in these formulas, and proved the important and deep
which theorem
about their number of zeros. In 1955 - 58, Turan and his pupils studied the interpolation", which prescribes the values of
"0 - 2
P nand
lacunary
P"
n
at
the
knots. They studied a very special selection of knots - zeros of derivatives of Lagrange polynomials - and obtained many beautiful results
t
Supported
in part by Grant MCS 77 - 0946 of the National
Science
Foundation.
*
Research supported
by Canadian National Research
A-7687. 187
Council, Grant
188
LORENTZ and RIEMENSCHNEIDER
(see [2 ) , [45 J and [41 J ) • In 1966, I. J. Schoenberg [39J asked when
the
interpolation
problem with a given structure is solvable for all possible
sets of
knots. This is the problem of regularity or poisedness of the interpolation matrix, which has proved to be exceedingly Atkinson and A. Sharma [1 J and D. Ferguson [7]
difficult.
K.
gave the basic the-
orems of regularity, Karlin and Karon [13] contributed
the
theorem
about coalescence, and Lorentz ([ 18 J, [19 J , [ 22J ), theorems of singularity. Among the applications of Birkhoff interpolation, we mention the uniqueness problem for monotone approximation
(Lorentz - Zeller
[29], R. A. Lorentz [30]), and the Birkhoff quadrature fonnulas (Lorentz and Riemenschneider [24]). In recent years several papers have dealt with the Birkhoff interpolation problem for spline functions (KarlinKaron [13] and others). The present report attempts to give an exposition of this theory for polynomial interpolation. For the sake of brevity, we omit spline interpolation, and "lacunary interpolation" with special knots. This paper
is based on the 1975 report [20] of one of us to the Center of
Numerical Analysis, University of Texas in Austin. A last remark: the name "Birkhoff" interpolation problem (rather than "Hermite-Birkhoff") seems to be completely justified
from
all
possible mathematical points of view; both historical as well as those of substance.
§2. BASIC DEFINITIONS AND THEOREMS 2.1. DEFINITIONS
Let
S = {go,gl"" ,gN} be a system of
n
times
continuously differentiable functions on a set A which is either an interval [a,b J or the circle T. A linear combination will be called a po!ynomial in the system S. A matrix
P
=
N ~
j=O
a. gj ]
189
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
(2.1.1)
m
E
n
(e ik )i=l, k=O' m > 1, n > 0
is an il1.te.ltpo.tatiol1. matltix 6011.
S if its elements
one and if the number of ones in
is equal to
E
e N
In general we do not allow empty rows, that is an k =O, ••• ,n. A
~e.:t
06
points of the set fined for
kl1.o:t~
A. The elements
IE
+ 1,
1= N + 1.
i for which e ill
or
= 0, ik
distinct
and the data
E, X, S
(de-
e ik = 1) determine a Biltkh066 il1.te.ltpo.tatiol1. problem which satisfying
P
p(k)(X.)
(2.1.2)
1) •
1.
N + 1
The system (2.1.2) consists of unknowns
a . The pair
is called Ite.gu.talt
E,X
j
linear equations
E, X is
~il1.gu.talt.
A pair
ik
with
equations
if
(2.1.2) have a (unique) solution for each given set of c the pair
are zero
X = {xl' ••. ,xm} consists of
consists in finding a polynomial
N + 1
ik
; otherwise
E, X is regular if and
only
if
the determinant of the system
(2.1. 3)
l}
D(E,X)
is different from zero. Formula (2.1.3) gives a row of the nant corresponding to the entry
e
= 1
ik
in
E;
the order of the rows
in (2.1.3) is taken as'the lexicographical ordering of By
A(E;X) we denote the (N + 1)
x
(i,k),
E, X
E. An interpolation matrix
E
is regular for each set of knots
consider several types
a
~
X
Xl <
(Qomplex Ite.gulaltity) when the knots
or Ite.gulalt
has this property if the in a given class.
of regularity; oltde.1t Itegu.taltity, if
and the knots must satisfy
eik=l.
(N + 1) matrix given by (2.1.3).
The basic notion of this report is that of a poi'& e.d matrix
determi-
<
are
x
< b; m-
arbitrary
One
pair can
A = [a,b]
Itea.t Ite.gu.taltity distinct
real
LORENTZ and RIEMENSCHNEIDER
190
(complex) numbers; and,
~~igonome~nie ~egula~i~y
lari ty on the circle, -
7T
< Xl < •••
means that the determinant
t-
D(E/X)
knots, while singularity means that shall distinguish between
~~~ong
< xm < 7T. The regularity
0
of
E
for all (admissible) sets of
D(E,X)
~ingula~ity
values of different sign, and weaQ
which is order regu-
vanishes for some
X.We
[6 I, when D(E,X)
takes
~ingula~ity,
when
D(E/X) vanishes
without a change of sign. A matrix is singular if and only if some non-trivial polynomial P
is annihilated by
E, X for some admissible
satisfies the homogeneous equations a singular matrix we consider matrix
p (k) (x.)
= 0
e ik = 1. For
for
1.
r(E), the lowest possible rank of the
A(E,X). Then
(2.1. 4)
deE}
d
~he de6eQ~
06
EXAMPLES:
A
N + 1 - reEl
E / is the largest possible dimension of the
of polynomials
P / annihilated by
E, X for some
Lag~ange in~e~polation mat~ix
in the column
O.
A
Taylon
inte~pola~ion
trix consisting of a single row of ones. e iO
X; this means that p
= ... = e i
,k
i are all zeros. An
= 1
has
m~x
subspace
X.
m = N + 1 is an
A He~mite
1
and
ones
x (N + 1) rra-
mat~ix
has blocks
of ones in each row while the remaining entries
Abel ma~~ix
prescribes each derivative
p(k)
at
exactly one point and thus has only a single one in each column.
2.2. THE ALGEBRAIC CASE
(2.2.1)
and
p
x
Here
A
[a,b I , S consists of the functions
N
N! ' ...
,gN-l (x)
are the algebraic polynomials of degree
k > N, we may assume that
n
~
1,
N. Since
N, and by adding columns of
P (k) := 0 for zeros
to
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
n = N; such matrices are called
E, we may set that
=n
N
191
no~mat.
We shall assume
in what follows.
Now formula (2.1.3) becomes
(2.2.2)
D (E, X)
x.n-k 1
=
{
(n _ k) : '
if we agree to replace lip!
, x
06 joint
m
by zero if
dete~m..i.nant
T he
PROPOSITION 2.1:
n-k-l xi (n-k-l) :, ••• ,
deg~ee
p < O.
..i.~
D (E, X)
a homo 9 eneou~ potynom..i.at
1 + 2 + ". + n - ~e
(2.2.3)
D(E,aX) = aPD(E,X), D(E,X + a)
(2.2.4)
D(E,X)
aX
-k xi (-k) !; e ik
ik
=1 k = P
which
D(E,X) ,
= {(Xi +a O) n-k (n - k): , •.. ,
{ax l ' ..• , aXm} ,
X
+ a
{Xl + a , ... , xm + a} .
In particular, it follows that the regularity (singularity) of E does not depend on the choice of the interval For a normal matrix column
m(k)
be
the number
are ones in column
M(k) =
O.
The functions
in
m (k), M(k) are called the potya.
6unc.t..i.oM 06 E. The condition
(2.2.5)
of ones
k
~ mer) be the number of ones in columns r=O For example, M(n) = n + 1, while M(O) > 0 means that there
k and let
0,,, .,k .
E, let
[a,b].
M(k)
> k
+ I,
k
o , ••• , n
LORENTZ and RIEMENSCHNEIDER
192
is called the P5lya condition, and the corresponding matrix iscalled a P5lya matILix.. Similarly, a Biltk.hofJfJ matlL..i.x. is a matrix E whose Polya function satisfies the BiILkho66 condition
(2.2.6)
M(k)
> k
+ 2,
O,l, ... ,n - 1.
k
These conditions play the following role. It is difficult to when
E is singular, that is when
easy to see when
THEOREM 2.2:
minant
0 (E,
=0
xl
O(E,X) is zero for all
for some
X, but it is
X:
(G. O. Birkhoff, D. Ferguson, and B. Nemeth)
O(E,X)
io
not ..i.dentically zeILO i6 and only
..i.~
decide
The deteIL~ati~6ie~
E
the P5lya condition. If nomial
M{k) < k + 1
P of degree
for some
k, then there is a non-trivial poly-
k which is annihilated by
This
E
proves the
necessity of the condition. The sufficiency was proved incorrectly by
G. O. Birkhoff [31
and later correctly proved independently by
~th
[33] and O. Ferguson [71 . For normal matrices, condition (2.2.5) is equivalent to the assumption that any
(2.2.7)
I t is
s
last columns contain at most
m(k) < n - n
(2.2.7) that we call the
l
+ 1,
o
sones:
<
P5lya condition for arbitrary
normal) matrices. Condition (2.2.7) holds if and only
if
E
(not can be
made into a normal Polya matrix by the addition of one or more
sup-
plementary rows. A normal matrix
E is decompo;.,able, [1
J,
can be split vertically into two normal matrices
E = El E
l
e
E
2
,
if it
, E . A matrix is 2
indecomposable if and only if either it satisfies the Birkhoff condition,
RECENT PROGRESS IN BIRKHOFF INTERPOL.ATION
193
or it consists of a single column. For each Polya matrix the maximal canonical
E, thereis
decompo~ition
(2.2.8)
E
where each matrix If
E
=
E
j
is either a one column or a Birkhoff matrix.
El $ E 2 , then the matrix
A(E,X) which appears in (2.2.2)
can be written, after a proper rearrangement of rows, as n -: r
r
(2.2.9)
+ 1
A(E,X)
n - r
As a corollary, we obtain
(2.2.10)
D(E,X)
Hence,
THEOREM 2. 3: i~
~e9ula~
(Atkinson and Sharma [1
i6 and only i6 both
2.3. REGULAR MATRICES
06
it~
I)
A
decompoMble ma..:tJUx E =E l e E2
component~
a~e ~egula~.
By Theorem 2.2, the Polya condition
(2.2.5)
is necessary for regularity. To obtain a workable sufficient
tion, we need the following notion. By a
in a row
~equence
condi-
i
of the
matrix E, we mean a continuous block of ones
(2.3.1)
1
which is maximal. Therefore, for a sequence either ei,k-l
=
0, and either
l = n
or
ei,l+l = O.
k = 0
A sequence
or is
else
odd
194
LORENTZ end RIEMENSCHNEIDER
(or even} if it has an odd (even) number of ones. A sequence (2.3.1) is -6uppoJtted if there exist two ones to the
NW and
SW of
e
ik
Already G. D. Birkhoff [3]
E
= l, in other words, if there are ones in
(i 2 ,k 2 ) in
positions (il,k }, l
in
E with
i
l
< i, kl < k; i2 > i, k2 < k.
noticed that odd supported sequences are
not good for regularity; he called a matrix
E eon-6eJtvative,
if
it
has no odd supported sequences.
A n.oJtma£. po£.ya matJtix
THEOREM 2.4: (Atkinson and Sharma [1] )
oJtdeJt Jtegu£.aJt i6 it ha-6 no odd
E
i-6
-6equenee-6.
~uppoJtted
The simplest proof of this theorem depends on an extended form of Rolle's theorem; another proof
is based
on the
properties
of
Birkhoff's kernel, (see Section 7.3). If
f
[a,b 1,
is continuously differentiable on
any two consecutive zeros of zeros of the derivative ber of zeros of
f'
a., B of
f
then
between
is an odd (or infinite) number
f'. Hence, if between
a and S some nurn-
of even multip£.ieity are known to exist, then f'
must have an additional zero (with a new value or
as an
additional
multiplicity of a known zero). Using this and induction, one proves
111
THEOREM 2.5:
f
i-6 an
n-t'<'me~
eontinuou~ty
diHeJtentiabte
6une-
tio n annihi£.ated by a po£.ya inteltpo£.atio n matJtix E co ntaining no odd -6uppoJtted
-6equenee~
and a -6et 06
~not-6
X = {xl, ... ,xm}, then
o
(2.3.2)
Applying this theorem to a polynomial we obtain
p=:o n
P
annihilated by
n
E, X,
and thus prove Theorem 2.4.
Since 2-row matrices have no odd supported sequences, we obtain
THEOREM 2.6 (Polya [36], Whittaker [46])
1
A
2
x
(n + 1) inteJtpo£.ation
195
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
The problem of order-regularity is much more complicated if the number of rows is more than two (even if it is only three, see For
compfex-~egufa~i~~,
when the knots
Xi may be arbitrary
numbers, the situation is different. D. Ferguson [7 ]
§6).
complex
characterized
complex-regular Birkhoff matrices (for another proof see [13]). Combining this with the decomposition formula (2.2.8) one obtains
THEOREM 2.7:
A Pofya ma~fLix
c.anon.£caf decompo-6i~ion
E if.> c.ompfex fLegufaJt i6 and onty
i6 ..Lt6
COMi-6tJ.> only 06 He.fLmite matfLice.-6 and
(2.2.8)
06 matfLice.J.> w-i.th at mO-6t ~wo non-zefLo fLOW-6.
Unfortunately, the known proofs are not simple. If
E is not regular, its "distance from regularity"
measured by its defect, given by formula (2.1.4). Using an
can
be
argument
from [ 5 J we can prove
THEOREM 2.8:
FOfL a Ylo~mal Polya matfLix w-i.th e.xactly p odd -6UppofLte.d
-6equence-6,
(2.3.3)
d <
[ ..£..:!:...!.. ] 2
This inequality cannot be improved.
2.4. EXAMPLES, SYMMETRY, AND TRIGONOMETRIC INTERPOLATION
Applying
Theorem 2.4, we see that all Hermitian (hence all Lagrange
and
all
Taylor) matrices are regular: they do not have odd supported sequences. Abel matrices are regular by Theorem 2.3 since they decompose
into
one column matrices, each with a single entry equal to one, and these are regular. Computing the determinant
D(E,X), one sees that in
196
LORENTZ and RIEMENSCHNEIDER
=(: : :) 'E2=(: : : : :)
(2.4.1) El
100
the matrix
10000
El is strongly singular, the matrix
E2
1
0 0 0
1
0
0
1
1
0
0
0
is weakly singu-
lar, while the matrix E3 is regular in spite of the fact that it has (two) odd supported sequences. Thus, Theorem 2.4 of Atkinson - Sharma cannot be inverted. Nevertheless, this inversion
is "usually" true.
Matrices which have exactly one odd supported sequence in one of the rows (with other sequences of this row being even or
not supported)
are necessarily singular (see Theorem 5.1). There are also other resuIts in this direction. In his "lacunary interpolation", P. Turan has studied symmetric matrices. A &ymmetJUc ma.tJt-ix
E should have an odd number, 2m + 1,
rows (to assure generality, we allow here
a
only of zeros). Thematrix E is symmetric if and all
central
of
row consisting
e-i,k=ei,k'
i=l, •.. ,m
k.
A symmetric matrix E is &ymme.tJt-ically JtegulaJt if the
pair E,X
is regular for each symmetric set of knots
Xo
wi th
= 0, x_ i = - xi' However, this notion can be reduced to
larity [22]: E is symmetrically regular if and only if both E2 are regular, where El (or
E ) consists of row 0 of E 2 all elements in odd (or even) positions have been replaced
reguEl
and
in which by
ones
while the other elements are left unchanged, and of rONS 1, •.. ,m of E. If the matrix
E
has some measure of symmetry, one
can
find
some simple necessary conditions for regularity which complement the polya condition. For example, again let the rows of
E
be numbered
-m, ••• ,-l,O, ••. ,m, and let qj' j =l, •.• ,m, be the number of k's which row
e_j,k = e jk
0 of
1. Also, let
Pe 'Po
E in even or odd posi tions.
for
be the number of zeros in
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
PROPOSITION
2.9
inequality i6 a
[23J:
Unde~
nece66a~y
197
the above a66umptianc, 6o~
condition
the
the
~egula~ity
06
6ollowing E
(2.4.2)
Only a little is known about Birkhoff trigonometric interpol ation. Here the system of functions
(2.4.3)
S
S is
{I, cosx,sinx, ... ,cosNx,sinNx},
The polynomials in
S
(a
k=l An interpolation matrix is a 2N + 1 ones. There is no a
minant for all
of
D(E,X) for
E
[-n,n).
are the trigonometric polynomials N l:
derivatives
x
TN
k
cos kx + b
m x (n + l) matrix
p~A.o~i
k
sin kx) .
E =
(e. k) m n with ~ i=l,k=O reason to assume that n = 2N since
of all orders are non-trivial. The deter-
E is translation invariant, D(E,X + a)
D(E,X)
a. The Polya condition (2.2.5) is replaced here by the con-
dition
(2.4.'1)
M(O) > 0 .
Also, an Atkinson-Sharma theorem holds for trigonometric terpolation. However, now one should consider cylindrical (wi th row
m of
in-
matrices
E proceeding row 1). In this case, when defining the
support for a sequence of row
i,
one can take supporting ones
from
the same row.
THEOREM 2.10:
A matJtix wLt.h
m > 2
Jtow.o
.oa.t.L~6ying
condition (2.4.4)
ic .t.JtigonometJtA.cally JtegulaJt i6 it hac no add 6equencec except thoce
198
LORENTZ and RIEMENSCHNEIDER
beginning in c.o./'.umn
O.
As further examples of results that hold for trigonometric interpolation, we mention:
PROPOSITION 2.11:
(i)
Let
i
l
, ..• ,i
pO.6e p .i.6 plt.ime. Let k > 0
60lt wh.ic.h the.Jte
be d.i66eltent ItOW.6 06 E and .6Up-
p
Rl (R ) be the .6et 06 even 2 aILe
(odd)
onu,.in PO.6.i.tiolU (ij,k), j=l, •.. ,p.
Then the 60Uow.ing inequa./'.itie.6 alte nec.e.6.6a1LY 601t the Itegu./'.aJtity 06
(2.4.5)
E
max (
(ii)
I Rl i , I R21)
<
(Johnson (12). A one ltoW matltix
.io .it ha.6
N +1
n
p
E i.6 ltegu./'.aJt i6 and only
one.6.in even pO.6.it.ian.6 and N ane.6 .in odd
po.6iUan.6 .
§3. COALESCENCE OF MATRICES
3.1. LEVELING FUNCTIONS AND COALESCENCE
The important
concept
of
coalescence for two adj acent rows of a matrix was introduced by Karlin and Karon (13). They also gave the Taylor formula, Theorem 3.3, though it was Lorentz and Zeller (29) C f
a
who firmly established
in that formula. Recently, Lorentz (22) put the method
althat
on
a
broader basis which allows multiple coalescence. Various applications of this method can be found in (13), (19) and (22) (see also §3.3, §5 and §6. 2) • Let
E be an
m x (n + 1) matrix, not necessarily normal, satis-
fying the Polya condition (2.2.7) E as a vertical grid of boxes. If i - th box in the
(see (3.1.1) below). We e
ik
interpret
1, then a ball occupies the
k - th col urnn. We place a tray of
n + 1 boxes under
199
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
the columns of the grid. Then the balls are permitted to
fall
from
the grid into the boxes of the tray in such a way that if the box immediately below is occupied, then the ball rolls to the first available box on the right. The condition (2.2.7) assures us that no ball will rollout of the tray. The distribution of balls in the tray constitutes the one row matrix obtained by coalescence of the m rows of E. It is to be expected that the final arrangement of the
balls
in
the tray is independent of manner in which the balls were allowed to fall. Here is an example of coelescence of a two row matrix:
1st row
1
0
1
1
1
0
0
1
0
0
0
2nd row
0
1
1
0
0
0
0
1
1
0
0
coalesced row
1
1
1
1
1
1
0
1
1
1
0
pre-coalesced 1st row
1
0
0
1
1
1
0
0
0
1
0
Fig. 1.
Let
m(k) = m , M(k) =~ denote the Polya functions k
of
some
interpolation matrix satisfying the Polya condition n ~
(3.1.1)
k=n
m(k) < n - n
l
+ 1,
o
< n
l
< n.
l
We shall use capital letters to denote the sum of a function,
=
k
°
e. g.
~ g(r). The level 6unQ~ion~ m , MO of m and M are the largr=O est functions g, G with integral values which satisfy
G(k)
(3.1. 2)
o
k
< g (k)
< 1, G (k)
~
g (r)
r=O
This is equivalent to the following: if
< M (k),
k
= 0,
•.• ,n.
200
LORENTZ and RIEMENSCHNEIDER
then
~{:
mO (k)
(3.1.4)
if
)Jk -> 1
if
)Jk
O.
a(M)
=
06 c.oUi.!>lon
The c.oe.66ic.ie.n.t
E, measures the distance of
aCE) of
M,orof the matrix
M to the level function
MO and is
de-
fined by n
(3.1.5)
E
aCE)
cdM)
(M(k) - MO(k».
k=O
(In the above interpretation of coalescence, this is the distance that the balls must roll.) A matrix
E ha-6 c.ot.e.l.!>ion-6 if
aCE) >
o.
The
basic properties of level functions are given in the following theorem.
THEOREM 3. 1 :
=
(M~ + M )0 2
(i)
(M
(ii)
«( M
(iii)
16 .the. 6une.tion
+ M )O 2
l
1
+ M )0 + M)o 2 3
=
(M
= 1
(M~ + M~)o
+ (M
2
+ M )0)0 3
Ml + M2 -6a.tl-661e.-6 (3.1.1),
.then
.!>o
doe.!>
M~ + M2 • (iv)
The
a (M
1
x
l
+ M2 )
(n + 1) matrix
EO with Polya functions mO, MO is called
the c.oa.e.e.-6ee.nee. 06 .the ma.tltlx E = El U E2 then the
E .to one ltow.
is a decomposition of
eoa.e.eM~enc.e
in E
06
generally,
if
E into two disjoint sets of rows,
.the ltow.!>
EO u E2 • From Theorem 3.1 we have 1
More
El
.to one ltoW
is
the matrix
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
201
(EO U E ) 0
(3.1. 6)
1
2
and
(3.1. 7)
E~ u E2
Moreover, the Polya condition (3.1.1) is 9assed on to
from
El U E 2 • In particular, we can consider the coalescence of two rows the matrix
in
E the i-th row and j - tl1 row. The i coalescence (E. U E.)o replaces these two rows by a single row, and l. J has collision coefficient a = a(E i U E j ). For a horizontal submaij trix
El of
E, say
E,
one row E2
we consider the coalescence (E
l
U
E~)
0
of
produced by coalescing a disjoint horizontal
to one row. The coefficient of collision
crease with the number of ones in
1. e. as
El
with
submatrix
aiEl U E~) canonlyinincreases in size.
(3.1. 8)
is called the
coe66~c~ent
06
max~ma!
co!!~~~on
06 El in the
matrix
E. We have deliberately not mentioned any ordering of the rows in E. Indeed, the disjoint sets of rows, El and E , could intertwine.This 2 is good for considerations involving real (or complex) regularity,but we shall need to consider the order of the rows for applications order regularity.
to
LORENTZ and RIEMENSCHNEIDER
202
3.2. SHIFTS AND DIFFERENTIATION OF DETERMINANTS: matrices
E1 and
E2 of a matrix
E,
+
l
M ) 2
°-
M~.
I
U
E ) 2
°
E\
example). We call
Then the two rows
-
~i < •••
El
and
°
the pJte-c.oa.le.6c.ed matrix with respect to
< ~~
respectively, then
an E2 •
E~ and E1 by II < ••• < lp
If we denote the positions of the ones in and
°
the row (E
aiEl U E ) = 0, and we obtain 2 from these two by simple addition (see Fig. 1 for
will be without collisions, i.e. row (E
s~
U E ) can be ob2 1 El be the 1 x (n + 1) matrix having
tained in the following way. Let the Polya function (M
For horizontal
~J'
< ~! , J
and
= 1, ... , p
j
by shifting the ones from positions ~. to ~~. J J The coefficient of collision for E1 U E~ is EO
El is formed from
I
(3.2.1)
+
a (E ) l
p 1: (JI, ~ j=l J
~
)
j'
Since the order of the rows is important for studying the order regularity of matrices, we now consider the placement of the coalesced rows in and
E
j
E and its effect on the determinant. Let be two rows of
alesced row
E.
E, and let
with respect to
~
E obtained from
(JI,i, ... , ~~)
Ei =
E .. The new (m - 1) x (n + 1) Ei and replacing row E by
Ei
j
.[n
are given
Xi by
x
j
X
J is
determinant
A(E,X) appearing in(2.1.3) ~l""
by derivatives of orders
we replace these derivatives by derivatives of ordersJl,i" .. replace
~
E. If
Xi omitted, we can obtain the
D(E,X) as follows. The rows of the matrix which correspond to
(E. UE.)O
j
is the matrix of c.oa.le.6 c.enee 06 Jtow Ei 1:.0 JtoW E X with
matrix
J
E by omitting row
the set of knots
be the pre - co-
'~~
'~p;
and
. The new matrix will have determinant (-l)O"O(E,X).
The .[n1:.eJtc.ha.ng e nu.mb eJt bring the rows of
is the number of interchanges required
A (E, Xl into the lexicographic order of
order inherited from Similarly, if
0"
E
to
from the
E. Ei
Ei is the pre-coalescence of
RECENT PROGRESS IN alRKHOFF INTERPOLATION
El with respect to
E\E , then we obtain the i
Qoa.ee¢Qe~ce
by omitting row
row
Ej
E
j
Ej
at
by
in6i~y).For
E
j ~i,
E , selecting some i
*
06
maxima.e
and replacing
practical purposes concerning regularity, this ma-
trix is essentially independent of
j. Indeed, by the nature of
~~, •.• ,~;
coalescence, the columns
submatrix of
mat~ix
Ei* (sometimes referred to as coa£.ucenceo6/(.ow Ei .to /(.ow
U
Furthermore, if
203
in
E \ Ei consist only of zeros.
E satisfies the Polya condition then each k, ~ * < k < ~ * l' q q+
E \ Ei of columns
the
vertical
( Q. 0* = -ll , is a Polya
matrix. Therefore, E * has a decomposition into Polya matrices having single column components in positions shows that the determinant the choice of
Q.i, .•• ,
The fo:rr.rula (2.2.10)
D(E*,X*) is independent (up to sign)
of
j .
EO to a pre-coalesced form 1
Motivated by the need to bring a row E , we define a ¢hi6t l
A : k ... k + 1
lows: a shift moves a one, position
Q.;.
ei,k+l
=
e
of a submatrix
1, of some row
ik
1. A shift is defined on
Q. q ,
As an example, again let
Q.'
q
e ik
El of E as fol-
i
=
in
El
into
the
1 only i f ei,k+l =0.
represent the positions of onesin
EO , El respectively; if Q.' 1 qO is the largest Q.~ with ~~ ~ Q.q' and if Q. is the first one of the sequence in E~ ending in ~qO' then ql there is a shift in some row of which ei ther increases Q.
or decreases
q
new matrix AE , 1 of
E
AEl
a(E ) by one unit with l reduces the
such that
Q.
ql
2. k 2. Q. qo
collision
Q.ql, •.• ,Q.qo unchanged. The coefficient
(3.2.ll
when
Such a shift is called a ~educi~g f..hl6t
•
A multiple ¢hi6t
A
= Al ... AS
simple shifts. It transforms
PROPOSITION 3.2:
Ei
k
Q. qO by one unit wi thout changing the remaining
(AE1)o replaces l
El with
= (.ei, .. · ,.e~)
Let
of order
El into a matrix
Ei = (.e , .• • ,lp) a~d l
be the p~e-coa.eef..ce~ce 06
Ej Ei
S
is a product of
S
AE 1 ·
be two /(.ow¢ 06 E, and
with Ite¢pect to
Ej .
204
LORENTZ and RIEMENSCHNEIDER
(a)
Fo~
Thell.e
hEi
ex~~t~
El
i.6 a
tiple -bhi6t mat~ix t~o
~n
o~dell.
S
C.oUL6~on
=
a
<
(Le.
U
i
S > y
06
(.bee
> 0) •
U E.)
J
a
c(
6OI!.
wh,[c.h
ll"" ,fp >1) ~ep~e-
-
= AI'" OM
the
j
A c.ctI!Jl..iu
E, then
= y(E l )
E ),
U
OJ!. d ell.
A
-bhi6t6:
ho~~zontal -bubmat~ix
(iiE ) 1
i
J.. i.6 unique, U hM
(~educ.ing)
A 06 Mde~
a(AE
A 06
have no c.oll~~ion.6. The ~h,[6t
by -bimpfe
a{E
-
mult~ple ~hi6t
only one
f l', ... ,f'p . Although
~entation-b
16
A 06
a~e
J
Ej
and
into
(c)
E.
AEi and
~ow~
(b)
muft~pfe ~h~6t
a
Ao.
any
(3.1.8)),
(E \ E ) doe-b not -ba.t,[-b6y the polya. l
mufthe
c.ondi-
n.
For a system
S of differentiable functions and the matrix A(E,X)
associated with
E, X, S, we want to find the partial derivatives
the determinant
D(E,X) of (2.1.3). To differentiate
respect to one of the parameters row of (2.l.3) which contains
D{E,X)
of with
xi' we have to differentiate
each
xi' This leads to
(3.2.2)
(and a similar formula for mixed derivatives), where the sum is taken over all representations of the multiple shifts of order i - th
row of
Xi as
in
Xi approaches another knot
D(E,X) as a function
x j • This behaviour can be de-
termined through the relationships between shifts, collisions, coalescence.
THEOREM 3.3:
Fo~
Xi
~
xj '
D{E,X)
ha..6 the
Taylo~
(Xi - x.) a (3.2.3)
the
E.
We would like to examine the behaviour of of
S
D(E,X)
a!]
(- If C D(E,X)
expa.n-b~on
+ •.. ,
and
206
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
whe.fc.e
C i.6 de6.i..ned in PJtopo.6Ltion 3.2(b), a
a (E.
~
U
E.).
J
and
o i.6 the inteJtchange numbeJt.
For polynomial interpolation, when
S is the system (2.2.1),we
have
THEOREM 3.4:
(ii)
(i)
16
El i.6 a hOJtizontai .6ubmatJtix 06
.i...6 a polynomial in the vaJtiable.6
x.
06 joint degJtee not gJteateJt than
y (E )· l
In a .6ingle vaJtiable D (E,X)
x!
with
coJtJte.6ponding to
El
coJtJte.6 po nding to a Jtow E. in E, 1
x.
~
ha.6 the highe.6t teJtm
~
D(E,X)
(3.2.4)
1
E, then D(E,X)
YT
y = y (E ) and i
(- 1)0* C* D(E*,X*) + •••
3.3. APPLICATIONS OF COALESCENCE
gularity, we take rows
Ei
and
de6ined by the maximal coale.6cence.
E*
For order regularity or order sinE
j
to be adjacent in (3.2.3) whereas
for real or complex regularity this is not necessary. This remarkapplies throughout this section.
3.3.1.
Suppose that
E is a normal Polya matrix. We can give
simple proof [19) of Theorem 2.2. If in (3.2.3), then the same is true of of two rows, we finally reduce
a very
D(E,X) is not identically zero D(E,X). Byrepeatedcoalescences
E to the one row form
{l,l, ... ,l},
for which the determinant is the Vandermonde determinant of the system
S
=
{go, .•. ,gn}. Therefore,
THEOREM 3.5:
16
E i.6 a nOJtmal Polya matJtix and the Vandvtmonde. de-
teJtm.i..na.nt 06 the .6y.6tem S i.6 not identica.lly Zl2.Jto, then not identically zeJto.
D(E,X)
i.6
206
LORENTZ and RIEMENSCHNEIDER
3.3.2.
If the determinant
changes sign, then so does
THEOREM 3.6:
16 one.
on
D(E,X) or
D(E*,X*) in (3.2.3) or (3.2.4)
D(E,X). Hence [13], [19]
E
the. c.oa.£.e..6c.e.d ma.tlt.ic.e..6
E* i.6
olt.
!.ltlt.OYlg£.y
.6iYlgu£.aJt, the.n .60 i.6 the. oJt.igina.e matlt..ix E.
3.3.3.
We can exploit the interchange number
(3.2.3)
(and even in (3.2.4) [22]), by comparing them after coalescence
0
occuring in formulas
of several rows in different ways.
THEOREM 3.7:
Le.t
It.OW.6 F l , ... ,F q 06 E be. c.oale..6eed q - 1 time..6, in two di66elt.ent way.6, to pnoduee. the .6ame. .6ingle. now. 16
0 ,,,, ,Oq_li 1
0i, .. .,o~_l
q.:. 3
a.Jte. the. eoJtJte.6ponding inteJtehange VlWI1be.lt.6,
and i6
(3.3.1)
*
0 1 + ... + 0q_l
0i + ••• + o~_l
(mod 2)
then E i.6 .6tJtongly Jteal .6ingula.Jt. 60.11. any .6Y.6tem S. The. .6ame e.eU.6ion ho£.d.6 60.11. .6tJtong oJtde.Jt .6ingulaJtity i6, in addition, F , ••• ,F l q
the JtOW.6
aJte adjaeent and a.l.e eoale..6eenee.-6 aJte in one diJteetion (Le.
to Jtow
It.OW i
c.on-
i +1, on Jtow
i
to Jtow
+ I
il.
The last statement is required since the coalescence of row to row
i+l contributes the sign from (xi - xi+l)Cl. This
i
contribu-
tion is the same on both sides if all the coalescences have this same direction. In general, a less simple statement, taking into the collisions for coalescences tions of the
q -1
i
to
i + 1, is true when the direc-
coalescences are free (see §6.2).
We give a more explicit formulation of Theorem 3.7 rows of
Ei
Fl
=
account
(R.i'···'JI.~), F2
=
(Jl.l'·"'JI.~) and
mean the posi tions of the ones of row
for
three
F 3 • By (Fl s ' we
F pre-coale:3ced wi th
respect
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
207
to row Fs' and by (F)st - the precoalescence with respect to (F Further, we adopt other
s the convention that two sequences following
mean that their elements should
be written out in the
order. By considering the interchange numbers for
PROPOSITION 3.8:
U
Ft)O. each given
the coalescences
The matltix E i-6 -6tltongty lteat -6ingutalt i6 it con-
tain-6 thltee ltOW-6 60lt which the two -6equence-6
and
Thus, a matrix can have three rows that are so bad that it singular for any arrangement of ones in the other rows, and for systems
3.3.4.
is all
S.
By properly selecting knots
PROPOSITION 3.9:
16 y. -
(3.3.2)
J.
i-6 odd, whelte
Yi
xi' Theorem 3.3 and 3.4 give us
= y(E i
06 cotti-6ion 60lt ltOW i
)
in
polynomial inteltpotation,
l: Il j~i ij
and
Il ij
E, then
= Il(E i E
U Ejl alte the coe66icient-6
i-6 -6tltongty lteat -6ingutalt 60lt
208
3.3.5.
LORENTZ and RIEMENSCHNEIDER
We have restricted ourselves in (3.2.3) to the expansion
D(E,X) in one variable for sake of simplicity. If several knots
of xi
approach
x., we obtain multiple coalescence. The expansion will then J contain, as its main term, a form of order a in several variables. If this form changes sign, the matrix
E must be strongly singular. This
requirement is particularly meaningful for real singularity when the values of the variables of the form are unrestricted.
3.4. EXAMPLES: trices
E and
Let
E be obtained from
E by coalescence.
The
E can be regular, weakly singular or strongly singular
in logically nine possible combinations. Theorem 3.6 rules combina tions:
ma-
E strongly singular wi th
E being regular
out or
two
weakly
singular. All the other combinations can occur. Indeed I by coalescing the matrices
E
l
, E2
and
E may be regular when examples of
E3 of (2.4.1) to two row form, we see that
E is any of the three types.
E. Kimchi and N. Richter-Dyn [16]
The
following
are less trivial.
o o o 1 o 1 o o o o o o o o
1
1
o o
o
1
o
1
o o o
o
1
100
0
o
0
1
0
1
o 1
0
0
1
0
0
1
1
o
I
0
0
0
o
0
I
000
0
0
00101
0
0
0
11000
0
1
(3.4.1)
E
5
0
0
=
The matrices
0
E and
lar; and the matrix
ES are weakly singular; the matrix E4 E6 is strongly singular.
is regu-
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
209
§4. INDEPENDENT KNOTS The connection between the concept of an odd supported sequence and the extended form of Rolle's Theorem was exploited in
§ 2.3
to
obtain a simple proof of the Atkinson-Sharma theorem. This simple connection suggests that a more detailed study of the information gained from Rolle's theorem is warranted. The method of independent was formulated by Lorentz and Zeller [28] and developed
knots
further
Lorentz ([18], [20]), in order to study singular interpolation
by ma-
trices. Let E be an
mx n +1
differentiable function on
x
=
(xl' ... ' x ) m
C
interpolation matrix, and f be an n-ti.rres [a,b)
which
E
and
and its derivatives specified by (4.1), we
can
[a,b 1, that is, let
(4.1)
f(k)
From the zeros of
f
is annihilated
f
(x.)
satisfy
1
1
by
in
E.
derive further zeros by means of Rolle's theorem. A selection
of
a
complete set of such zeros is called a "Rolle set" of zeros. A Rolle. .6e.t
R 601t a 6u.n.c.t.ion.
f
annihilated by
E, X is a col-
, k = 0,1, ... ,n, of Ro.e.le. .6 e.t.6 06 z e.ltO.6 (with mul tiplicik ties specified) 601t e.ac.h 06 the. de.lt.ivat.ive..6 f(k) selected inductively lection
R
as follows: The set (4.1). If
Ro
consists of the zeros of
f as specified
in
RO, ... ,R k have already been selected, then we select Rk + l
according to the following rules: 19
A zero of f(k) in also a zero of
29
All zeros of
Rk of multiplicity greater than one
f(k+l) with its multiplicity reduced by one. f(k+l)
(including multiplicities) as slJ8cified
by (4.1) are included in 39
is
For any adjacent zeros
ct
Rk + l • I S of
select, if possible, a zero of
f(k) belonging to R , we k f(k+l) between them subject
210
LORENTZ and RIEMENSCHNEIDER
to the restrictions: (a) If the new zero is one of the
xi' then it is not liste1in
(4.1), or (b) there is an additional multiplicity of f(k+l)
zero of
t
is the multiulicity of
is defined as follows.
=
f(k+t) (xi)
as a zero of
xi
as a
We add to (4.1)
the~
0, and determine the multiplicity of xi
f (k+l) from these equations. This may connect
two sequences in than
of
f(k+l) given by (4.1), then the multiplicity of xi
Rk + l
tion
as a zero
which is not acknowledged by (4.1).
(c) In the event of (b), if
in
Xi
E and prescribe
a
raul tiplici ty
larger
t + 1.
If a zero does not exist subject to the restrictions in 39, then we say that a
lo~~
occurs at step
k + 1. A Rolle
set
constructed
without losses in any of its steps is called maximal. The function f may have many Rolle sets, some of them may be maximal, while
others
are not maximal. Some properties of Rolle sets are immediate consequences of the selection procedure. First of all, the only multiple zeros of f(k+l) in
R + k l
are among the points
xi
in
X. Secondly, the extended fom
of Rolle's theorem shows that a loss will not occur if the rows of E corresponding to
xi between adjacent zeros of
ported sequences.
LEMMA 4.1:
Rolle
~et~
Rk contain no odd sup-
(This was the connection used in §2.3.) We have
16 -the
mabtix E
06 a 6unction
f,
ha~
no odd 6uppolt-ted
annihila-ted by
~equenc.e~,
E, x, a.lte
t:henaU
ma.ximal.
The number of Rolle zeros in a maximal Rolle set can be determined by induction:
LEMMA 4.2:
16
f
i~
annihilated by E,X, then 6011. eac.h k, k=O,l, ••• ,n,
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
the numben 06 Rotte zeno~ 06 i~
6uncUon f S
Let
=
at
tea~t
211
in a maximat Rotte ~et
f(k)
be a system of n-times continuously dif-
ferentiable functions which are linearly independent [a,b]. A set of knots
with respect to the system every polynomial
P
in
the
M(k) - k.
{go, ••• ,gn}
subinterval of
60n
S
[a,b]
XC
on
each
is called independent
S if for each interpolation
annihilated by
open
matrix
E,
has a maximal Rolle set
E,X
of zeros. Using a weak form of Markov's inequality, which is
valid
each system S, it is possible to show that Rolle zeros for can be selected away from the zeros of
for
p(k+l)
p(k). More precisely,
(see
[37) for algebraic polynomials)
LEMMA 4.3:
~uc.h that i6 60n
~ome
i~t~
Thene
i~
i3 - a
inMea~ing
a monotone
2. R., a .::. a
6unc..t.i.on 6(,I',),O'::'6(R.) < %R,
< i3 .::. band
potynomiat p in Sand
p
=
(k) (a)
p
=
(k) (13)
k, k = O, ••. ,n -1, then thene
~, a + c(R.) .::. ~ < 13 - cU,) OM which
0
ex-
p(k+l) (~) = o.
For simplicity and without loss of generality, we take [a,b] = [-1,1.1. From Lemma 4.3, one derives
p, 0 < p < 1, thene i~ a ~equ.enc.e
THEOREM 4.4:
Fon eac.h
with
having the 6ottowing pnopenty. Let
p < Yl
[-p,p] u {± y }, and s ~upponted ~equenc.e~
Then each potynomiat RoUe
E
be
~u.b~et
a
be an intenpotaLion matnix which ha~
in the p
X
{±YS};=l
now~
c.onne~ponding
in S, annihitated by
to
knot~
E,
X,
06
no odd
Xi' - p,::,xi,::,p. ha~
a
maxima!
~et.
For the proof, the points
ys
are chosen inductively very close
to 1 so that the selection of Rolle zeros in step 39 is always sible. It is essential for the proof of Theorem S.l - indeed, the main idea - that the "harmless" knots
Xi
posit is
can be made variable in
212
LORENTZ and RIEMENSCHNEIOER
an interval
(- p, p), arbitrarily close to (- I, 1). Clearly, any knot
set X contained in
{±
Ys}
is independent with respect to the sys-
tem S. Theorem 4.4 gives another simple proof of Theorem 2.2 (Windhauer [47] or [20]). Assuming that
E is a normal Polya matriX,
we
take
X c {± y } and show that the pair E, X is regular. Indeed, a polys nomial Pn annihilated by E, X is identically zero by a standard ap-
plication of Lemma 4.2. As has been pointed out in [19], Theorems 2.2 and 2.4 extend to equations of the form
(4.2)
where
1) ,
D.
J
are certain differential operators of order I, and S is the
Chebyshev system connected with these operators (for a definition of S, see [15, p. 9, p. 378- 379]).
§5. CLASSES OF SINGULAR MATRICES
The Atkinson-Sharma theorem provides only a sufficient
condi-
tion for the regularity of matrices; the condition is not necessary. However, a good guiding idea is that. this condition
is
"normally"
necessary, or at least necessary under some simple additional conditions. All theorems of this section refer to
inte~poiation
by
aige-
b4aic poiynomiai6 and 04de4 6inguia4ity.
THEOREM 5.1:
An
mx (n+l) nMmai Bid.h066 mat4ix i-l> -I>tMngiy 6ingu.-
la4 '<'6 i:t: co ntain,b alLOW w.<.th plLeci6 ely a ne odd 6uppoued 6equenc.e (aU
othelL ,bequenc.e6
06
thi6 lLOW being even alL not 6uppolLted).
The simplicity of the statement of Theorem
5.1
belies
the
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
213
difficulties of its proof. This theorem was first proved in the special case when the interior row consists of a single supported one by Lorentz and Zeller [28 I, and in full generality by Lorentz
in
[ 18]
and in [20]. The theorem also appears in Karlin and Karon [13] as
a
consequence of the method of coalescence, but the proof offered there contains an essential gap. A proof may be based [20] on Theorem 4.4 of the method of dependent knots. The row containing a single odd supported is assigned a
variable knot
taken among the zeros of
±y
s
the root
sequence
y, - p < y < p, while the other knots are
of Theorem 4.4. It is necessary to study
k
p (k) (x) = _d_ p (x,y), where k dx
depending on the parameter
in-
P is some polynomial in
y, and to show that for some
x =y. This is done by examining all zeros of
y
the x,
it
has
p(k) (x)
in
(- 1, 1) •
The matrix
E3 in (2.4.1) shows that Theorem 5.1 fails to hold
for matrices with two odd supported sequences in a row. For matrices with an odd number of odd supported sequences in one row some special cases are known (see Section 6.2). However, we have the example
of
the matrix
1
(5.1)
1 1
o o o o o 1 o 1 o o o o
which is weakly singular. This example appears in [5 I
and has maxi-
mal defect, d = 2 (see (2.3.3)). It would be interesting to krDw whether a three row Birkhoff matrix with an odd number of odd supported
se-
quences can be regular. A second class of singular matrices is found by restricting all but one row. A row of the interpolation matrix
E is ,6.[mple i f it con-
tains a single entry one and remaining entries zero. The matrix E is
214
LORENTZ and RIEMENSCHNEIDER
atmo~t ~lmpte
if all its rows but at most one are simple.
In the canonical decomposition (2.2.8) of an almost simple matrix, each component matrix is almost simple. More interestingly, the almost simple matrix E is strongly singular if (and only if) one of its component matrices is strongly singular. This observation is used in two ways. First of all it is enough to study the strong singularity of almost simple Birkhoff matrices. Secondly, one can assume the existence of a "smallest" regular or weakly singular al1lost simple Birkhoff matrix EO containing odd supported sequences. Theorem 5.1 and methods of coalescence are used to obtain a contradiction for the matrix EO' In this way one obtains (Lorentz [19]):
THEOREM 5.2:
no odd
An atmo~t ~lrnpte Bl~Rh066 mat~1x 1~ ~eguta~ 16 1t
~uppo~ted ~equenee~
ha~
1~ ~t~ongty ~1ngula~ othe~w1~e.
and
An interesting special case of this was established by K. Zeller in 1970. Let E be a normal interpolation matrix with at least three rows and which has non-zero entries only in an interior row io andin
positions of all zeros in row
o
< ••• < k < n be the o p i o ' As a corollary to Theorem 5.3, we
column zero of the other rows. Let
< k
have
THEOREM 5.3:
A
rnat~1x
06 the type
ju~t de~e~1bed 1~
k] , - k]'-l , j = l , , , . , p
~eguta~
16 and
a~eodd.
As long as the question of regularity of an interpolation
ma-
trix has not been solved, i t is natural to ask whether it can be solved "most" of the time. That is, what is the probability that a normal interpolation matrix is regular? A first step in this direction is to count the number of Polya matrices, or Birkhoff matrices, among
all
m x (n
and
+l)
normal interpolation matrices. Let
B(m,n) denote the number of all
A(m,n),
P(m,n)
m x (n + 1) normal interpolation,Polya
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
and Birkhoff matrice.s re.spectively. Here
215
we allow some rows
of the
matrix to consist only of zeros, and assume 1 < m < n. vIe shall outline the results of Lorentz and Riemenschneider [25J.
THEOREM 5.4:
One
ha~
~he
equat~on~
m(n+2»)
1 =n:+2'"
(5.2) A(m,n) = (m(n + 1») , P(m,n) n +1
(
I
B(m,n)
n+l
=n1
In particular, one has the strong asymptotic relations
P(m/n) -
! 2 [m ~ 11m A(m/n),
n
-
n
~
00
j
(5.3)
Ir ~11
B(m,n) -
'-
m
2m
P (m,n), n ~
DO
.1
The first relation (5.3) shows that (for large
n)
almost all normal
matrices do not satisfy the Polya condition; the second relation shows that there is a large proportion of Birkhoff matrices among the Polya matrices. Theorem 2.2 implies now that almost all normal matrices are singular. However, whether a given matrix satisfies Birkhoff condition is easy to determine. Thus, should be: What is the probability that a normal
the
the proper
Polya
or
question
m x (n + 1)
Polya
(or Birkhoff) matrix is regular? We assume that our
x
n
normal matrices satisfy the condi ti6n
(I + 6) n Ilog n < m < n,
(5.4)
where
m
a
> 0
is a constant.
THEOREM 5.5 [25]:
Fo~
each
E > 0,
the~e ~~
an
nO
216
LORENTZ and RIEMENSCHNEIDER
60llow~ng
(5.4) but
w~th
n > nO ' at m04t
eB(m,n)
00 them have
THEOREM 5.6 [25]: 60~
m04t
Among all
p~ope~ty.
n ~ no' eP(m,n)
a~e
eB(m,n)
B~~kh066
mat~~~e4
a~e ~egula~.
What
~4
mo~e,
all
4uppo~ted 4~ngleton4.
Fo~ ea~h
among all
B(m,n)
e > 0, the~e ~4 an
P(m,n)
Polya
mat~~ce4
nO
= note)
40 that
4ati40ying (5.4),
at
4egula4.
§6. THREE ROW MATRICES
6.1. ALMOST HERMITIAN MATRICES
It is not clear in what respect the
theory of regularity becomes simpler for three row matrices. The theorems on coalescence are not strong enough to reduce the general case to this one. Furthermore, we shall see that even very simple three row matrices present considerable difficulties. The results of
§6 refer
to order regularity. We shall study
3 x (n + 1)
normal Birkhoff matrices with the
following placement of ones
elk
I, 0 < k < p; e 3k
=1,
0 < k < q;
(6.1.1) 1.
Then
p + q + 1 = nand
also assume that
k2 < n; without loss of generality, we
shall
kl < k2 - 1, P .5. q. For the knot set, we shall take
X={-l,x,l}. One of the smallest matrices of type (6.1.1), E3 of (2.4.1),has served to show that regular matrices can have odd supported
~aes.
Generalizing this example, several authors (DeVore, Heir and
Sharma
[6 ] , Lorentz and Zeller in [19), and Lorentz, Stangler, and
Zeller
217
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
[26
J)
studied matrices of the form (6.1.1). It was hoped that in this
way the problem of regularity could be completely solved for at least one nontrivial case. The incomplete success of this attempt leads one to believe that i t is hardly possible to express the property of regularity in terms of simple properties of elements The method of the paper [6 classical Jacobi polynomials
1
e
ik
of a matrix E.
was to apply known facts about the
pea,S) (x). In [19], n
the
alternation
properties of zeros of derivatives of the polynomial (1 + x) p (1 - x) q were used. The first method gives more detailed information while the second method is applicable to wider classes of matrices.
THEOREM 6.1 [26 nec.e~~aJty
J:
In oJtde.Jt that the. matlL-tx (6.1.1) be. Jte.g ulaJt , li.u.,
that
p + q + 1,
(6.1. 2)
k2 > q
(6.1. 3)
In the
c.a~e.
(6.1.2),
E -t~ JtegutaJt
-t6 and onty -t6
(6.1.3), the. ma.tJt-tx c.an be. eitheJt JtegulalL oJt taJt-tty
06 .the. ma.tJtix
tlLix w-tth
paJtame.te.Jt~
p = q. In the. c.a~e.
~ingutaJt,
(6.1.1) -tmptie~ .the lLegulaJtity -tn~tead
ki' k2
06
k l , k2
but the Jtegu-
06 a
~imilaJt
ma-
-t6
OJt
{Note that inequalities (al have been stated incorrectly in the paper [26
J, namely with
ki
~
kl < k2
~
ki. This error occured in the
last lines of the proof in [26, p. 435J. The inequality (5.5) should
218
LORENTZ and RIEMENSCHNEIDER
be replaced by the reverse one:
"(5.5)
YI+1(A) 2. yiP,) for some 1.")
The proof of this theorem is by the "chase method". As a didate for the nontrivial polynomial annihilated by
can-
E, X, we take
P(x,A)
We let
A change continuously from - '"
to those zeros of
P
(k ) l
and
P
(k ) 2
to +
00
and study what happens
whose existence is guaranteed by
Rolle's theorem. The matrix is singular exactly when one of these zeros overtakes the other at some
xO' for then
P
(k ) l
(xO,A)
=p
(k ) 2
(xO,A) =0.
The second part of Theorem 6.1 means that in the triangle given by
P
y =
A(X),
~
x, Y
~
q, x + 2
~
y, there exists a monotone increasing
with slope at most one, so that
on the curve, and regular below it. For
is singular above
E
p
= 1,
~
and
this curve was
dis-
covered in [6 I, and was shown to be the upper branch of the ellipse
(6.1. 4)
(q
+
2)
(x + y -
1) 2 -
4 (q
+
1)
xy
0;
moreover, E is weakly singular on this curve and strongly
singular
above. For some values of the parameters, the statements
of
6.1 were proved also in (6 I and [19]; in addition, it was
Theorem possible
to distinguish between strong and weak singularity. One general case of weak singularity has been found to date, namely when
q = p + 1,
kl + k2 = P + q + 1. For more details, consult the paper of
DeVore,
Meir and Sharma [6 I.
6.2. CRITERIA BASED ON COALESCENCE Polya matrix. For the knot set
X
Let
E
be a
3 x (n + 1)
{O,x,l}, the determinant
is a polynomial in x. Clearly, E will be strongly singular sign of
D(E,X) is different in (0,£) and (n,l) (e:, l-n
normal D(E,X) .if
the
sufficientlysmall~
RECENT PROGRESS IN BIRKHOFF INTERPOL.ATION
219
This simple observation is the essence of several criteria strong order singularity of
E,
although the statements
for
the
themselves
appear totally unrelated. There are several equivalent forms in which this comparison of signs can be carried out. One of them is given by the special caseof Proposition 3.8 when the matrix E consists just of the three ordered rows
F I' F 2' F 3 (of course, the interest of Proposition 3.8
is not
limited to this case). Another form is one given by Karlin and Karon (.t ,.t , ... ,.t ) = F2 be the positions of the q l 2 1 1 3 3 (.tl, ... ,.t ) and (.tl, .•. ,.t ) be their posiq q
([13, Theorem 2.3): Let ones in row 2, and let
tions in the pre-coalescence of row 2 with respect to row 1 and
row
3 respectively.
PROPOSITION 6.2 (Karlin and Karon) : ma..tJr.).x, :then
E
16
E
)..6 a. 3 x (n + 1) nOJtma..t po.tya.
)..6 .6.tJr.ong.ty .6).ngu.tcOt when
.t~-l
.t~-l (6.2.1)
q
~
j=l
{
J~
M(.t. -1) +.t. +.t.1 +.t.3 + J~ J J J J k=l. J
In 6a.c.t, :th-i..6 l>um need only be :ta.f<en oveJt
} e3 k
k=.t. J j
:the 6-i.Jt.6t e.temen:t.6 06 odd l>uppoJtted .6eque.nc.el>
OM
06
::: 1 (ood 2) •
'
wh-i.c.h :the.t j
a.Jte.
Jtow 2.
The method of Karlin and Karon was to analyze the signs of the determinants involved by using arguments from the
theory
positivity due to S. Karlin. For the last statement of the one verifies that adjacent ones, contribute
0 mod
2
.tj+l
of
total
theorem,
.t. + I, or unsupported ones, J
in (6.2.1).
Both Proposition 6.2 and Proposition 3.8 are consequences
of
coalescence and the use of the Taylor's formula (3.2.3). This can be. explained best of all if we define "directed coalescence" as follows. If
F , F2 are two adjacent rows of E, we define the directed l alescence F 1 => F 2 as the matrix derived from E by replacing
corow
LORENTZ and RIEMENSCHNEIDER
220
F1 by its pre-coalescence,
°1 ,2'
number,
Pl ,
with respect to
of the coalescence
F1
~
F2
F 2 . The
interchan~e
is the number of
inter-
changes needed to bring the sequence of integers
F1 , F2 into natural order. (Here a row F is represented by the positions of the ones as in § 3.) In a similar way,
F1 .. F 2
replaces row F 2 by its pre - co-
alescence wi th respect to F1 and has the interchange number Then (by (3.2. 3)
° 2,1 .
)
(6.2.2)
where
,2 = u(F U F ) is the coefficient of collision. For calcu2 1 l lating the interchange numbers of further directed coalescences, the u
positions of the ones in
F1 " F2
and
F2
F1
=>
are assumed to bein
their natural order; then, for example, cr 3, (2, 1) = cr 3,( 1, 2 ) directed coalescences F3 ~ (F .. F ) and F3 ~ (F l ~ F ) l 2 2 tive1y. Let
Ci
for the respec-
be the sum of the exponents of powers of (- 1) giving the
signs mentioned at the beginning of this section. We can give several equivalent expressions for
Ci
(mod 2) by means of directed
coa1es -
cences and Theorem 3.3. For example, to obtain Proposition 3.8, estimate the sign of
O(E,X)
near
0 by means
F ) * F , and near 1 by means of 2 3 this way (F l
~
(6.2.3)
Fl
=>
of
the
we
coalescences
(F 2 * F ) andobtainin 3
o - 01,2 + cr (1,2),3 + 02,3 + 01, (2,3) (mod 2).
To obtain Proposition 6.2, we consider the coalescences (Fl " F2) .. F3 and F " (F => F ); this gives 2 l 3 (6.2.4)
221
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
(to see the equivalence of (6.2.3) and (6.2.4) directly, (6.2.2) and an extended form of (3.1.7». 8
=1
Equations
one
uses
(6.2.1)
(mod 2) can be shown to be equivalent by the careful
and
computa-
tion of the collision and interchange numbers of (6.2.4) by means of the quantities in (6.2.1). Similar ideas give a special case of Propositions 3.8 and found by Sharma and Tzimbalario (42). Let Birkhoff matrix with ones in positions and let
Fl
E be a
3 x (n + 1)
=(ll,···,t~),
6.2
normal
F3=(tl', ••• ,l~'),
be the positions of the zeros in row 2.
PROPOSITION 6.3:
16
kl > max(l~ - p, C' r
r)
a.nd i6
p ~
(6.2.5)
j=l
.then
E
(k r + , - kJ') + pr - 1 (mod 2), J
i-6 -6.tJtongly .6inguta.lL.
Here we use the coalescences (F 1 ~ F 2) ... F 3
and F 1 "* (F 2
<=
F 3)
to obtain
(6.2.6)
° 1 ,2 + a l ,2 + ° 3 , (1,2) +° 3 ,2 +°1, (3,2) +a l , (3,2) =8 (mod 2).
Under the assumptions of the theorem, it is easy to verify that (6.2.5) and
8
=1
(mod 2) are equivalent by (6.2.6). Sharma and Tzimbalario
proved Proposition 6.3 by using properties of special determinants. An in teres ting special case of Proposi tion 6.3 is when the first and last rows of
E are Hermitian. If, in addition, there is only one
element in the Hermitian first row, the last row is Hermitian,
and
there is an odd number of odd supported sequences in E, then Proposi tion 6.3 implies that E is strongly singular (see Passow (35).
222
LORENTZ and RIEMENSCHNEIDER
§7. BIRKHOFF'S KERNEL 7.1. DEFINITION AND PROPERTIES
Birkhoff's kernel, K(t)
= KE(X,t)
a spline function (piecewise polynomial) intimately connected
is
with
Birkhoff interpolation by algebraic polynomials. It appears in the integral identities (7.1.4) and (7.1.6); the second of the these gives a remainder for the interpolation formula. The deepest
theorem
of
Birkhoff in his famous paper [3 J was an estimate of the number of zeros of his kernel. It is given here as a theorem about the number of changes of sign of splines with discontinuities linked to the matrix E.
As usual, let
for
r > 0
mean
ur
if
u < 0, except that it is not defined if both the kernel
u
u .:: 0, and
= 0,
r
= O.
We
o
if
obtain
D(E,X) of (2.2.2) by ren-k-l placing the elements of the first col\.llU'l in (2.2.2) by (xi-t) + /(n-k-l)!:
(7.1.1)
If
KE(X,t) from the determinant
K(t) = {
n-k-l (xi - t) + (n - k - l)! '
n-k-l -k xi xi (n - k -1) ! ' •.. , (-k)!
Dik(X) are the algebraic components of the first column elements
of the determinant (2.2.2), defined for
.=
1, then
n-k-l ( xi _ t) +
(7.1. 2)
(n - k - 1) !
If the knots are ordered, xl < '" is a polynomial in
e ik
t
K(t) m then the determinant in each of intervals (-00, Xl) , (~,x2) , .•. ,(xm' +(0) ,
hence a spline. One sees that
< x '
K(t) is zero outside of [xl,xml. {The
same applies to the derivatives of the ken'lel. '!hus, K(j) (t), j =0, •.. ,n-l, [A j ,B j J , where A. (or B j ) is the smallest (or ] } = 1 for some k ~ n - j - 1 the largest) of the x.~ with e ik Integrating (7.1.2), we obtain
is zero outside of
.
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
(7.1. 3)
Let
223
D(E,X) .
An denote the class of all (n - 1) -times continuously dif-
ferentiable functions
f
on
[a, b) for which
f (n-l) is
absolutely
continuous.
THEOREM 7.1 (Birkhoff's Identity [3) ): ZeJWf., in. .the laf.,.t column..
FOIL ea.ch
f
E
Le.t
E be a. YWItmcU'. mcU'lUx w.uh
An an.d each f.,e.t 06 k.n.o.tf., X in.
[a,b) ,
tf(n) (t)K(t)dt.
(7.1. 4)
a The simplest special case of (7.1.4) is Taylor's formula
with
integral remainder. From this theorem, we can obtain mean f E Cn[a,b). If
value
formulas.
Let
K(t) does not change sign, then by using (7.1.4) and
(7.1.3), we can obtain
The same is true if
K is of arbitrary sign, but
of degree not exceeding e
ik
n.
f
is a
polynomial
In both cases, the relations f(k)(x.) = 0, ~
imply fen) (~) =0 for some ~, xl < ~ < x ' m Suppose now that E is a normal Polya matrix without any
= 1 and D(E,X)
strictions and
to
X is a set of knots for which D(E,X)
there exists a polynomial
p(k) (x.)
n
~
P n of degree at most
f(k) (x.) , ~
1.
F
re-
O. If fE An+l'
n for which
224
LORENTZ and RIEMENSCHNEIDER
We would like to get a formula for the difference extend
E by adding a O-th row with only a single one I e OO
E to
and by adding an (n + 1) -st column of zeros. Let the
xi
kernel
then
I
f(x) - Pn(x).
=1
I
x be different fran
X is the set of knots obtained by adding
= KE(X,t)
K(t)
We
x to
X. The
is the Peano ke~nel of the interpolation.
One ha~ 60~
THEOREM 7.2 (Birkhoff):
1
(7.1. 6)
D(E,X)
X C (a,b)
Jb f (n+l) (t) K (t) dt a
{A similar formula holds for the difference f(k} (x)
- p!k} (x)
if we insert the one in the new O-th row in position k, Le. e
O,k
I
=l.}
7.2. NUMBER OF ZEROS OF SPLINES
The deepest theorem of Birkhoff in
[ 3 I counts the number of
06
chang e~
I.>-ig n of a kernel
KE
estimate is also valid for other splines (D. Ferguson [8 eralization by Lorentz [21] concerns the numbe~ A function
S
on (-
00 ,
if there are points gree
~
< xm
Xl <
so that
THEOREM 7.3 [21]:
be a l.>p.Une 06
= (e ik )
= 1, ••• ,m,
at
x .• J.
I6
Let
be an
and that
S
m x (n + 1)
e
ik
=1
P -il.> the numbeIL
-i6 the numbe.IL 06 one.-6 -in the-iJr. mult-ipli.c-it-iel.»
(7.2.1)
gen-
of splines.
S
is a polynomial of den
for
at
i , and is zero outside of (xl,X ). Let [a,b] be the smallest m S vanishes.
i
ze~O-6
A
+ (0) is a spline of finite support of degree n
interval outside of which
E
1).
n on each interval (xi,x + ), has exactly degree i l
least one
Let
06
(X, t). This
06
matIL-ix
deg~ee
06
wheneveIL
06
n -1 w.i.;th "nato Xl < ••. < xm•
ze~ol.>
and onu
1.>0
s(j), j =n -k -1, hal.> a jump
odd ~uppolL.ted 6e.que.nce.6 06
E, and
Z -il.> the
S -in (a,b), then
Z
that e.J./n =0,
numbe~
E,
N + 1
06 zeIL06 (c.ount-ing
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
225
An essential part of this theorem is the definition of the multiplicity on which
of a zero
z may be a maximal interval (x. ,x.) ~ J S vanishes, or a point, which mayor may not be one of the CI.
z. A zero
knots. The definition of on
z,
then
CI.
=
CI.
in [211
min(i3,y), where
S to the right and to the left of for
min,
z, then
is a follows. If S is i3
and
z.
yare the multiplicities of
(In [21], max
but this was corrected in [201.) CI.
1
=
if
S changes sign on
z,
If CI.
stands erroneously
S is discontinuous on = 0
otherwise. In [111 ,
Jetter gives another apparently larger count for
CI..
A different ver-
sion of Theorem 7.3 is given in Schumaker [401, with defini tion of
CI..
But also his
p
continuous
still
another
in (7.2.1) is different, so that his
formulation does not contain Theorem 7.4. Of course, for point zeros
z of
S that are not also knots, a
is the ordinary mul tip1ici ty; and each change of sign of
S is a zero.
The proof of Theorem 7.3 uses a type of Rolle's theorem for splines.
7.3. APPLICATIONS OF THE BIRKHOFF KERNEL
Let
E again be a
normal
Birkhoff matrix. From Theorem 7.3 we derive
THEOREM 7.4 (Birkhoff):
the.
~e.Jtne.l
KE(x,t)
.6e.que.nc.e.;., 06 Let
The. numbe.Jt 06 c.ha.nge.;., 06 ;.,-ign -in
doe.;., not e.xc.e.e.d the. numbe.Jt
p
06 odd
[a,b]
on
;.,uppoJtte.d
E.
p =0
and
X be given. If
from (7.1.3) we see that
R(X,t) is not identically zero,
D(E,X) I O. Thus, the pair E,X is regular.
Since a simple additional argument shows that the case
K(t)
impossible, this gives a new proof of the Atkinson-Sharma
=0
is
Theorem,
Theorem 2.4. Thus, this theorem is essentially contained in Birkhoff I s results of 1906. An interpolation matrix E is called .6tJtongly Jte.gulaJt each
X,
the kernel
K(X,t) is of constant sign and does not
identically. We have the implications:
if
for
vanish
LORENTZ and RIEMENSCHNEIDER
226
E conservative
E strongly regular
~
~
E regular,
and they cannot be inverted. For the last two statements, an example is given by the matrix E3 of (2.4.1), which is regular but rnt strongly regular. Another example (Jaffe (10)
is the strongly regular ma-
trix
(7.3.1)
1
1
1
1
a
a
a
1
a
1
a
a
1
a
a
a
a
a
(
E
:)
which is not conservative (contains odd supported sequences). See (10) for other examples. It should be noted that the strong regularity of E is equivalent to the validity of the extended Rolle's theorem (Theorem for
E,
for all
X, all
f
c(n)[ a,b),
E
and some
2.5)
a < E; < b.
§8. APPLICATIONS OF BIRKHOFF INTERPOLATION 8.1. RESTRICTED RANGE APPROXIMATION a function
In the uniform approximation of
f E C,[a,b) by algebraic polynomials, we may want to
re-
strict the approximating polynomials in some way. For example, we may assume the approximating polynomials ally, we may restrict the
P
n
P n are increasing. More gener-
to satisfy [27]
(k.)
(8.1.ll
where
£ .P
J n
e:. = J
±1
J
(x)
> 0,
a < x < b,
j = l, ••. , p,
are given signs, and
given integers. In analogy with the case
< n
kp
kl
= 1,
are
this is still
called the problem of monotone approximation. Even more generally, one can restrict the ranges of the derivatives
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
227
by [38] (k .)
(8.1. 2)
L(X)'::'P ]
n
]
For the bounding functions u.
=+
and that ei ther
R,j
that either
]
(x)
R.
j
a<x
, u ' it is necessary [38]
00, or that
=-
j=l, •.• ,p,
j
u.
is differentiable for
]
00, or that
R.. ]
to
assume
a < x < b,
is differentiable.
Problems
concerning the uniqueness of a best restricted approximating polynomial are intimately related to Birkhoff interpolation.
Fo/t. each
THEOREM 8.1:
fEe [ a, bl
.theJte ex-i-6.t-6 a un-ique pol.ynom-ial.
06 the c.ta.-6-6 (8.1.1) 06 be-6.t un-i60Jtm a.ppJtox-ima.t-ion.to This was proved by Lorentz and Zeller [27) for R. A. Lorentz [30] for
p > 1.
For the
proof,
one
f. p =1, first
and
by
finds
a
Kolmogorov-type theorem, characterizing polynomials of best approximation of the family (8.1.1). These polynomials possess certain
ex-
tremal properties; among them there exist minimal. pol.ynom-ia.l.-6of best approximation, with the smallest number of extremal properties.
For
the proof of Theorem 8.1 it is sufficient to show that the difference,
Q, between any best approximation polynomial and minimal
polynomial
is zero. This is achieved by means of the Atkinson-Sharma Theorem if p = 1, but requires a more sophisticated proof by induction for p > 1 [30) • A similar proof can be given [38]
to establish the
uniqueness
in case of the more general problem (8.1.2). In [30], some additional restrictions are required for indication in [17]
f
E
if
k
j+l
k j + 1. There is an
that these restrictions can be avoided.
8.2. SIMULTANEOUS APPROXIMATION a function
R.j' u j
k l C + [ a,b I
The simultaneous apprOXimation
and of its derivatives of order
kj
of
by a
L.ORENTZ and RIEMENSCHNEIDER
228
polynomial Pn and its respective derivatives means approximation in the metric
(8.2.1)
We assume
II f II
1 ~
max : f a<x
kl < '"
~
< kp
(k. ) J (x)
I+
max a<x
1f(x) I·
k. This problem is again related
Birkhoff interpolation. The set
to
B(f) of the polynomials of best ap-
proximation does not necessarily consist of one pOint. Usingilie minimal polynomials of (8.1.1), one can find (31] the dimension of
B(f)
(see Chalmers [4], R. A. Lorentz [31), and also [17]).
8.3. CHEBYSHEV SYSTEMS ON [-1, + 1]
a Chebyshev system on
kl k2
When is the system S ={x
,x
k , •.. ,x p},
[-1, +ll? Using Theorem 5.3, we obtain almost
immediately [181, (see also Passow [34]):
THEOREM 8.2:
Fo~ in~ege~¢
0
~
kl < •.• < k ' p
Che.by¢he.v ¢y¢te.m art
[-1, + lJ i6 af'ld Of'l.ty.ttl
6e~eVlce¢
a~e
k j +l - k j
~he
¢y¢tem
S
i¢
a
kl = 0, aVId i6 aU d.i.6-
odd.
8.4. OTHER TYPES OF INTERPOLATION
Among other types of interpolation
which are of considerable interest and should be investigated, we rrention Birkhoff interpolation by polynomials of several variables, and Birkhoff interpolation by integral functions. For the last there is the dissertation
problem,
[9].
8.5. BIRKHOFF QUADRATURE MATRICES
Corresponding to every
Birkhoff
interpolation problem, there is a quadrature problem formulated follows:
as
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
Given the
m)( (n + 1) interpolation matrix E with N + lones,
a set of knots dg (8.5.1)
229
X = {xl < ••• < x } C [0,1), and a measure m
on [0 ,lJ, when does there exist a quadrature formula of
the form
f:
f(x)dg
which is exact for polynomials of degree
n?
The literature on this subject is meagrei besides the
inter-
esting paper of Micchelli and Rivlin [32J, which treats some special cases, we have an attempt of a general
theory
in
Lorentz
and
Riemenschneider [24 J and Stieglitz [42J, [43]. What follows is mainly based on
[24 J .
A pair E, X is called q-4egula4
wi~h 4e~pee~ ~o
l
problem (8.5.1)
is solvable. Let
11. =
J
be the moments of trix obtained from
dg, and let
J0
.
x J /j!
dg (x), j
B(E,X,dg) be the
(N
dg
if
= 0,1, ... ,
A(E,X) of (2.2.2) by adding a last row consisting
(Il
PROPOSITION 8.3:
n,
+2) x (n +1) ma-
, ... ,ll )' The following result compares, in o n ticular, regularity and q-regularity:
of the moments
the
A pai4
E, X
i~
q -4egula4 wLth
4e~pee~
to
par-
dg i6
an.d on..e.y i6
rank A(E,X)
(8.5.2)
in. pa4tieu.e.a4, i6
E, X i~
rank B(E,X,dg);
4egu.e.a4, then.
A matrix E is q-4egula4 with is q-regular for any set of knots restriction
xl = 0, xm = I
N =n
4e~pee~
X with
to
an.d E, X
~
q-4egu.R.M.
dg if the pair
0 =x I <
is necessary to obtain
<
a
x
m
E, X
= 1. The
new concepti
230
LORENTZ and RIEMENSCHNE IDE R
without this restriction, q-regularity (at least when
N =n)
would
be identical with regularity. The regular matrices are strictly contained in the class of q-regular matrices. Indeed, the following singular matrices (Theorem 5.1) are q-regular with respect
to
dx
on
[0,1]
(8.5.3)
1
o
o
1
o
o
1
o
1
o
1
o
o
1
o
The emphasis on the measure
o o o
dg in the definition of a q-regu-
lar normal matrix is essential ([24], [43]). The matrix E is q-regular wi th respect to all non-zero measures dg = w(x)dx..:: 0, w(x) if and only if A pair
less than
Ll[ 0,1]
is a regular interpolation matrix.
E
E
E, X can be q-regular even if the matrix
n +1
E
contains
ones - it is sufficient to remember Gaussian quadra-
ture formulas. Thus, a pair
E,
X can be q-regular wi thout the matrix
E satisfying the Polya condition - Theorem
2.2
is not
valid
for
q-regularity. We have, however,
THEOREM 8.4: i~
not M.te.ty
16
dg
(t
~uppMted
0)
i~
a non-negative
on the two point
E, q-negu.tan with ne~peQt to
~et
mea~une
on
{O,l},
[0,1]
whiQh
then any mctbU.x
dg, mu~t ~ati~6y the po.tya Qondition.
This was first proved by Stieglitz [42] for
dg = dx. The gen-
eral case needs a different proof, which is based, among other things, on differentiation of determinants similar to
D(E,X).
A normal Polya matrix may be so badly singular that it is
not
q-regular for any reasonable measures. For simplicity, we assume that the measures are of the form (8.5.4)
dg
w(x)dx, w(x) > 0,
w(x) E L 1 [0,1).
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
THEOREM 8.5 [24]: .6
Let E be a nOltmal. pol.ya matILix with exact.e.y p odd
e.que.nce.6 in the ILOW.6 cOILILe..6ponding to
an
231
xi' 0 < xi < 1.
I6 theILe. i.6
.6uch that
X
(8.5.5)
rank A(E,X)
n+l-
[~J,
then E, X i.6 not q-ILe.gul.aIL with ILe.6pect to any mea.6uILe (8.5.4). If
contains eXactly one odd sequence in its interior
E
which is also supported, then Theorem 8.5 and 5.1 imply
rows
E
that
not q-regular for any measure (8.5.4). The examples in (8.5.3) that the theorem does not hold if
is show
E contains other (non - supported)
odd sequences in its interior rows. A matrix
E is called Gau.6.6ian if
is q-regular for some set
I6
and yet the pair
X of knots and some measure
E,X
dg.
X
6oIL
P additional. one.6 aILe. ILequiILed .60 that
E
PROPOSITION 8.6 [8 ] , [24]:
.6Ome. me.a.6uILe. (8.5.4).
N
Le.t E be Gau.6.6 ian with knot
woul.d not have. odd .6e.que.nce.6 in !LOW.6 cOILILe..6ponding
to
.6
e.t
0 < xi < 1,
then N>n-p.
(8.5.6)
In paILticul.aIL,
N .::
1 "'2 (n
-
1).
A deeper and harder problem is to determine classes of Gaussian matrices. The known cases involve Hermitian (see [32],
and quasi-Hennitian matrices
[14]), when Proposition 8.6 is essentially invertible. The
following interesting theorem is due to Micchelli and Rivlin
THEOREM 8. 7 :
IJ I
=
.e.,
and
Let
I, J
k + l. -
be .6ub.6e.t.6 06
2s
=
n + 1.
I6
{o, 1 , ... ,
n}
with
I
w(x) ;.6 a.6 in (8.5.4).
[32].
I
I
=
k,
the.n
LORENTZ and RIEMENSCHNEIDER
232
the~e
ex~~t~
wh~Qh
the
(8.5.7)
~~
a
06
knot~
0
< xl < ••• < Xs <
1
60~
60~mula
J:
s a,f(i)(O) + ~ b,f(j)(l) + ~ c f(x) iEI ~ jEJ J r=l r r
~
f (x) w(x) dx
(un~quely)
mo~t
un~que ~eleQt~on
~olvable
to be exaQt
60~
all
polynom~a~06
deg~ee
at
n.
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[ 1]
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[2]
J. BALAZS and P. TURAN, Notes on interpolation. II,
III,
IV.
Acta Math. Acad. Sci. Hungar. 8(1957), 201 - 215; 9(1958), 195 - 214: 9 (1958), 243 - 258. [3]
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[ 4]
B. CHALMERS, Uniqueness of approximation of a function and its derivatives. J. Approximation Theory 7 (1973)
[5]
I
213 - 225.
B. CHALMERS, D. J. JOHNSON, F. T. METCALF and G. D. TAYLOR, Remarks on the rank of Hermi te-Birkhoff interpolation. SIAM J. Numer. Anal. 11(1974), 254 - 259.
[ 6]
R. DEVORE, A. MEIR and A. SHARMA, Strongly and wE.akly non-poised H-B interpolation problems. Canad. J. Math. 25
(1973),
1040 - 1050. [7]
D. FERGUSON, The question of uniqueness for G. D. Birkhoff interpolation problems. J. Approximation Theory 1 - 28.
2(1969),
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[8]
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D. R. FERGUSON, Sign changes and minimal support properties of Hermite-Birkhoff splines with compact support. SIAM Numer. Anal. 11(1974), 769 - 779.
[9]
C. S. FLOWERS-SHULL, Mean interpolation and interpolation entire functions. Dissertation, Texas A December 1977.
[10]
J.
of
&M University,
L. JAFFE, Rolle regular Birkhoff matrices. In App!toxhna.-Uon
The.-
otty, II. Academic Press, New York 1976,397-404. [11]
K. JETTER, Duale Hermite-Birkhoff-Probleme,
J.
Approximation
Theory 17(1976), 119 -134. [12]
D. J. JOHNSON, The trigonometric Hermi te-Birkhoff interpolation problem. Trans. Amer. Math. Soc. 212(1975), 365 - 374.
[ 13]
S. KARLIN and J. M. KARON, Poised and non-poised Herrnite-Birkhoff interpolations. Indiana Univ. Math. J. 21(1972), 1131-1170.
[14]
S. KARLIN and A. PINKUS, Gaussian quadrature formulae withmultiple nodes. In Stud.i e.~ .in S p-t.ine. Func.t.io n~ and App!todmat.ion The.otty. Academic Press, New York 1976, 113 -141.
[15]
S. KARLIN and W. J. STUDDEN, Tc.he.byc.he.66 t.ioM .in
Ana-ty~.i~
and
Stat.i~t.ic.~.
Sy~te.m~:
W.ith App{.i.c.a-
Interscience Publishers,
New York 1966. [16]
E. KIMCHI and N. RICHTER-DYN, An example of a non-poised interpolation problem with a constant sign determinant.
J.
Approximation Theory 11(1974), 361 - 362. [17]
E. KIMCHI and N. RICHTER-DYN, On the unicity
in
simultaneous
approximation by algebraic polynomials. J. Approximation Theory 18 (1976), 388 - 389.
[18]
G. G. LORENTZ, Birkhoff interpolation and the problem of matrices. J. Approximation Theory
free
6 (1972), 283 - 290.
234
[19)
LORENTZ and RIEMENSCHNEIDER
G. G. LORENTZ, The Birkhoff interpolation problem: New methods and results. In PlLoc.eeding-6 Int. Con6elLenc.e ObelLWol6ac.h, Birh1iuser Verlag, Basel, 1974 (ISNM25), 481 - 501.
[20)
G. G. LORENTZ, BilLRh066 IntelLpolation PlLoblem. CNA Report-l03, The University of Texas at Austin, July 1975.
[21)
G. G. LORENTZ, Zeros of splines and Birkhoff's kernel. Math. Z. 142 (1975), 173 -180.
[22)
G. G. LORENTZ, Coalescence of matrices, regularity and singu1arity of Birkhoff interpolation problems. J.Approximation Theory
[23)
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G. G. LORENTZ, Symmetry in Birkhoff matrices. In the PlLOc.eeding-6
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G. G. LORENTZ and S. D. RIEMENSCHNEIDER, Birkhoff matrices. Birkh1iuser
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!4ath. Sci. Hungar.
[26)
G. G. LORENTZ, S. S. STANGLER and K. L. ZELLER, Regularity
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[29)
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[30)
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by
monotone
polynomials. J. Approximation Theory 4(1971), 401-418.
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[3lJ
R. A. LORENTZ, Nonuniqueness of simultaneous approximation algebraic 17 -23.
[32J
236
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C. A. MICCHELLI and T. J. RIVLIN, Quadrature
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and
Hermite -Birkhoff interpolation. Advances in Math. (1973),
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[36]
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I
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[42]
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Approximation Theory and Functional Analysis J.B. FroUa (ed.) ©North-HolZand Publishing Company, 1979
APPROXIMATION POLYNOMIALE PONOt:Rt:E ET PROOUITS CANONIQUES
PAUL MALLIAVIN Insti tut
Henri Poincare
Paris,
Soint sur
E.
E
France
une partie fermee de
On dit que
R+, p(x) une fonction
definie
p(x) est un paid.6 de BeJtH.6tein si notant par Co(E)
les fonctions continues sur
E et tendant vers zero
pour tout entier
a
n > 0
l'infini
alors
et
est totale dans
On note par
H(q,E) la classe des fonctions reelles
niques dans le complementaire de
h(z)
> 0
h(x)
> q(x)
On sait que [ 3 I {x
n+2
p (x) } n > 0
E et verifiant
pour pour
H(q,E) 1
¢
est non totale dans
Z E ((; \ E
X E E.
avec
q = -
entraine
que
~
d' etudier
E et la croissance de pro-
duits canoniques de la forme
IE log 11 - ~ 1d]..l (t)
237
log p
Co (E). On se propose
le lien entre les poids de Bernstein sur
w]..l (z)
h harmo-
238
ou
MALLIAVIN
d]l
est une mesure de Radon sur
E.
(Cf. [3
1 pour une premiere
etude dans cette direction.) Les deux problemes paraissent differents, ils assez lies. II est d'abord evident que nique dans dier si
a:: \ E
sont
toutefois
W]l(z) est une fonction hamn-
et par sui te qu' elle fourni t un candida t pour etu-
I ¢. Les resultats interessants vont dans l'autredi-
H(q,E)
rection. Les Theoremes principaux sont enonces ci dessous. On dira qu' un point fonction continue
Xo
de
E est Jte.gut1.e.Jt si etant donne une
h (x) quelconque definie sur
bleme de Dirichlet
E la solution du pro-
H(z) pour cette fonction satisfait
a
lim H (z) z=xo
L'ensemble des points irreguliers de
E est de capacite nulle.
On a
alors l'enonce:
1.1 TH~OREME:
Suppo-6on-6 que. E e.-6t une. pa.Jtt1.e. de. t' axe. Jte.e.t p0-6ili6,
que. t' oJt1.g1.ne. appaJtt1.e.nne. t'1.nte.Jt-6e.c.t1.on de.
E
a.
E
ave.c. un ouve.Jtt -601.t ou b1.e.n de. c.apac1..te. pO-61.t1.-
ve. ou b.ie.n v.ide.. SUPP0-60Yl-6 que.
AtoJt-6
H(q,E)
e.t -601.t un po1.nt Jte.gut1.e.Jt de. E, et que. o q(e ) -601.t une. 6onct.ion conve.xe. de. o.
e.-6t non v1.de. -6.i e.t -6e.uteme.nt -61. .it e.x.i-6te. une.
]l pO-61.t.ive. -6UppoJtte.e. paJt
w]l(x)
En approchant
me.-6uJte.
E, te.tte. que.
> q(x)
pouJt .tout
x E E,
]l par une mesure somme de masses de Dirac on ob-
tient
1.2 PROPOSITION:
So1.nt E une. paJtt.ie. de. R+ et -6UPPOMn-6 que. H(-log p,E)
APPROXIMATION POLYNOMIALE POND~RI!E ET PRODUITS CANONIOUES
MJ..t /ton vJ..de (-log plea) dL6Qltete
El
C
E
239
etant convexe). AlolL!.J Lt exJ..!.J.te une pa11..tie
telle que
p
ne .6o-Lt pa.6 un po-Ld.6 !.JUIL
E . l
On considere l'ensemble des fonctions croissantes constantes sur Ie complementaire de
E; on dira qu' une telle fonction
m(x)
pour tout
x
E
localement
:>
pour tout xEE [resp. m(x):::.O
0
EJ .
On note d'autre part par
~E
la fonction caracteristique de E,
par a(x)
ou
1
2logA
fxA -1 ~E(tl ~ t xA
A designe une constante fixee. D'autre part on aura
a
supposer
dans certains cas que
log p (x) log p (txl
1. 3.
0(1)
(x,tx
E
Eli
Ilog tl
0(1) •
On ales enonces suivants donnant des conditions necessaires et des conditions suffisantes dependant de fonctions arbitraires de
1.4. m
E
SUPP0.60M que
.6at,(.46a4.6e.
p
it (1.3) et qu'on
pu,(.ue
H+
tltouvelL
H+(E) tel que
f
E
log pix)
> -
al0lL4 on peut tILouvelt une pat-Le d-L4Mete
El
.6o-Lt non v-ide.
1..5.1.
00
,
a(x)m(x)
lim inf
- m(-x) logp(x)
< +
00
C
E,
teUe que W(El,p,l)
240
ato~~
MALLIAVIN
W(E,p,Ol
REt-1ARQUE:
e~~
vide.
On peut remplacer 1.5.1 par la condi tion
lim inf
1.6.
-1 log p (xl
[2
J;
m(tl
~t
Les enonces 1.4 et 1.5 ramenent le probleme
de fonctions d' epreuve des classes
H+ et
- m(xl] <"'.
a
la determination
H-. Ces fonctions pourront
etre choisies adequates au cas particulier considere. Donnonsquelques exemples: D'abord un resultat general.
PO~OM
1.6.1.
A (xl
1/2 (
atM~
eA(x)
E
H+.
Ce resultat peut etre precise moyennant des hypotheses supplementaires.
1.6.2.
Suppo~on~
que
lim a(xl
0,
po~on~
A* (xl
ato~~
1.6.3.
quette que
Suppo~on~
~oi~
que
ta
c.on~~an~e
k,
e kA*(xl
on
+
H •
pe~t t~ouve~
a > 1
Ceci montre qu'un resultat de Katznelson [2] ne peut pas
etre
tet que
lim sup a(x)
< 1,
ato~~
E
eaA(xl E H+.
sensiblement ameliore.
1.6.4.
Suppo~on~
que
q~e
E
~oit
ta
~eunion d'inte~valte~
[a k , b k ],
tet~
APPROXIMATION POLYNOMIALE PONOeReE ET PROOUITS CANONIQUES
Ce resultat est
a rapprocher
241
de (4.1).
2. BALAYAGE AVEC LE NOYAU DE WEIERSTRASS
La theorie de balayage en potentiel logarithmique est developpee soit pour des compacts, soit en utilisant les fonctions de Green par exemple celle du demi-plan. On se propose d'utiliser ici la rie classique pour un cas non compact le noyau etant Si
log
11
theo-
-zt-ll.
est une mesure, on considerera l'integrale (que l'on supposera
fl
toujours absolurnent convergente)
Wfl (z)
D'autre part si
fl
est
a
support compact on peut considerer egalernent
On a alors Ufl(O)
-
Ufl(z).
Nous allons commencer par ecrire la fonction
LEMME de factorisation:
2.1.
que
Sol.t
q(x)
q
comme un potentiel.
une 6onction pO.61.t1ve, te1.te
q(eo) .601t une 6onctlon convexe, ~ cnol.6.6ance llnealne poun
a.6.6ez gnand, et telle que 1
2.1.1.
J
q(t)t- 3 / 2 dt
o
aR.M.6 on peut tnouven une 6onctlon
q p0.61tlve. ven161ant
x
242
MALLIAVIN
2.1. 2.
q (x),
PREUVE:
L' hypothese de convexi te fai te sur
q(x)
2.1. 3
x m(t)..£t t fo
ou
x
q
m(t)
> O.
implique que
est croissante.
2.1.1. implique que
f Soit
_dm(tl < tl/ 2
co
•
Q(z) la transformee de Mellin de
q.
Transformons
for-
mellement les deux membres de 2.1.2. par Mellin, remarquant que
cotg 11 z z
- 1 < Rez < 0
on obtient
o(z) 2.1.3. donne, notant par
z tg ( 1TZ) Q ( z) .
M la transformee de Mellin-Stieljes de
Q(z)
M(z)
---;r-
d
Notons par l1Z
l1Z
h(x) la fonction ayant -
~
< Rez <
~
• h(x)
'ou
• Q(z).
o(z)
- tg
dm
pour
transformee
de
Mellin
se calcule par residus et on trouve
sur cette expression il est evident que
APPROXIMATION POLYNOMIALE POND~R~E ET PRODUITS CANONIQUES
2.1. 4.
h (x)
a
>
(Inegalite fondamentale)
2.1. 5.
h
2.1. 6.
L
E
243
"',a
1 - "2
< a <
1
- -1.2
1 -< a -< 2
"2
,a designe l'espace des fonctions sommables par rapport a la a-I ~ -a mesure x dx; L l'espace des fonctions bornees par x Ceci "',a etant, justifions les operations formelles effetuees ci-dessus. ou
Ll
Posons:
2.1. 7.
Alors
q
dm E Mo.
q
~
<
a
dm
-
est bien defini et
Comme d'autre part
duit de composition -
*
espace des mesures sommables pour xo.
d'ou d'apres 2.1.5.
2.1.8.
h
q
E
L1 , a ' -
si
a >
1 < a
<
2"
logll - xl E Ll ,a ' - 1 < a < 0, Ie
k = log 11 - xl
*
q
est bien defini et
E
< O. On a enfin
- cotg
K (z)
7TZ
Q(z)
Z
si
a, -
Rez
1 2"
< B < 0,
et Q(z)
H(z)M(z) •
D'ou K(z)
=
M~Z)
et
z
k(t)
=
r
M~X)
dx
presque partout
o
les deux membres etant continus ceci vaut partout d'ou 2.1.2.
1
-2"; 1
"2.
proL 1 ,a
MALLIAVIN
244
Pasons
r(x)
il resulte de 2.1.8. que
lim r (x) x=oo
2.1. 9.
existe
Nous allons monter un lemme elementaire sur l'allure d'un
patentiel
d'une mesure portee pour l'axe reel.
2.2. LEMME:
Soit
pa~ R..' axe ~eel,
d~
u~e me~u~e po~itive
a ~uppo~t
compact
po~tee
f log I z - tl d ~ (t) = - u~ (z). SUppO~OM que
lim U~(x + iy) o y=o Alo~.6
o~
a
lim U~(x + iy) o y=o
PREUVE:
U~(x) est semi-continue inferieurement donc
D'ou l'integrale
- flog
11 -
que les points reguliers de
-1
Xo t i d ~ (t)
E
lim h(x + iy) y=o d'ou en utilisant 2.2.
q(x)
est convergente. Remarquant
APPROXIMATION POLYNOMIALE POND~R~E ET PRODUITS CANONIQUES
en tous les points reguliers de tout dense sur
E et
E,
246
ceux-ci formant un ensemble par-
WjJ (x) etant semi-continue superieurement, q (x)
continue, on obtieni
I10gil - xt-li djJ(t) > q(x)
3.
3.1.
pout tout
x
E,
E
c.q.f.d.
Nous nous proposons dans ce paragraphe de demontrer enonces 1.2.
TH1!:O~ME:
ul1e meJ.>ulLe
dA
S-i
H(E, - log p)
eJ.>t
I
3.1.1.
v-ide, a.tolLJ.> 011 peut tlLOUVell.
ayal1t POUIL J.>uppolLt UI1 eMemb.te d-iJ.>c.lLet
I
o ,
D1!:MONSTRAT10N:
11011
Soit
~¢.
H(E, -logp)
logil - xt-ll djJ(t)
i dA
i
<
00
El
E,te.t que
•
Il existe d'apres
> - log p(x)
C
X E
I
1.1.
E •
t
11 resulte du fait que cette integrale est> -
00
que
Io djJ=jJ(t)
est
une fonction continue. Soit n(t)
partie entiere de
jJ(t)
et soit exp [-
I
log(l - zt-l)dn(t)] = F(z).
F(z) est une fonction meromorphe n'admettant que des poles simples. D'autre part, posons
s (t)
3.1. 3.
I
o
jJ (t) - n(t)
log 11- zt -11 ds(t) =Re
I
< s (t)
< 1
t _z z S(t)dtt = ReIooS(XU) l+iT u-l-iT du U
o
246
OU
MALLIAVIN
T = yx- l ; soit
a
alx +
fo
tel que
J1 / 2
+
a/x
< I, a > 0
pta)
+
J2 1/2
J+ 2
OO •
La premiere integrale est inferieure
Cll(t)x~t La seconde
a
La derniere Reste s
=
a
0(1) .
t
log x + 0(1).
a
0(1).
evaluer la 3eme integrale
on
Ie
fera
en
posant
! s11l~I:r' d' ~ ou
:r
sl + 1
Re
dt
a
2 s(xu) 1 l + 'iT ~ d 1 < -2 u- -n u
J1;'2
+ / Re
f
1 + i T u-l-iT
f112 2 IRe
1 + i T jdu +-' 1 J2 Re~ l' dU/ u-l-i T 2 I 1/2 u-l-n
~)
(1 -
du / .
La premiere integrale < - log T + 0 (1), la seconde et la tro:l..sierne sont 0(1), d'ou en tenant compte de 3.1.1.
xA !F(X + iy)! < Bp(x)y,
x
E
E,
!y! <
~
un entier > A + 2, b l , .. " b ' r pOints de E distincts; r alors on peut trouver une fraction rat:l..onnelle H(z) ayant les bk Soi t
r
pour poles simples et telle que
H(z)
o(z
-r
),z+co
F(z) H(z)
verifiera
APPROXIMATION POL YNOMIALE PONO~R~E ET PROOUITS CANONlaUES
3.1. 3.
IF1(Z)I
-1
Ix
-2
247
x + iy) .
(z
On a
( d P (t) J z- t ou
p(t) =
~
Le m~
Residus de
Fl(x) < t.
residu a verifiant
Ia m I < B l 2 n(t) = 0(t / ) en vertu de la formule de Carleman
D'autre part on a
f
d'ou
dn(t) t2
< 00. Par suite si l'on pose
d;\ (x)
3.1. 4.
J
(p (x) )
-1
p(t) d;\(t) z- t
dp
(x)
Y z -n +
n
alors
J-
r
J
d;\
< 00
et
n l
t + zn+l(z_t)
p (t) d;\ (t)
ou Y est le premier moment de dp different de zero. Prenons z=x+i, n x E Ei cette egalite contredirait 3.1.3. c.q.f.d.
4.
FACTORISATION DU NOYAU
logll - ul
Nous allons decomposer dans l'algebre de composition sur (0,00) le noyau
logll - ul dans le produit d'un noqau po~~t~6 et d'un ope-
rateur differentiel. C'est un fait bien connu que l'evaluation
des
produi ts de Weierstrass est complique par le fait que le noyau Jog 11- ul est
(0
si
u)
positif sinon la repartition des
zeros
intervenant
localement et globalement : globalement par la fonction
a
m sur le cercle R. localement par R les perturbations au voisinage des zeros a l'evaluation donnee par la
lie par Nevanlinna
moyenne
m • R
la moyenne
248
MALLIAVIN
Nous allons donner une famille de telles factorisations du noyau log 11 - u 1 dependant d' une fonction arbi traire.
4.1. PROPOSITION:
nee, de.6..[n..[e
Wit
So"[t
r 0,
+
00
net) une 6onct..[on
f, telte que
a:
va./t..[a.t..[on
tocalemen.t bolt-
n (tl
Xo tet que
O(log
4.1.1.
So..[t
set) une fioltct..[olt
de plu,o que
~at..[46a...[~altt
s (tl ... + "', que
-1
h).
aux meme4
cond"[t..[on~.
net) ~o..[t nut au vo~..[yJa.ge de zVto. So..[en.t
x -T Idn (tl
CI09ll 0
CIOgll - .1i. t
y(x)
i ds(t).
PO~OIt~
F(x,t)
=
x
C~ u-x
~
4..[
0 < t < x,
r-ilil
~
.0..[
x
0
PREUVE:
F (x, t)
=
p(x)
n(x) "S(X')
x
t
x- u
Suppo~on~
u
u
L'hypothese 4.1.1. permet d'ecrire
APPROXIMATION POLYNOMIALE POND~R~E ET PRODUITS CANONIQUES
X -£
lim r £=0 J O
r+ 00 x) x+£
+ 00
+
0
Jxo + dt
n(t)s-l(t)~~t~ t
xn(t) x-t
E
249
Jtt. t
(+ 00 =F(x,x + e:)p(x+£) +)' F(x,t)dp(t) x+£
d'ou 'P (x)
lim [F (x, x + £) p (x + £) - F (x , x - £) P (x - £) £=0
J
X
lim £=0
-£
+
+ 00
+
F(x,t)dp(t)
Jx+£
o
p(x) satisfait en
1
Xo
1a condition 4.1.1. ce qui permet d'ecrire
Ie
premier crochet
p(x)
lim [F(x,x + £) - F(x,x £=0 X
p (x) lim [
d'ou
Jo
-
E
+
J +00 x+£
xs(u)
x - u
£)l
J!!!. u
l=p(x)y(x)
4.1.
4.2. COROLLAlRE:
PREUVE:
On a
'P (x)
x1/ 2
P(t)
V.P.
J
P (lS) d( n(t) ) t1/2 t
(t
u- 1 / 2 du u - 1
Jo
u- 1 / 2 du v.P. roo t 1 - u
App1iquons 1a proposition 4.1. avec
s(t)
log
11- t 1/ 2 1/2 1• l+t
1
1 t / 2 , 0 < t <+00.
A10rs une integration par residus donne
y(x)
V.P.x
+oo u1/2
J o
x - u
J!!!. u
0,
x > O.
D'autre part
260
MALLIAVIN
l 2 u /
F(x,t) =x Jot
u -
X
..ill!. U
t < X
en tenant compte de la deuxieme expression donnee de Le fait que
y(x)
Pest egale
a
o
P dans l'enonce.
montre que la deuxieme expression proposee pour
la premiere. L'application de 4.1. etablit alors 4.2.
Nous allons donner une premiere application du resultat de factorisation 4.1.
SaLt PI une nonc.-tion nega-tive defr(n-ie .6uk E
4.3. TH1!:ORtME:
,t-<-.6na-<-.6an,t
a 1.9.1.
a-i,t
dt
-r
sit) une 6onc.,t-<-on c.kO-<-.6.6an,te. .6a,t-<-.66a-<-.6ant 1.9.1, E
.6a-
PO.6on.6
4.3.1.
So-<-,t
e,t
POUk .6UppOJr.,t
que
e,t
J logll
4.3.2.
te.Ue que. ds(t)
~
-
Ids (t) > Pl(x)
X E
E
•
SUPP0.60n.6 que.
4.3.3.
J
> -
a(x)s(x)
00
•
E
Alok.6 -<-l
eX-<-.6te
une. 6onc.t-<-on
op (z)
< 0,
hakmon-<-que. dan.6 le. compleme.n-
ta-<-ke. de. E, te.lle. que. op(x)
REMARQUE:
Si
< PI (x),
X E E.
E est compose d'intervalles de longueur logarithmique > a> 0,
APPROXIMATION POLYNOMIALE PONoilRilE ET PROOUITS CANONIQUES
on peut dans 4.3.3, prendre
~lors
Ie produi t canonique construi t
PREUVE:
avec
ds
= 1;
de plus 4.3.2 a lieu si
est simplement posi tif.
Posons
nIx)
Alors
a (x)
251
dn
s (x)
r
PI (t) a(t)s(t)
J E,,[x;+oo]
dt t
a pour support E, et
dp
PI (t) a(t)s(t)
d(...!!.)
s
dt t
tEE
o Escrivons 2.1, remarquant
J
>
'I-
t
F(x,t)dp(t) +
pIx)
a
E •
0(1) et utilisant 4.3.2,
(Pl(x».
On a si
F(x,t)
ou
f
s(xS;) S; - 1
..9.£
< 8s(x) logll -
~I
S;
8 > 0, et une evaluation analogue pour
t
E
3
[x'2 xl
d'ou
J F(x,t)dp(t)
ou
8
1
Pre nons
et
8
n
l
sont deux constantes numeriques positives d'ou
2
8
-1
3
n
et posons
J logll
-
~
I dnl(t) + A
262
MALLIAVIN
ou - A
sup [ PI (x) -
tp
(x)
f,
0 < x < Xo
Le resultat suivant classique pour les fonctions entieres d'ordre
~ s'etend a 2
tp(z):
i l existe une suite infinie de cercles
tels que
e.
uniformement en
Dans
{iz i
tiere, donc
tp(z)
< ~} () CE, tp(z) est harmonique negative surla fron-
z.
quel que soit
< 0
5. Nous allons donner dans ce paragraphe des conditions pour que la suite
{xnp(x)}
suffisantes
soit non totale dans l'espace
CotE) des
fonctions continues sur E nulles a I ' infini. Etant donne x
E
I x'
Ix
C
E.
x
E
soi t
Ix Ie plus grand intervalle tel
inf { 1,
J!t t
Lee natatianc etant eellec de 4.3, cuppaconc
lee hypothecec de 4.3, cont catic6aitec
pou~
Pl(x)
= log
pluc cuppoconc ou bien que
5.1.1.
lim inf(- PI (x)
~ a. (x) ) > 0
au bien que log a.* (x) = 0 (PI (x))
5.1. 2.
que
Posons
a. * (x)
5.1. PROPOSITION:
E
- PI (x) a.*(x) (I-log a.*(x)) sex)
et que
<
00
p(x);
que de
APPROXIMATION POLYNOMIALE PONO'R'E ET PROCUITS CANONIQUES
OIL
263
bien que ~
<
5.1. 3.
o.*(x)s(x)
PREUVE:
Nous allons construire un produit cononique
tp (z)
f 109 11 -
=
x
-1 zt
OD
I dn (t) 2~
dans lecam-
Supposons par exemple que 5.1. 3, est
satisfait.
tel que la fonction conjuguee soit uniforme plementaire de E.
•
modulo
Posons:
n(x)
5.1. 4.
- PI (t)
sex) f
A(t)...!ll.
0.* (t) S (t)
t
En[x,+oo]
OU
a
A(t) sera une fonction positive localement constante sur E c'est dire constante sur chaque Ix' On a puisque PI et s satisfont
a
1.9.2,
OU cette quantite tendant vers -
00
on peut determiner la fonction A tel-
Ie que entier,
Soit telleque
A(X)
-+ 1,
x
-+
co.
g(z) la fonction holomorphe dans Ie complementaire de 10glq(z)1 ="o(z),
Alors la formule 4.1, donne
E
264
MALLIAVIN
d'ou I gN (X) I < p (X)
pour
I
N entier fixe assez grand, x E E .
Indiquons rapidement comment modifier un raisonnement
class i-
que pour conclure
J
.il.!l z- t
dt
ou f(t)
lim y=o
D'autre part la formule de Nevanlinna donne
f
.e
log I g(re~ ) Ide =
Puisque
Soi t
q
n(t)
+
-
00
f2
n(t) t dt.
cette derniere quantite tend vers -
le premier entier tel que
alors Z -q-
1 aq + z-q- 2
J
q l
f(t)t +
(z - t)
et d'autre part
2TI
Jo
log (1 + r
-2-1
sin
e b) de
0(1)
d t
00
•
APPROXIMATION POLYNOMIALE POND~R~E ET PROOUITS CANONIQUES
ce qui contredirait la formule de Nevanlinna.
a
thogonale
tn(n ~
0) :
f(t)dt
255
est ainsi or-
f (t) < ~ d'ou 7T 2 pet) = t p(t) ce qui entraine que tnp(t)
d'autre part
est non totale dans
CotE) ou
est non totale dans
CorE) .
Lorsque les hypotheses 5.1.1, et 5.1.2, sont satisfaites on definit encore
n(x) avec la meme formule, les hypotheses
5.1.2, ayant pour effet
que l'on peut
determiner la fonction
A(t) soit bornee sur (0, +
f dn = entier et que
telle que
5.1.1,
On peut d'autre part remarquer que les conditions 5.1.3 remplacent la condition 2.3
et A{t)
00).
5.1. 2
qui n'a plus besoin alors
et
d'etre
verifiee. Nous allons maintenant donner diverses evaluations de la fonction
set) qui combinees avec 5.1, donneront des conditions
saires pour que
xnp(x) soit totale dans
5.2. PROPOSITION:
SoLt
11 (x)
p
e
11 (x)
o
,si
dp < 0 P
<
alo~6
on peut
p~en-
.
et appliquons
(xl
dp
d'oii,
dt
~,
En [O,x]
sex)
Soi t
CotE)
; J
dlte dan6 5.1.
PREUVE:
neces-
X E
E
x ¢
E
4.2.
Alors
0, on conclut P
*
dp > 0
la fonction
ell {xl
est une fonction
COROLLAIRE:
So..i.t E c.omp06e.
s.
d'..i.nt~vaUM
de. longue.Wt logaM.-thm..i.que > h>O;
266
MALLIAVIN
I
dx
-log p(x) e-fl(x)
<
x
00
Naus allons donner sous des hypotheses supplementairesde meilleures evaluations des
sex)
possibles.
Pasans dt t
5.3. PROPOSITION:
S~
pO.6on.6
0,
J
vex)
aioft.6 qu.eiie que
lim ~(x)
dt
1
Ell [0, xl ex (t) /log (i (t) I
.60~:t
ia c.ol't.6:tan:te
t
on peu.:t dan.6 5.1
I-
sex) = el-v(x) .
PREUVE:
Posons alars d p
11
11
on en deduit que
=
11 0 (x)
1
-2'
au
~x x
~ E
1
~ /log
til
- 2'
X E
E
pftendfte
APPROXIMATION POLYNOMIALE POND~R~E ET PRODUITS CANONIOUES
p(xO
ou
o Remarquant que Ie noyau et
decroissant sur
3X/2 [ x/2
257
introduit en 4.2
P
[0, 1
< b < A (- log a)
-1
est croissant sur [1,+00]
J, on a
x bM 1 3/2 dl: p(-}e - t - p(x}[ [3/2 t a P(£;- Fc, c,
P( ~}dp(t} > p(x}>..
e2 a
e-
2
+ [1
P U,:-l} E;-3/2 d E;] (l + a).
'2 La premiere integrale est> -
p(X)A
d'autre part si
tier posi tif ou negatif }
p (x)
- -2-
et toutes ces integrales etant positives d I ou N designant un entier fixe,
on a
J P ( ~ ) dp (t)
> p (x) ( -
A + (1 +
cette integrale se calcule et est assez grand et
REMARQUE: pu prendre
N > A,
d'ou
Au lieu de fixer
>
eAV(x)
N
d' ou
J
P(
N = N (x). Ce calcul ne peut etre mene
poids de la forme
eA(x}v(x} ou
) dp (t)
> 0
si
x
est une fonction poids. c.q.f.d.
N et de faire tendre
des hypotheses supplementaires sur
~
x
a
->-
00
on
aurai t
bien que llOyermant
E. On obtient alors des fonctions
A(X)
->-
00. Quel que sOitE,ceprocedS
268
MALLIAVIN
de calcul ne permet pas d'obtenir
croissant plus vite
).(x)
quelogx.
On aurait pu d'autre part definir a(x) en considerant l'intervalle 3x) ~ -. _ X [ x B- 1 , xB) au lieu [ '2' -2- au B desl.gne un nombre fixe B > 1. Un cas interessant est celui ou lim vex) <
00
x ..
,
00
ceci est en particulier le cas au la serie
<
00
ou In designe 1e n e interva11e de l' ensemble
10 g
E
et
(! In .9..t.) t
On pourrait comparer cette condition avec 1a condition de Wiener d'effilement
a
l'infini. On a l'arne1iaration suivante de 5.2.
5.4. PROPOSITION:
5.4.1.
S~
lim sup
a(x)
e
< 1
s (x)
PREUVE: p(x)
s(x)x- 1 / 2
a10rs dp
1
- '2
dx
p(x)--x-
(~)p(x)~ 2
x
X E
E
¢
E
x
APPROXIMATION POLYNOMIALE POND~R~E ET PRODUITS CANONIOUES
269
et d'autre part
p (x)
Eva1uons 3x
2 (
~
P(
Jx
1 - Al > p (x) ( 2 11 +
) dp (t)
2
choisissant
Y
verifiant
y
I
YX
o
< 11 <
-1
Y x
e
<
2e
~
I
2
) ,
log 3 oil
-P(~)~ t t
Y
1/2
et 3 / 2X
I2 >
interval1es
Jyx [~,
l'integrale sur
3n x
5.5. REMARQUE:
3X 2
1
3n +1 x
[--2--'
x
dt
t
t
si A <
-P(-)-
2
sera pos i ti ve; i 1 en sera de meme pour les d' oil
e AII (x) es tune fonction
On dira que l' ensemble
E est dense
a
s.
l'infini
dans
F si posant a{x,B,E)
on a,
J[xB -1 ,xBlnE
~ t
B fixe, lim inf a(x,B,E) a(x,B,F)
5.5.1.
>
0
pour tout
B > 1.
Posons 1l*(X)
A1 ors s i
~
5 .5.1. vau t ,
J
F
e
11
[0,
All* (x)
xl
~ t
est une fonction poids pour E, quelle
260
MALLIAVIN
que soit 1a constante A ,verifiant
A
< 1. La demonstration
s'ef-
fectue comme en 5.4.
Suppo~on~
6.1. TH2oREME:
que
H(E,q) ~oi~ non vide, alo~~ quelle que
~oit la 6onetion
r E R+(E) , on a
6.1.1.
lim
PREUVE:
1 q(x)
wr
+ 00 •
(- x)
Supposons que 6.1.1. ne soit pas satisfaite la limite infe-
rieure du premier membre de 6.1.1. sera egale construire une fonction vide. Soi t
q1
te1le que
q
a
= a (q1)
b <
00.
Nous
et
H (E,ql) soit non
la mesure harmonique du complementaire de
0
E
allons
dans
C,
telle
que
alors
d x < 00.
Jql (x) (0 (00) - a (x»
Soit
n(x) une fonction croissante tendant vers l'infini
J on prendra
q1
n(x)q' (x) (0(00) - a(x) )dx < 00 ,
nq'
tel que
.
Alors 6.1. 2.
Soit
0.
lim inf
h(z) une fonction harmonique appartenant
a
H(E,qI)' et
~
Ia
mesure associee par 1.1. lim inf
r W ( - x)
o
Wll (x)
Remarquons que Ie maximum de Wr x < 0, son minimum sur
sur
Iz I
x > 0, on obtient
= Rest atteint sur I' axe
qu'il
existe
une
suite
APPROXIMATION POLYNOMIALE PONO~Re:E ET PRODUITS CANONIQUES
Rk
~
00
261
telle que
DI autre part on a sur E
X E E
r d'ou en remarquant que W
et Wll sontharrroniques dans {z; Izl <~}IIa:\E.
On obtient
Iwr(z)
1
<
~
Wll(z)
1
z
1
< ~
k etant arbitrairement grand
Appliquons la formule de Nevanlinna t
o
<
fa
r(u)
~u
r Moyenne de w
sur
I
z
i
t < 0
contradiction. On peut remarquer que l'on aurait l'hypothese
pu
remplacer
simplement
H(E,q) non vide par celIe qu'il existe une fonction F(z),
holomorphe dans Ie complementaire de E, telle que IF(X) 1 < wq(x) II suffisait de remplacer dans Ie raisonmement ci-dessus la minorante harmonique de
6.2. PROPOSITION:
Dr:MONSTRATION:
- logIF(z)
On a Wr(-x)
<
2
J:
1 .
ret) ~t - Wr(x).
I
Wll(z)
x
E
E.
par
262
MALLIAVIN
wr
+OO
(-X)
= ou
S(u)
= u,
si
fo
=
J:
0 < u < 1; et
x
dt t
- - r(t)X
+t
log(l + S(
~»dr(t)
.1.
S(u)
si
u
1 < u < +
00
•
De meme on a
-C
r(t).E.t t
L'inegalite
I1
- log
- ['" log 11 - 8 (
o
~)
1+ 00
Jo
log [1 - S ( ~ ) I dr (t) .
- vi > log i 1 + vi
I dr (t) > rex> log 11 + 8 ( Jo
donne
~ ) I dr (t) ,
d'ou 6.2.
BIBLIOGRAPHIE
[1 I
G. COULOMB-COURTADE, These, Paris 1976.
[2 I
Y. KATZNELSON, Comptes Rendus 246 (1958), p. 281.
[3 I
P. MALLIAVIN et S. MANDELBROJT, Sur l' equi valence de deux problemes de la theorie constructive Sci.
[4
~co1e
Norm. Sup,
(3)
des
fonctions,
Ann.
75(1958), p. 49 - 56.
I s . MANDELBROJT, GeneJLal :theOJteJM 06 Clo"'uJLe, Rice Institute Pamphlet. Special issue (1951).
[5 I
L. NACHBIN, Element", 06 ApPJtox~mat~on TheoJty, D. van
Nostrand
Co., Inc, 1967. Reprinted by R. Krieger Co " Inc. 1976.
Approximation Theory and Funational Analysis J.B. FroUa (ed.)
©North-Holland Publishing Company, 1979
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
REINHOLD MEISE Mathematisches Institut
der
Universit~t
D - 4000 DUsseldorf, Universitatsstr. 1 Bundesrepublik Deutschland
PREFACE
When the author started to deal with the subject
of
the
present article, he did not know too much of the various ways how one can do calculus in (real) topological vector sIJaces. He was rrainly interested in a generalization of the theory of distributions to infinite dimensional locally convex (l.c) spaces, by means
of
duality
theory and with relations to infinite dimensional holomorphy. Therefore he began studying spaces of differentiable
functions on
1. c.
spaces by analyzing the definitions of Cn-functions used by Aron (3J, Bombal Gordon and Gonzalez Llavona (10 J and Yamamuro [24 J • Then he (re-) invented the notion of
n
tions on an open subset arbi trary covering of
times continuously y-differentiable funcrl
of a 1.c. space
E
(2.4), where
y is
an
E by bounded subsets. As he realized later, this
notion had been introduced with a slightly different definition
by
Keller [18) already. Further investigations showed that many results known to be true on open subsets of
:rn.N carryover at least to Frechet
spaces or strong duals of Frechet-Montel spaces i f one takes as y the system of all compact subsets of
E. In order to be precise, we will
now sketch the main results of the article. In the first section most of the definitions are stated as well as some results which will be used in the sequel. In the second part 263
264
MEISE
we introduce the 1. c. space en W, F) of
n
y
ferentiable functions on an open subset 11 values in the 1. c. space y
F, where
n
times continuously
y -dif-
of the 1. c. space
E with
is a natural number
is a system as introduced above. For open subsets
complete l.c. space and only if
E~o
E,
we show that
COO W) co
11
or
and
00
in a quasi-
is a Schwartz space if
is a Schwartz space.
In the third section we give
a
sufficient condition
for
the
approximation property of Cn (\l). The proof of the corresponding theco orem 3.5 is a generalization of the proof given by Bombal Gordon and Gonzalez Llavona [10] inthe case of Banach spaces. (Actually an analysis of [10] ledto the result presented here.) Since the proof
uses
a criterion for the a.p. due to Schwartz [22], we first characterize n Cn (ll) E F as a topological subspace of C (Il,F). The main lemma (3.3) y y for theorem 3.5 goes back to [10] as well as to Prolla and Guerreiro [20]
(in the case of Banach spaces). It has also some further appli-
cations which generalize parts of the results of [20]
and which may
be of interest in connection with the theorem of Paley-Wiener - Schwartz
C~o(E,~) '.
for the elements of
In the last section we prove that, for open subsets in certain 1. c. spaces
El and
E2
III and 112
respectively, there exists
a
na-
tural topological isomorphism between C~O(1l1,C~oW2» and C~o(\ll x 11 2 ). By the results on the a.p. of tation of
C~o(1l1
x
11 2 ) as
While finishing
his
C~o(m this also implies
(-tensor product
a
represen-
C~o(1l1);( C~o(1l2)'
investigations, the author received
the
preprint [12] of Colombeau, where spaces of COO-functions on Schwartz bornological vector spaces are studied. Colombeau has pointed out to the author that any function in C~o(Il,F) is a Cn-function in Silva's large sense if the quasi-complete space
E is given the compact bor-
nology. For Frechet spaces and for strong duals of Frechet - Schwartz spaces
E
both notions coincide. This shows that
the
bornological
setting is more general as far as the Schwartz property of C~o(ll) is
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
265
concerned. However, it is not known how to prove the results of this paper with bornological methods if
E is the strong dual of a Frechet-
Montel space only. Let us come back to distributions again: It should be remarked that obviously some results in this article can also be regarded propositions on the dual of
as
c:o(n), a space which is a natural gen-
eralization of the distributions with compact support. mentioned in remark 4.8 will be the subject matter of
The a
results
subsequent
paper.
ACKNOWLEDGEMENT:
J. F.
The author thanks R. Aron, K.-D. Bierstedt,
Colombeau, H. Jarchow, H. H. Keller and L. Nachbin for some
helpful
discussions or correspondence on the subject of the article.
He also
gratefully acknowledges partial financial support from GFD and IMU.
1. PRELIMINARIES In this section we shall fix the notation, recall some definitions and state some results which will be applied later.
We
shall
use the theory of locally convex (l.c) spaces as it is presented e.g. in the books of Horvath [17], Kothe [19] and Schaefer [21] . Throughout this article, a l. c. space always means a
Iteat Haus-
dorff l.c. space, because we only want to deal with real differentiable functions.
1.
Let
E and
subsets of
F be 1. c. spaces and let
E which cover
tinuous linear maps from
y be a system
E. Then, on the space E into
bounded
L(E,F) of all con-
F one can introduce the correspond-
ing y-topology of uniform convergence on the sets in ing l.c. space is denoted by
of
y; the result-
L y (E,F). It is well known that this ta.
pology does not change if the system is enlarged in such a way
that
MEISE
266
it is directed under inclusion and that any subset of a set in y belongs to
y. We shall always assl.Ulle that
y has these properties.
finite Yb we denote the systems of all dimensional bounded, all compact, all precompact and all boundeds~ By
Ya' y co ' y c
and
sets of E. The corresponding spaces
Ly (E,F) are denoted by La (E,F) ,
Lb(E,F). We write and
2.
Ly(E) instead of
DEFINITION:
a)
By
E and
F be 1. c. spaces, y a system of bounded
n
IN.
E
£(En,F) we denote the linear space of all n-linearmap-
pings from b)
·E n
into
F.
n > 2; u E £(En,F) is called y-hypocon~inuou~,
Assume for any
k with
1
~
hood W of zero in in E such that 0)
~
k
n, any
{u
£ (En ,F) for y
E
£(En,~)
S
E
Iu
is
neighbour-
n = 1 by
zero
L (E,F) and for
case
sn
for
£y(En,F) with the topologyof ~-
y, we can endow
of
y-hypocontinuous}.
u E £y(En,F) is bounded on
form convergence on the system continuous
£a(En,F)'£CO(En,F), ••• if
yn
linear
{sn I s E y}. mappings
As
we
in
write
Y = Ya' Yco ' •••• The elements
£cr (En ,F) are called ~epalLaA;e.ty con~inuou4 n-.tinealL map-
pi.ng~.
d)
Y and any
u(Sk-l x V x Sn-k) c W.
Since, obviously, any
of
E
if
by
n > 2
the
S
F there is a neighbourhood V of
We define the space
any
instead of
Y
Ly (E,E).
Let E and
sets which covers
E'
We write
£ (En) instead of y
u E £(En,F) is called 4ymme~lLi.c, if for any permutation of
n elements and any
x = (xl' ••• , Xn) E En
we
~
have
SPACES OF OIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
U(X'lT (I}""
267
,x'lT (n))
t~(En,F} of
The closed linear subspace
ty(En,F}
is
en-
dowed with the induced topology. The proof of the following lemma is an easy exercise.
3. LEMMA:
a)
Fon any Fo~ any
u E ty(En,F} th~ 6ollow~ng hold¢ tnue:
S E Y the ne¢tn~et~on
u
I sn
~~ un~6o~mly eon -
tinuou¢. b)
any k. w~th
FM ~4
1 ~ k <
n,
and a.ny
S
E
y,ul!f- l
y
a system of bound-
E x Sn-k
x
eontbtuou.¢.
4 . DEFINITION:
LetE and F be 1. c. spaces and
ed subsets of E covering E. a)
The l.c. spaces by
b)
L~ (E,F) :
F
L~{E,F) and
n
I
E
IN, are defined
inductively
n Ln+l (E,F):= L (E,Ly(E,F». y
y
There is a. unique sequence (.pn) n e IN of linear .pn : Lnm,F) ~ £(En,F)
satisfying
y
=:
is injective for any
n
E
IN
1 = 1~(E,F) 'd_
[
X
L n (E, F) as a linear space of n-linear mappings on y
values in
c)
Let
an d
J n 2 '·· "xn +1 . Since
.p n+l Cu) [ xl'" "xn +1 }
n
mappings
En with
F.
n e IN be fixed. An element
U E Ln(E,F}
y
¢!lmmet4~C. if the corresponding n-linear mapping symmetric. We define {u e Ln (E,F) y
Iu
is symmetric}
is
called .pn(u) is
268
MEISE
and endow this linear space with the l.c. topology induced n
by
Ly (E,F).
5. PROPOSITION:
~y~tem 06 bounded ~u.b~et~
PROOF:
~pa.ee.~
Le.t E and F be. l. e.
and le.t y be. a eoveJL.i.ng
06 E. Then .the mapping <(in: Lns(E,F) -+£s(Ef,F) y y
The proof is by induction on n. For
obviously true. Hence, let us assume that morphism for
n =1
<{ij
the statement is
is a topological iso-
I < j < n. We shall prove that <(in+!: Ln+I'S(E,F) +£s(Ef+1,F) -
-
Y
y
is a topological isomorphism. This will be done in several steps. a)
For any Let Then
u E L~+I,S(E,F) .,on+l(u)
S E Y
is y -hypocontinuous:
and a neighbourhood W of zero in F be given.
U~,w:= {m E £~(En,F)
of zero in
I m(Sn)
c W} is a neighbourhood
£s(En,F).By induction hypothesis
<(in
y
pological isomorphism. Since
is a to-
u: E -+ Lns(E,F) is oontinoous, y
in E, such that there is a neighbourhood V of 0 n c W. But '{!no u(V) c Us,w' Le. u(V)[ Sn this implies '{!n+l (u)[ V x Sn] c W. Since u is symmetric, this shows <{in+l (u) E £s (E n + l ,F) . y
b)
'{!n+l
is bijective:
The injectivity of tivity of
,,,n+l T
'{!n+l. Take any
is clear. Let us show the surjecmE £s(En+I,F) and define, y
for
any xl E E, u(x I ) :~-+F by u(xI)[~,,,,,xn+l] :=m(x l ,··· ,xn +1)' Then U(X ) is in 1 of zero in E we have Thus we have defined a mapping
u : E -+ £sy (En , F) , which
is
linear and continuous, because for any neighbourhood W of
o
in F there is a neighbourhood
u(V)[Sn] = m(V
x
Sn)
C
W, Le.
v u(V)
in C
E
U~,w'
such
that
By induction
SPACES OF 01 FFERENTIABLE FUNCTIONS ANO THE APPROXIMATION PROPERTV
hypothesis cJ
tp
n+l
u :
is in
269
L~+l,s (E,F) and
tpn+l(u) =m.
is a topological isomorphism:
This is easy to see, since
{U~:~ ISEy,Wneighbourl:x:lod.ofOinF} £~(En+l,F)'
is a fundamental system of neighbourhoods in
u~:~ = {u E L~+l,s (E,F) I u(S) C (tpn)-l{~,wJ}
while the sets
describe a fundamental system of neighbourhoods inLn+l,s(E,F) y
if
S runs through
hoods of
6. DEFINITION: a)
0 in
y and W runs through
the
neighbour-
E e; F:
Le (F~,E)
F.
Let E and F be 1.c. spaces.
The e-p1todue-t 06
E and F is defined as
I
where e denotes the topology of uniform convergence on the equicontinuous subsets of
F'
(cf. Schwartz [22]or Bierstedt
and Meise [5 J ). n
b)
is inj ecti ve, hence
E
E:
F
induces via
E @ F, called the injee-tive or
on tion
E 0e; F
pltO du.et (} 6 E 0)
n
The mapping j :E0F .... E £F , j( l: ei@f.)[y']:= l: (f./Y')·e i i=l ~ i=l ~
of
j
a 1. c. topology
e;--toPQ~ogy.
The comple-
E 0e; F is called the injec.tive or £--te.J'UI01t
and F.
E has the approximation property (a.p.) in the Grothendieck, if
E' 0 E
is dense in
Lc(E)
sense
of
(cf. Schaefer
[21], III, 9.1).
We shall use the following result of Schwartz [22], Ch. I,
§l,
Prop. 11, in the form stated in Bierstedt and Meise (6].
7. THEOREM: A quui.-eomple-te l.c . .opaee
Banach .opaee
8.
k
and k
F
IR
E
ha.6 .the a..p.
-the algebJtaie teMOit pJtoduc.t
L66 601teve1ty
E 0 F i.6 del1.6e in
-.6pac.e.6: A completely regular topological space
E e: F.
X
is
270
a
MEISE
k-~pace
(klR-~pace)
if for any topological
space
Y
equivalently Y a completely regular topological space) f :X
-+
Y
is continuous iff
f
IK
lR
(y =
a
or
function
is continuous for any compact sub-
set K of X. By Arhangel ' skU [1 J (Blasco [8 J )
open
subsets
of
k - spaces (k lR -spaces) are k - spaces (k -spaces) again. m
2. SPACES OF DIFFERENTIABLE FUNCTIONS In this section we introduce 1. c. spaces of n times continuously y-differentiable functions and investigate their
topological
properties. Because of the applications in section 3, we are interested in the completeness and the Schwartz
property
mainly of
such
spaces. We begin by recalling some definitions.
1. DEFINITION:
f a function on
Let E and F be 1. c. spaces, n an open subset of E,
n with values in F and y a system of bounded sets
in E which covers
E. f is called y- d.i 66eJtent.iable
at a po.int a E n
if there exists uE L (E,F) sudl that for every S E Y lim ~ (f(a+th) - f (a) - u(th» = 0 t+o uniformly in h E S (i.e. for any S E Y and any continuous semi-norm q on F, there is
/) > 0
such that for any
t E lR, with
o
Obviously y-
(f(a+th) -f(a) -u(th» < 1).
u is uniquely determined by
deJtivat.ive 06 f .in a. We write
f
and a; u is called
f' (a) instead of
the system of all bounded (finite) subsets of E, f (Gateaux-) d.i66eJtent.iable at if
f
a. f
is y-differentiable at any
u.
If
the Y
is
is called FJtechet-
is called y-d.i66eJtent.iable on
n,
a E n.
For Gateaux-differentiable functions there exist several generalizations of the classical mean value theorem (see e.g.
Yamamuro
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
271
[24], 1.3). We shall use the following one, which is a consequence of
the Hahn-Banach theorem and a result of classical calculus.
2. LEMMA:
Let E and :f' be l. c • .6pace.6, n an open .6ub.6et in
a,b E n
let
tained in
be
:
=
S[a,b]:= {a + t(b
n. Auume 6uJttheJtmOlte that
Uable at anlj g (t)
.6uch that
f:
n ....
x E S[ a,b] and tha.:t :the mapping
a)
F
It
E
E
and
[O,l]} i.6 con-
i.6 Ga.teaux- di66e.Jteng: [0,1]
.... La (E ,F) ,
ff (a + t (b - a», i.6 con:tinuou.6. Then :the 60Uowing
hold<> .tJt.ue:
1
f (b) -
J
f (a)
ff (a + t(b - a})[b - aldt.
o
The following lemma indicates that y-differentiability of a function f is already implied by Gateaux differentiability and
a
continuity
property of the derivative (see also Keller [18], 1.2.1 and Yamamuro [24],1.4.4),
3. LEMMA:
f :
n ....
Let E and F be Le. . .6pae.e.6, n an open .6ub.6et 06 Ga.:teaux diH eJten:tia.bte ann.
F
tinuou.6, thelt f
PROOF:
i.6 y-di66eJtentiable on
Let a be any point in
bounded subsets of
E and let
n,
16
ff : n .... L (E,F) y
E
and con-
n.
S any element of the system
y of
q be any continuous semi-norm on F. By
the oontinuii:;y of ff in a, for e: > a there exists a convex balanced neighbourhood X E
a +
U of zero in
E such that
a + U c n and such that for any
U
sup q«ff (x) - ff (a»)[s 1 ) < e:. SES
Since 2 we
S is bounded in E, we can find have for any
t
wi th
a
<
I t I .::.
0 > 0 <5
with
and any
oS c U. By lemna h E S:
MEISE
272
~ (f(a + th) - f(a) - f' (a) lth])
~ (J 1 (f'
(a + 1: th )[ th 1 - f' (a) [ th 1 ) d
1:
o
r
This implies
1
Jl
Supq(E(f(a+ th)-f(a)-f'(a)[thj).=:.sup hES
Hence
f
is y-differentiable at
n
E
0
a.
Let E and F be 1. c. spaces, n f {2l an open subset of
4. DEFINITION: E and
q«f'(a+Tth)-f'(a»)[hl )dT < E •
hES
y a system of bounded subsets of
E
which
IN 00 (:= IN U {oo }) we define the .6pa.c.e 06
y - d-i 66e.ltent-iabte
6unc.t-io Yl.6
e~(n,F)
0
n n w.I.th value..6 -in
:= {f : n ->- FI
for any
fj E eW,L~(E,F»
with
o.=:. j < n
on nand
The vector space c.Of1veltgef1c.e
j
E
lN
o
t,tme.6 F
E.
a
with
O.=:.j < n +1
(fo :=f) and for any
is
fj
subset of F,
For
c.ont-inuou.6ty
j E lN o
Gateaux - differentiable
f~ )
enen,F) is endowed with the topology y
un-i 6Oltm
06
06 the. de.lt-ivat-ive.
n. This topology is given by the system {po K S } "-, , ,q
norms, where
norm on
n
covers
l
is any integer with
n, S is any element of and where
Pt,K,S,q
y
0
~
of semi-
l < n + 1, K is any compact
and q
is any continuous
is defined as
semi-
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
Pt,K,S,q (f) :-
sup sUP j q (f . (x)[ Y 1 ) • J x€K yes
sup
O~j.:::.t
In the sequel we shall write
REMARKS: a) By lemma 3,
o .: :. b)
A
j
273
f (j) instead of
fj
is really
f
.
j
y-differentiable
~
on
for
< n.
function
f is in
C~W,F) iff it is of class
in the sense of Keller [18], 2.5.0. The advantage of def1nition 4, however, will become clear pretty soon (see e.g. proposition 5). Obviously we have for any semi-Monte 1 space E:
0)
S:(I'l,F}.
Q 06 the l.. c. 6pac.e. E be a
5. PROPOSITION:
Lez zhe open
and te.t
be. g,[ve.n. Auu.me. that
j
E lN
n
E IN,,,
o w.<.th
PROOF:
Let
o .: :.
< n + L
j
~.1 and any j with
C~ Ul,F)
for any
C~(n,F)
0.:::.
60Jt
a.ny
complete.
Cn(~,F). The definition y
implies that for any compact subset K of
j < n + 1, (f!j) IK) 1s a Cauchy net in C(K,L~ (E,F».
j
(lEA
with
0.:::. j < n + 1
there is
C (Q,Lj (E,F»
E
such that
J
derivative of gj equals
gj+l: Let
Then there exists an open interval +
gj
converges to 9 . uniformly on every compact subset of O.
Now we shall show that for any
va : t
,[~
"'R-~pctce.
L~(E,F) is complete by hypothesis and since 9 is a kE-spaoe,
Since
(f(j» (l
Then
L~(E,F} L6 comptete
be any Cauchy net in
(fa) a E A
of the topology of
~ub~et
f~ j) (a +
th)
j
with
0 < j < n
a
E S'l
and
I in
is defined for any and
a
v~ (t)
h E E
be given.
lR on which the
function
€ A.
Obviously
= f{j+l) (a +th)[h I. (l
274
MEISE
For any two l.c. spaces
= u(x)
e: (u,x)
X and
Y the evaluationmape::Ly(X,y) xX .... y, (v~)
is separately continuous. Hence
towards the function
w : t .... gj+l (a + th) [h I,
a
E
A
converges
uniformly
on
every
compact subset of v'
=
I. Thus, v : = lim v is differentiable on a.... ex w. Because of vet) = gj (a + th), this implies
lim tl (g.(a+th) -g. (a» = lim v(t)-v(o) t-+o] ] t .... o t
I
and
v' (O)=w(O) = gj (a + th),
Le.
This shows
n
6. REMARKS: a) Let us recall that any open subset
of a metrizable
l.c. space or a (DFM)-space is a klR-space. b)
Concerning the completeness of be remarked: If
F is complete and
L~ (E,F) is complete for any plete, and
E~c
equals
complete for any space
j
E
Let E.
any
n
PROOF:
E lN co
E
bornological,
j E JN o ' If
JN
o
j
then
F and E~ are ~
E topologically" then
L~ (ErF)
is
' Especially for any (F) -or (DFM)-
E and any complete 1. c. space
is complete for every
7. COROLLARY:
L~ (E,F) the following should
E
F, the
spare L~ (E,F)
JNo '
E be any (DFM) -.6pac.e, F any (F) -.6pac.e and
n
an
C~(n,F) =C~o(n,F) =c~(n,F),if.. an (F)-.6pac.e6oJ!.
'
As it was shown by Dineen (13), Prop. 1 and prop. 5,
hemicompact k-space. Hence proposition 5,
n is a
C~o(n,F} is metrizable. By remark ~b)and
C~o(n,F) is complete.
The following lemma will be useful in the sequel.
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
Le.t E be a i.e. • .6pae.e, y a .6y.6.tem 06 bouYlded .6ub.6e.t.6
8. LEMMA:
E wh-<.e.h e.OYl.ta.iYl.6 .the e.ompae..t .6e.t.6, 11 aYl opeYl .6ub.6e.t 06 E~
TheYl
PROOF:
CnW) is defined only for
Since
Y
A : c~ (11)
continuous linear map
a
y(j) =
A(f)(j) =0
by
06
C~(Q).
11 'I (lJ, we can choose a Ell.
A(f) : x ->-. Since any
y on E coincides with its own derivative and
for
for
C~(Il)
-+
06
E aYld n E IN",.
-<..6 a e.OYlUYlUOU.6.ty pJtojee..ted .topo.tog-<.e.a.t .6ub.6pac.e
Then we define
since
275
j
~
2, A projects
As
j ~ 2, the continuity of
A follows from the esti-
mates:
(1)
sup [ A (f) (x) [ xEK
sup [[ < Pl,{a},K(f) xEK
for any compact subset
( 2 ) sup sup [ A ( f) xEK hES
I
(x) [ h 1 [
for any compact subset Similar arguments show that
9. DEFINITION:
Let
K of
11.
sup sup[
of
E'
Y
11 and any
S E Y .
is a topological subspace
E be a 1. c. space. A subset
K of
E is
veJty e.ompae..t, if there is a Banach disc B (i.e. a convex bounded subset that
B of
E for which
K is contained in
10. REMARKS:
EB
EB
is a Banach space) in
of
called balanced E such
and compact there.
a) The notion of very compact sets was introduced (with
a different definition) by de Wilde [23],Chap. III, 4. Def. b)
By a consequence of the Banach-Dieudonne theorem a K of
subset
E is very compact iff there exists a convex balanced
276
MEISE
compact subset 0 of E such that K is contained
in
EO
and compact there.
11. PROPOSITION:
Let E be a .e. c • .6pace .in wh.ich elleft.y compact .6u.b-
.6et .i.6 veft.y compac.t. Then
C~o(n) .{..6 a SchwaJt.tz .6pace 60ft.
any
open
.6U.bHt n 06 E.
PROOF:
By a well-known characterization of Schwartz spaces, it suf-
fices to show that for any compact subset K of set
0 0 of E, and any
n , any compact sub-
n E IN there is a compact subset 0 of E such
that any sequence
COO (n) with sup Pn+l K Q(f D co .eElN" <-
<
}
1
contains a subsequence which is Cauchy with respect to the semi-norm Pn , K, Q0 • Since the closed convex hull of a compact set in E is compact again,
K can be covered by a finite number of compact convex
Hence, w.o.l.g., we may assume that
K is convex. By hypothesis
by remark lO.b) there is a balanced convex compact subset such that
K U
Q C 0 o
sets.
and K as well as
Now take any sequence ( f.e) .e E IN in
Q
of
0 0 are compact in
c''''co (n)
and E,
EO'
with sup Pn+l K O(f.e)
.eElN' ,
~l.
In order to show that (f.e).e. E IN contains a subsequence which is Cauchy with respect to the semi-norm fix j with 0
~
j
,
.
gj,.e(X'Yl""'Yj} := f.e]
the topology induced by
j+l Eo
we proceed as follows:
We
.e E IN, gj,l : K x oj ... lRby
C(K x Qj), where
K x Qj
is given
.
Let (a,y), (b,z) E K x oj implies
0
(x)[Yl'''''Yj)' Then we show that {gj,.e.I.eE~}
equicontinuous subset of
f?) (x)
,
~ n and define for any ( )
is an
Pn K Q '
be given. Then multi - linearity of
277
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
Ig·J , .e.(b,z) - g.J , <-o(a,z)1 +
j
+ k:l Igj.,.e.(a,zl'·" ,~, Yk+l"" ,Y j ) - gj,.e.(a,zl'··· ,zk-l' Yk ,· .. ,Y j ) I.
The general mean value theorem (lemma 2) gives
sup sup '+1 1f lj+l) (x)! wI 1 ~ Pj+l,K,Q(f.e.) ~ 1, this implies (obxEK wEQJ serve that only for a ~ b there is something to prove) :
By
Ig·J , <-o(b,z) -g J, . .e.(a,z)1
1 (f(j) (b) - f(j) (a»)[ .e. .e.
Z
I 1< -
lib - a liE • Q
Concerning the other terms in (1), the following 1
~
k
~ j
(observe that only for
Thus we have shown
zk
~
Yk
holds
true
for
there is sanething to prove) :
278
MEISE
is equicontinuous on K x Qj C E6+ l . Since , , '+1 K x Q~ is a compact subset of K x QJ in E6 ' and since {gj, I' I E IN } {gj ,I , I E ::IN}
Hence
is uniformly bounded on pact subset of
K x
Q~
, {gj ,I' I E ::IN}
is a relatively com-
C (K x Qj) by the theorem of Arzela - Ascoli. But then
o
it is possible to choose (inductively)
a
subsequence
of
which is a Cauchy -sequence with respect to the semi-norm
p
n,K,Qo' By the considerations at the beginning, the proof is now complete.
REMARK:
A similar argument as in the proof of Proposition 11
was
used in the article of Bierstedt and Meise [7) , theorem 7. (a), where it was shown that the space subset
compact
K of a metrizable Schwartz space is a Schwartz space again.
Let E be a qua.6i- c.omplete I. c..
12. THEOREM:
ing
H(K) of holomorphic germs on a
a~e
.6 pac.e.
Then the 60Uow-
equivalent: ih a
(1)
E~
(2)
Eve~y
(3)
FM
Sc.hwa~tz
.6pac.e.
c.ompac.t .6ub.6et 06 E i.6
any open .6ub.6et n 06
ve~y
c.ompac.t.
c~o (!'l) = c~ (n) ih a Sc.hwa~tz
E,
hpac.e. (4)
The~e
ih a
PROOF:
exihth an open hub.6et
E~ and hence
E
OM whic.h
COO
co
(1)
E and
E. Then
KO is a
KOO is equicontinuous in (E~) , •
E is quasi-complete, the topology
the duality between
06
K be any compact subset of
neighbourhood of zero in Since
¢)
hpac.e.
Sc.hwa~tz
(1) ~ (2). Let
n (F
A(E',E) is compatible with
E', thus, KOO is equicontin\Dus1n(E~)'=E.
By the dual characterization of Schwartz spaces (see e.g.Horvath [17)
3, §lS, Prop. S) , there is a compact subset Q of
E such that
OO K
• compact in (E ') I Since E is quasi-complete, QOo c QOo is compact by the theorem of bipolars. But then K, being compact in
is relatively
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
E, is compact in the Banach space
E
QOo
( 2)
'*
( 3)
by proposition 11.
( 3)
".
(4)
(4)
.
trivial.
(1)
by lemma 8 .
REMARK:
279
Using the concept of bornology and the notion of Silva dif-
ferentiability, Colombeau [12] gives independent proofs of proposition 5 and theorem 12 in a more general setting.
13. COROLLARY:
Let:.
E be any (F) -lIpac.e
be an al!.bI:tJr.al!.Y open lIubllet 06
Oft
E. Then
any (DFS) -lIpac.e and let
COO (12) co
oo
= c c W)
n
III a Sc.hWlVLtz
lIpac.e.
It is a consequence of the Banach-Dieudonne theorem (see e.g.
PROOF:
Kothe [19], §21, 10. (3»
that any compact set in a Frechet space
is
very compact. Since (DFS)-spaces can be represented as compact injective countable inductive limits of Banach spaces, by [14], §25, 2.2, every compact subset of a
14. REMARK:
Floret - Wloka
(DFS) -space is very ClCIt{l8.ct.
It would be interesting to know whether, concerning nu-
clearitY'C~o-functions behave similar as holomorphic functions complex 1. c. spaces (cf. Boland [9 ] ).
Since nothing in this direc-
tion seems to be known, let us remark that for any open subset E =
Ell
JR
the space
n e IN
jn: mn
->-
E
n
of
n e
m
C~o W) is nuclear.
But this result is essentially finite dimensional: For let
on
denote the canonical embedding. Then
is an open subset of
mn, hence
COO (rl ) is nuclear for any n co n
Em.
Now it is a consequence of Yamamuro [24 L(1.6.1) ,that COO (m =proj COO (Sl ). co +-n co n Since the projective limit of nuclear spaces is nuclear, this proves the nuc1earity of
C~o(n).
280
MEISE
3. THE ROLE OF THE APPROXIMATION PROPERTY The aim of this section is to derive a
condition for
suffici~
e~o (0). This will be done by an application of theorem
the a.p. of
1.7. Therefore, we first give (under appropriate hypotheses) a charen (0) and a quasi-carplete l.c.space.
acterization of the E-product of
y
Le.:(: E and F bel. c.. .6 pac.e.6, let y be a .6y.6tem 06 bounded
1. THEOREM:
.6ub.6et.6 06 E wh..(.ch conta..(.n.6 the compact that E
j
..(..6
a
kJR -.6 pace
qua.6.(.-c.omplete. Then
60Jt
en (0) y
polog..(.cal l..(.nea.IL .6ub.6pace
E
1 F
~j .(..6
and let 1
.6et.6
~n ~ "'.
c~ Ul) and
+ 2 and that
M.6ume F
Me
topolag..(.c.aUY..(..6omo.ILph..(.c. to the ta-
e~P(n,F)
06 C~(n,F) whe.ILe
p.ILecampact..(.n F}.
PROOF:
The proof is similar as in the finite dimensional case,
but
becomes more involved, since we have to deal with total derivatives. The general idea is the following: Define /j. (x) : = Ox and show that the mapping morphism between
f
-+
e~ (0) E F ... Le (e~ (0) ~,F)
by f
0
/j.
and
is a topological isoenp(n,F). This
will
y
be done in several steps. a)
For
0 < j < n + 1
( /j.j (x,y), f) By hypothesis Blasco [ 8 ], n x E /j.
IK
x oj
j
define the mapping /j.j·nxEj-+en(O), • y c by
: = f (j) (x)[ y 1. Then Ej +1
/j.j
is a kJR -space, hence
is continuous. by
the result
of
is a kJR -space. Thus, it suffices to show
that
is continuous for any compact set K in n and any
cxmp!ct
subset Q of E. From the definition of the topology of obvious that
/j.j (K x Qj) is an equicontinuous subset of
SPACES OF DIFFERENTIABLE FUNCTIONS ANO THE APPROXIMATION PROPERTY
equicontinuous subsets the topology of
coincides with
the
J "'"
d'1n)' y cr
iK x ~
weak topology, hence we only have to show that is continuous. Let ti ve of
f
f(j) : K
->-
f
is synunetric, by proposition 1.5 we have the continuity of
L~s(E,lR)
continuous on
£~(Ej).
For
For
S E Y
Then
(x,y)
-+
P~: £~(Ej)
let
I
continuous,
is
f(j) (x)[y]
= P6
K x Qj. Thus the continuity of
0
6
(f(j)
j
CB(Sj)
-+
especially
I K)
(x,y)
is
is proved.
1 < j < n + lone can define a continuous linear map-
ping
6j:n"'£~(Ej,e~(n»~}
~j(x):
by
t.j(x,·).
By part a} and by the synunetry of the derivative it is obvious j j 6 (x,.) is a synunetric j-times linear mapping from E into
that
cn(Q) , • Let us prove that
c
we have shown, that
a
j
t.j(x, .) is y-hypocontinuous: In part a)
is continuous in (x,o), hence for any neighCn(n)' there is a neighbourhood
bourhood W of zero in in
deriva-
y
and hence the mapping
y
: 1< x
en (Q) be arbitrary. Since the j - th
E
denote the restriction.
b)
281
c
y
U
of
zero
0
such
E such that
for any that
u E u j . If
S E Y
S C AU. Hence for
the synunetry of
is arbitrary, then there is
V:= A-j+lu
j 6 (x,.) this proves
pology of
K in
n and any
en W), the set a j (K y
CnCQ)', By the coincidence of y
x
S
>
one has aj(x,v x Sj-l) c ",j(x,.)
E
w.
By
£~(Ej,C~(n)~). that
y, by the definition of bhe to-
sj) - a
A(Cn(Q) y
E
Aj, First we observe
Now let us show the continuity of for any compact
A
I
j
(K
x
, CnCQ» y
sj) is equicontinuous in
and
282
MEISE
c~(n) I, for any convex balanced neigh-
on equicontinuous subsets of
e~ W) ~
bourhood W of zero in
there are
f l' ... , fm
e~ W) such
in
that
Since
f~j): n ~ !~(Ej) is continuous for
there exists a neighbourhood any
k wi th
1
~
sup. yESJ
U of
1 ~ k ~ m, for any
x such that for any
I f~j)
(x)[y I -
x' E U
f~j) (x'}[y I
and any
denotes the gauge of
~j c)
and
y E sj
W. Since
is continuous for any compact subset hence
U
< l.
By our first observation this shows for any
qw
E
k < m
Hence we have for any
where
x'
x E K
x'
Un K
E
S E Y
n.
K of
was
But
n
arbitrary,~jIK is a k
m-space,
is continuous. For any
u
F
E
E
e~(rl)
L
(en(n)
I
eye
,F) the mappingf :=uoi'>: n+F
u
e~P(n,F).
belongs to
It is easy to see that for
u
0
0 < j < n + 1
the mapping
~j
is continuous. Hence we have proved
can
show
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
f (j+1) u
any <5
u
0
~j+1
h E E, any
x E
such that
0
>
{~(L'1j (x + th,y) t
and any 1
n
E
<5 }.
is equicontinuous in
< n. To do this, take j with 0 .::. j < n,
j
and an arbitrary
~
x + th
{x + th l i t I .::.
K :=
o .::.
for
for
I
t
0
f
with
t
h E S. Choose and
I
<
I
< <5 }
<5
put
lI
j
(x,y) lyE Qj, 0
C~(~) "
<
I
t
since by 2.2 we have for any f E C~(~)
y E sj
'
I
t
with
S E Y
Then remark that the set
,
I
1
=
283
('+1)
J
(x +Th)[h,y]dTI .::.
! (f(j) (x +th)[y] -
t I Pj+l,K,S(f) <
<5 •
f(j) (x)[y] ) I
Pj+1,K,S(f)·
coincides with the weak topology
Since the topology of
on
equicontinuous subsets, we have lim ~(t:;j(x + th,y) - t:;j(x,y) - t:;j+1(x,(th,y))) t ..o
in
Cn(~), y c
in
y
E
0
t
uniformly in
Y E sj
if this holds in
Cn(~), uniformly y a
sj. But the latter is a consequence of the defini tion of cPy (n),
since for any
f E C~(~)
<~ (t:; j (x + th, y) - t:; j (x, y) - t:; j + 1 (x, (th, y) ) ), f ) t
.l. (f ( j) (x + th) t
tends to zero uniformly in we get by induction
y
E
sj
f (j)
(x) - f ( j +1 ) (x)[ th ]) [y]
i t t tends to zero. From
this
284
MEISE
lim ~(f(j) (x + th) - f(j) (x) - u o ~j+l(x)[thl) t~o t U U in
j F). Hence £s(E y ,
f~j+l) (x)
u
0
f(j) u
0
is Gateaux-differentiable in
Zj+l (x). Since
u
0
~j+l
x E 0.
and
is continuous, by 2.3 we have
fu E e~(n,F).
shown
In order to show that we even have compact
K in
0.
pact in
F.
y
I
c
~
0
Alaoglu-Bourbaki that for en (0.)
take
any
S E y. Then it follows from the theorem of
and any
tively compact in
fu E e~P(n,F),
,
j < n + 1
hence
u
~j (K
the set
x Sj)is rela-
~j (K x sj) is relatively com-
0
But we have shown above that
u o
d)
The mapping
k : F
£
e~ (0.) ... e~P (n,F) defined by
k(u)
:= f
u
is an injective topological homomorphism. Let any compact
K
in
n,
any
S E Y
and
t < n + 1 be given.
For
v.e.,K,S,l we have .e.
u j=o
~j(KxSj»O
.e.
n (~j (K x sj»
0
j=o J!.
n {f E c~ (n) I sup sup. !f(j) (x)[y J
j=o
I
xEK yESJ
V.e.,K,S,l·
By the theorem of bipolars this implies
-<
l}
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
--;;-9-------
9u
r
j=o Since any
U
li
a
j (K x sj).
E F e: en (n) is weakly continuous on equicontinuous suby
U E
q(u(v»
VEV o
U
j=o
sets, we have for any continuous semi-norm q on
sup
286
F
and
any
en w) y
FE
sup
q(u(v»
,9-
.e,K,S,1
vE U lij(K j=o
x
sj)
sup xEK
sup. q (u (li j (x, y) ) ) yES]
sup xEK
sup. q(f(j) (x)[y I yES] u
Po .... , K , S ,q ( fu) •
Hence the result claimed under d) is proved. e)
The mapping
k defined in d) is surjective, Le.
a
topo-
logical isomorphism. The surjectivity of inverse
j
of
k will be proved by constructing a rightf E e~P en, F) and any
k. Take any
y'
0
f E e~(n). Hence
uf(y' ) := y'
0
f. Let us assume for a moment that
an easy exercise to show that u
f
F'
+
e~W)
by
y' E F', then it is we
can
define
w) by E en (n) E F holds. Then we can define j : enPen y , F) + F E en y y f j (f) = t Uf , since by our hypotheses transposition is a topological
u
isomorphism between
e~w)
is now proved, if we show
and
EF
k
0
j
=
en(n). The surjectivity of k y id But this is a consequence e np Y W , F) Fe:
286
MEISE
of the following identity which holds for any
~,
y' E F, any x E
and
any
(k
0
t·
j(f)[x I ,y')
(ufo/'.,
0
(x},y')
< f (x),y' ) .
Hence the proof of the theorem is complete, if we show u Let any compact subset
K
By hypothesis, the set
Lf
hence La l
of
,
any
~
f (j) (K)[ sj I is
~
:=
S E Y / and
sup
Pf,K,S (uf(y'»
sup
o.::j~l xEK
sup
sup
o~j~l xEK
.
a i.e. Uf(L.e} c V .e,K,S,l
16
2. COROLLARY: ~y~t:em
Yeo
~ub~et
~
On
06
the
sup. yES]
pre compact
F'c • For any
in
F,
y' EL~ we have
u (y,)(j)(x)[y] f
sup. I y' (f(j) (x)[y] ) I -< 1, yE S]
Thus we have shown
hifPothe~e~
EC~W) e:F.
f < n + 1 be given.
j=o
is a neighbourhood of zero in
f
06
theo~em
u
f
1
E
L(F~,C~(Q}) •
a~e ~at~~6~ed
E, then we ha.ve nOlL
alt c.ompac.t: ~u.b~e.t~ 06
oo~
the
a.ny open
E:
C~o W) e: F
Th~~ ~~
the c.aH
c.omplete f. c..
PROOF:
C~;(~,F}
OM
~pac.e
anif (F) F,
-~pac.e OJ!.
al1d al1Y
al1if (DFM)
-~pac.e
E,
al1if
qu.a~~-
n E IN,,,,
The first part of the statement follows from theorem 1,
=
C~o(~,F). To prove this identity, let
f E C~o(~,F)
and be
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
given. Then for any and any compact
1 ~ j
E IN with
j
Q in
287
< n + 1, any compact
K
in rl,
E, by part a) of the proof of theorem 1,
we
have
Now observe that the evaluation map A: C(Qj,F) xQj .... F, A(g,x) :=g(x), is continuous. Hence
B: K x Qj .... F, B(x,y)
= A(f(j) (x) ,y),
is con-
tinuous and thus
is compact in
F.
The second part of the statement follows from remark 2.6.
REMARK:
Corollary 2 generalizes a result of Bombal Gordon and GJnz.3.lez
Llavona (10), who characterized for Banach spaces of
E, Aron (2)
C
n
co
(E) £ F
for Banach s,paces E. Also
gave a somewhat different description
C (E) £ F. b
Now we come to the main lemma for many of the results presented in the sequel. For Banach spaces
E
it goes back to Bombal Gordon
and Gonzalez Llavona (10) as well as to Prolla and Guerreiro (20).
Le.t E be a qua-6i-eomple.te baltltelled i. e. -6paee wi.th
3. LEMMA:
.the
6011owing pltopelt.ty: (CFA): Folt any eompae.t -6ub-6e.t EK
Fult.theltmolte le.t rl n
E
E .thelte exi-6.t-6 an
wi.th a.p. and a eon.tinuou-6 linealt
jK : EK .... E -6ueh .tha.t K
and le.t
K in
IN,,,,
C
f
E
--6paee
mapping
jK (EK) and j;l (K) 1-6 eompac..t in ~.
be an open -6ub-6e.t 06
and
injee.tive
(F)
E, le.t F
CgoW,F) be given.
be a noltmed -6paee,
Then 601t any
eompae.t
288
MEISE
I.> ubI.> £
e.t
n,
06
Ko
> 0 the.ll.e. -<-I.>
that
f
any c.ompac.t
u E E'
u E C~o(W,F)
0
sup xEK
sup
°2j'::'R.
PROOF: Put
and an ope.n ne.-<-ghboull.hood
€I E
W
06
U
Q
1-
(fo u) (j) (x)[y
and choose - according to
o
EK
(qs) s E IN of semi-norms. Since
m
E
any
K,
!.>uc-h
111
< £
o
(CFA) -an (F) - space
for which there is a continuous embedding jK:
w
and
and !.>uc.h that
Assume that the topology of
can find
o
sup. Ilf(j) (x)[y yEQJ
K: = Ko
EK with a.p.
R. < n + I,
-<-n E, any
Q
(EK) , €I EK
is given
by
EK + E.
the increasing
EK has the a.p.,
for any
system
m E IN
we
such that
I
m
By the quasi-completeness of vityof
jK
that
approximating
wm
tjK(E')
is
EK
and
E
i t follows from the inj ecti-
K
>'((E ) ',EK)-dense in
appropriately we can find
(EK) '.
Hence
urn E E' €I EK C E'
by €I
E
such that
sup qm(um(x)
xEK
Consequently there is a sequence for any
(x)).::.
~
m
such
that
K in C(K,E ). Since the
eva-
(urn) mE IN
in
E' 0 EK
s E IN
lim m+ oo
i.e. urn restricted to luation map set
-
K K
A: C(K,E )
sup qs(um(x) xEK
tends to x K +
id
-
K
EK, A(f,x)
x)
0,
:= f(x),
is continuous, the
SPACES OF DIFFERENTIABL.E FUNCTIONS AND THE APPROXIMATION PROPERTY
L
U u (K) mE IN m
K E , hence also compact in E. Since K
is compact in set of
:= K U
K the same arguments show that for any
u
KO U U m>s
EK and
K
:= L
Since
+ W0 c fl. Since
0
So
1
f(j)
:
j
with
(K ) 0
E 0
n
-+
id
K
in
C(K,E)
IN such that for any s>s
-
0
E
in
and
since
and any x E K
0
then it follows
and since for
fl
0
2. j 2. £. the func-
L~o(E,F) is continuous, there exists a convex
anced neighbourhood any
s
lim u m-+ oo m
Lo is a compact subset of
tion
the set
IN
E
E.
Ko C K, there exists
Put
m
s
Wo be a convex balanced neighbourhood of zero
Now let for which
is a compact sub-
o
is compact in
289
U
0 2. j 2.
.e.,
of zero in any
E with
x E Lo
Lo + U C rl such that
and any
z E E
with
balfor
x-z E U
the following estimate holds
(1)
s up',
yELJ
For
1 < j < £. the set
II f ( j)
(x)[ y ) -
f ( j) (z)[ y ) II
<
e: £. + 1
f (j) (L) is compact and hence bounded in o '-1 ' Lj (E,F) L (E,LJ (E,F». E is barelled by hypothesis, hence fJ(Lo) co co co j-l is equicontinuous in Lco(E,L (E,F». This implies that there is a co
290
MEISE
neighbourhood Wj of zero in and any
Since
y' E Lj-1
E such that for any
x E Lo,any Y1 EWj
we have
f(j) (x) is symmetric for any
x, this means that we have
IIf(j)(X)[Y]II~
(2)
for any
y = (Y1' ... 'Yj)
one of the Yk is in
Ej, where
E
j -1 of the
Yk are in
Land
Wj l
The set
j-1 ( n (W. n U) is a neighbourhood of zero in K j=1 J
hence there exists
s
{x E
IN with
E
EK
1
I q s (x)
Now we define
s > So
-1
u(x) E Ko + U C n
w : = u -1 (rl)
w we can define the mapping
is easy to see that with
j < n + 1
f
0
and any
for any
and by
x E Ko(from now
u as mapping from E into Ko and on
f () U : w ... F. By our definition 2.4
u E en (w,F) and that for any . co y
s
is an open neighbourhood of
E
Ej
x
E
w, any
it j
the following holds
(fo u) (j) (x)[y]
(3)
J/,
< --} C jK ( n (W. n U». s j=1 J
on let us omit the map jK ' Le. we regard E). Then the set
I
such that
and observe that by the choice of
our construction we get
EK
f(j) (u(x»[ u(y)] •
In order to prove the desired estimate, we observe first that we have U(Ko )
C
Lo
J
and that for any
x
E
Ko ) u(x) - x
(1) :
(4)
IIf(x) - f(z)lI~
t! 1
E U.
Hence we get fran
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
Then we observe that for any x E Ko
hence for any
j
with y E Q~
and any
IIf(j) (x)[y j- (fou) (j) (xj[y III
1
2. j 2.'£
291
we
have u (Q ) j C L., o J we get from (1), (2) and (3)
< Ilf(j) (x)[y j-f(j) (u{x»[u(y»)11
2. IIf(j) (x)[yj-f(j) (u(x»)[y) II + IIf(j) (u(x»)[yj-f(j) (u(x»)[u(y»)11 (5)
(.e. +
<
1) •
£
.e. +
1
E.
By (4) and (5), the proof of the lemma is complete.
4. REMARK:
Let us recall from Bierstedt and Meise [6)
ductive injective system
QompaQt.i'.y subset
~e9u.e.a~,
K of
subset of
if
E
{Ea.,ja.S}a. EA
= 0.-+ ind
E there exists
that an
of l.c. spaces
Ea.
is Hausdorff and if for any corrpact a. a. E A such that K is already a c:x:npact
Ea.' Then it is obvious that any 1. c. space
{Ea.' jo.S}a. EA of (F)-spaces
Ea. with a.p. has
(CFA). Hence in any of the following cases i)
E is a
ii) E
= *n~
E which can be
= n'" ind
inductive
the
property
E has (CFA):
(F)-space with a.p. En ' where
of (F)-spaces
iii) E
is called
E
represented as an inductive limit of a compactly regular system
in-
{En,jnm} is a strict inductive
system
En with a.p.
, where {E,j } is a compact injective inducn n nm tive system of {F)-spaces En with a.p. For brevitywe shall E
call any space of this type (DFSAl-space. Using a trick which goes back to Aron and Schottenloher [4 l,we can now prove the desired result on the a.p. of
C~o(n).
292
MEISE
5. THEOREM:
Let E be
and
n E 1N00
a.U the hypothe.6e.6 06 theoJtem 1 onE a.nd
e~o
(Q) E:
@
F
for any Banach space
en (n) co
identify F
E:
F
and
Q in o
any compact
E, any
rna 3, there exists such that
f
0
Let us define
u
E
F.
e~o (n,F). Hence we only have to show
f
en (n,F») any compact subset K of Q, co 0 £. < n + 1, and any e: > 0 be given. By lem-
u E E' @ E and an open neighbourhood
Cco(Qo,F)
~
~£
(finite
dimensional)
C~o(Eo) is dense
h E C~o(Eo) @ F
g
hou E C~o(E) @ F, and for any
IIf(x) - g(x)1I
in
£
3
C~o(n),
•
x E Ko
~
IIf(x) - f o u(x)1I + IIf o u(x) - h o u(x)1I <
<
E:
+ IIf o (u(x»
Furthermore we have for any any
result
such that
<
Then
Ko
F. Since it was shown in the proof of lemma
U(K o ) C Q n Eo = Qo' and since
there exists
of
Then
and
o
= Cco(Q)
w
C~o (w,F) satisfies the estimates given in lemma 3.
E
n
that
E
fo E e~o(Qo,F)' and by a classical
n
By corollary 2 we may and shall
e~o (Q ,F) for any Banach space F.
is dense in
To do this, let any
that
cf1 (n) co
e~o(Q) can be proved by showing that e~o(Q) @ F is dense
the a.p. of
e~o (Q)
in lemma 3 and a.6.6u.me oUlLthvtmoJte that OM
e~o(Q) is quasi-complete by hypothesis. Hence, by theoreml.7
PROOF:
in
Ct.6
x E Ko
- h(u(x»11 < 2£ .
I
any
j
with
1 < j < l
and
293
SPACES OF DIFFERENTIABL.E FUNCTIONS AND THE APPROXIMATION PROPERTY
IIf(j) (X)[y l-g(j) (X)[y) II < Ilf(j) (x)[y)- (f
+ II (f
0
u) (j) (X) [y I -
(h
0
0
u) (j) (x)[y) II +
u) (j) (X) [y I II
~ e: + IIf~j) (u(x»[u(y»)- h(j) (u(x»[u(y»)11
Hence we have shown of
E
0'
0
C~O(E) @ F in C~o(n,F).
6. REMARK: n
Q (f - g) < 2e:, which proves the density
Pi K ,
< 2e: .
All the hypotheses of theorem 6 are satisfied
lNco and any open subset
with a.p. or a
n of
E, if
E is either
for
any
an (F) - space
(DFSA)-space. This follows from 2.5, 2.6 and remark 4.
We shall show now that for Frechet spaces
E with a.p. this result is
optimal.
Fa!!. Il F!!.ec.het .6pllc.e
7. THEOREM:
C~o(n) hll.6 the Il.p. 6o!!. IlYlIJ
a)
n t- ¢ b)
06
c)
E
(a) (b)
~
n
E
lN co
E.
The!!.e eX-<-.6t thllt
PROOF:
E the 6o£.eow-<-Ylg Il!!.e equ-<-vllieYlt:
n E lN co IlYld IlYl opeYl .6ub.6et
c~o(n)
n t- ¢
06
E .6uc.h
hll.6 the Il.p.
ha.6 the Il.p.
~
(b): trivial
(c): By 2.8,
E~o
is a continuously projected topo-
C~o(n), hence
logical linear subspace of Frechet space
= E~
E the a.p. of
E~
E~ has the a.p. But for a
is equivalent to the a.p. of E, hence
E has the a.p. (c)
REMARK:
~
(a): This is clear according to the remark 6.
For Banach spaces
E theorem 7 was shown
by Bombal
Gordon
294
MEISE
and Gonzalez Llavona [10] for
= E.
rI
slightly different version (using
Again for Banach spaces
C~(rI) endowed with
the
E
a
topology
induced by Cn (rI)) of theorem 7 was presented by Prolla andG~eiro co [20) and also by Aron [ 3 I .
TheILe ex.t~t~ aft (FS) -~pac.e
8. COROLLARY:
ftot have ;the a.p.
601L
aftY
n
E
lN co
E
do el.>
I.>uc.h that
aftd aftY ftoft-empty opeft I.>ubl.>et
rI
06 E. PROOF:
This is a consequence of theorem 7 and the existence of (FS)-
spaces without a.p. The existence of such (FS) -space
follows
from
Enflo's counterexample, as Hogbe-Nlend proved in [16) . Because of lemma 3, the method applied in the proof of theorem 5 can be used also to derive some further density results
just
by
"lifting" density relations known in the finite dimensional case. Before stating them let us recall that a continuous n-homogeneous polynomial
p on E is called 6.tft.tte, if there
exist
such that n
n
p(x)
j=l
By
Pf
(E)
9. THEOREM:
(CFA). Theft (~\Il)
PROOF:
)
x E E.
for any
we denote the linear hull of all continuous n - homogeneous
polynomials on
rI
J
06 E
E, n
Le;t
lN
n
of
qua~.t-eompie;te
f
E
balLaiied
lN co aftd i. c.. I.>pac.e
E
Pf (E) 0
the I.>pac.e
Q
o ' It is easy to see that
E be a
nOlL aftY
Let any
paet subset
E
n
F
P (E) C COO
i.e.
F aftd any
.tl.> deftl.>e .tft
(E).
I.>paee
opeft
I.>ubl.>et
n
Ceo W,F) •
Ceo (rI,F), any compact subset K of
E, any
co
f
r!, anyo:::m-
i < n + I, any continuous semi-norm
q on F,
SPACES OF OIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
E >
and
0
be given. We shall show that there exists
g
295
E
Pf(E) 0 E
with
P.e,K,Q,q(f - g) <
Let
Fq
denote the completion of the canonical normed space
and let TI 0
f
E
E •
denote the canonical continuous linear map.Since q en (fl,F ), according to lemma 3 there exists u E E' ® E such co q TI
:
F
F /ker q
+
F
that
sup xEK
sup O~l~l
sUR· II (1T
0
f) (j) (x)[y]- (1T
0
f
0
u) (j)[y JII <
Now we proceed as in the proof of theorem 5 and define rlo := rl n Eo and fo: (1T
0 f) Irlo·Then fo
e~oWO,Fq)
E
Since the polynomials on Eo are dense in in
F q; and since
E •
yEQJ
u (K) is contained in
proof of lemma 3), there exists
rlo
=
EO := 1m u,
C~o(flo) @E
Fq •
C~o W ) ; since
1T(F) is dense o (this was shown in the
ho E P(E ) 0 1T(F) = Pf(E ) 0 o o
1T( F)
such that
Pl,U(K) ,u(Q) ,II' " (fo - h o ) <
E.
m Assume that
ho
i = 1, ... ,m. Then
~ Pi ® 1T(Yi) , where Pi E Pf(E o ) and Yi i=l m is in Pf(E) o F and h := ~ Pi 0 u 0 Y i i=l
E
F for as
in
the proof of theorem 5 it follows
P.e,K,Q,q (f - h) < 2e: .
Hence we have shown that
is dense in
Cn (fl,F) • co
The following corollary is an immediate consequence of theorem 9.
296
MEISE
10. COROLLARY:
Let
E be a qua-6-<--complete bcutJr.elled l.c. -6pace w.tth
(CFA) • Then 60ft an!! ~(F ~)
06
n E IN 00
an!! l.c. .6pace
,
C~o(E) ® F
E the .6pace
F, and an!! open .6ub.6et
c nco W,F)
-<-.6 d en.6 e -<-n
.
Looking at theorem 5 and corollary 10 and their proof
in
finite dimensional case one has the impression that condition (or more or less the a.p.) together with finite dimensional
the (CFA)
results
can be used instead of Coo-functions with compact support. The following theorem is of the same nature. Before we state it, let us remark that an easy calculation shows the following: Let and let
E'
subsets of
E be any 1.c. space
denote its (continuous) dual. For any system y of bounded E
(covering E) and any
e
y EE', the function
y
C~(E). Using this and the classical theorem
belongs to
of
Paley-
Wiener-Schwartz the proof of theorem 9 also gives
11. THEOREM:
Let
E be a qua.6-<--complete baftftelled i. c . .6pace
(CFA). Then 60ft an!! ~(F~)
06
E
n
E
lN oo '
the l.tneaJz. hull
F, and an!! open
an!! l.c . .6pace
06 the.oet
{e
y
'
wUh .6ub.6et
flY EE', f E F}
.t.6
den.6e -<-n
4. A KERNEL THEOREM FOR FUNCTIONS OF CLASS
C~o hypotheses)
In this section we shall show (under appropriate that any functions in
C~o(~l x ~2) can be regarded as an element of
COO (~l'Coo (~2)) and vice versa. Using theorem 3.5 this also co co a tensor product representation for
C~o W l x Q2)' Before we can prove
our result we need several lemmas. The first lemma is consequence of definition 2.4.
implies
an
immediate
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
Le;t E,F and G be l.c.. l.lpac.e.6, le;t !"I be an open
1. LEMMA:
06
E, le;t
be a e.ovelL-Lng !.>y.6;tem 06 bounded !.>ub!.>e;t
y
a)
Folt any
b)
FOIL
f E C~(!"I,F) ;the oune.;t-Lon f E C~{rI,F) an.d any
any
be.tong!.> ;to
16
2. LEMMA:
.6 ub.6 e;t 06 f(x , l
->-
PROOF:
j
00
f E C~o{rl,F)
wUh f{rl)
be.tong!.> ;to
C G
i2 : E2
continuous linear map
(i~)*(m)[xl
->-
El x E2
.j
=
j
E
IN. Thus
(i~) * : £~o «E I
m([i~(x)l).
: E
~2
by
E CcoWI x
-+
2
.
2
gives rise
~2
x E ) j ,F} ...
to
£~o (E~,F), defined
C~o WI x !"I2'£~o «EI x E2 )j ,F).
Q2,L~0(E2,F}}.Letus
nl the function
f
denote the function
C~oWlx !"I2,F) thatfor
E
gj(x I , , ) :
!"I2'"
L~0(E2,F)
is Gateaux-differentiable and that its Gateaux-derivative is gj+l (~" This proves that for any
xl E!"II
C~oW2,F), hence the function
a
(i J')*0 f(j) is in COO (nIx n ,£s (E J',F)), 2 2 co 2 co'
gj' Then i t follows from
and any xl E
2 .j
by
If now f is any element of C (QIX!"I2,F}, co
Then i t follows from lemma l.a) that
(i~}*of(j)
j
open
(x ) = (0 ,x ) is 2 2 (E x E )j is con2 I i
defined by
then by lemma lob), 1. 5,and lemna 1.c), f(j) is in
any j E lN
;then.
F,
c~o W,G).
g E C (n 'C~o (!"I2 ,F» l
on.e de6-Lne.6 a 6une.;t-Lon
The mapping
(i~)*O f(j)
f(j)
;the 6un.e.;t-Lon
Le;t E , E2 an.d F be l.e.. .6 pac. e.6 and le;t !"I. be an l ~ co E CcoWl x Q ' F) , E. 60lt i =1,2. Then 60lt any f ~ 2
. ),
•
C {rI,L (E,F». y y
tonuous and j-linear, for any
hence
le;t
belong!.> ;to C~W,G)
uo f
j E lN o
obviously linear and continuous, hence
by
E, and
G J....6 a e.lol.led l-Lnealt ;topo.togJ...e.al .6ub!.>pac.e 06
any
xl
00
l.lub.6e;t
be g-Lven.
u E L(F,G)
c}
297
g:
the function
).
f(x ,·} belongs to l
nl ... C~0(!"I2,F), g(x l )
= f(x l ,·)
can
be defined. In order to show continuity of E >
g on
gj
I
let any
xl
E
!"I, any
C"" W 2 ,F) of the fonn Pi K_ Q q co '-'-;,' 2' is uniformly continuous on {xl} x K2 for any j,
0, and any continuous semi-norm on
be given. Since
!"II
298
MEISE
there exists a neighbourhood for any (x ,x 2 ) E {Xl} x K2 l any
j
VI x V2
of zero in
El x E2 such that
and any (h l ,h ) E VI x V 2 2
we have
0.::. j .::. l.
wi th
<
(where
p. Q denotes the semi-norm ) I 2,q 'lhis implies for any hI E VI
hence
for
u + sup. q (u (y»
yEQ~
£
on
L~o (E,F».
g is continuous.
FOIL
3. PROPOSITION:
1,2
i
1
ni be an open .&ub.&e.t
and l.e.t
ie.t Ei be. a qua.&-i.-c.ompie.te. .e.. c. • .&pac.e.
-i..& c.ompiete. and wh-i.c.h equat.& (E.)
(E ) ~
60ft wh-i.c.h
i =
06
I
I
cc
topo£og-i.c.atty,
E . A.&.6ume 6ufttheftmofte. that i
E2 i.&
a klR-.&pac.e.. Then thefte e.x-i..&t.& a c.ontinuou.6 iineaft and injec.tive map A
C~O(rll
60ft any
PROOF: tion
x
f E
O2 )
+
C:W l ' C: o ( 2»
C~o(Ol x
I
de.6ined by
Let
Xl i
l
+
+
°2 ),
Let us show first that for any g
A(f): Xl
f(x l , .) belongs to
f
E
C~o(Ol x 02)
the func-
C~o(Ql,C~oW2»'
denote the linear continuous mapping
i l (Xl) = (xl,O) and let f E C~O(nl of lemma 2 one shows that for any j
f(x l , • )
i
l
: El
+
El x E , 2
n 2 ) be given. As in the
x E
IN
proof
the mapping \OJ:= (il) * 0 f(j)
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
belongs to
c:O(n l x
n2,L~o(El,IR)).
by our hypotheses
Hence, by lemma 2 the
cUtl,C~O(~2,L~o(El,lR))).
gj :x l .... 'Pj(x l , 0) is in
299
it follows from 3.2
mapping
Nowobserve
and general
results
that
on the
E-product that we have natural isomorphisms
Using this isomorphism, we get from gj the napping gj for any gj+l
j
IN
E
• Obviously
j
k
E
IN
OJ
C(nl'Lco(El'Cco(n2))
go = A(f), and we shall prove
is the Gateaux-derivative of
first remark that for any
E
now that
gj' In order to do this,
let us
we have
and
that
follows from the proof of lemma 2. Hence we get
Now let ~2'
xl
E
~l'
any compact
hl Q
E
E , any l
2 of
l
E
:IN
, any compact subset
E 2 , any compact subset
be given. We have to prove that there exists any
t
wi th
a
<
I
t
I .s.
Q
l
0 > 0,
of
K2 of
El and
E>
such that
0
<
(2)
By (1) and 1.5 we have to estimate for
O.s.
k <
~
E •
a
for
MEISE
( 3)
Since
~l
f(j+k+l) is continuous on
x
~2' it is uniformly continu-
ous on a suitable neighbourhood of the compact set uniform continuity of
o
> 0
{xl} x K . By 2 f(j+k+l) and (3) it is clear that there exists
satisfying (2). Consequently we have shown that
9 = go
is an
C~O(~l,C~O(~2»'
element of
Linearity and injectivity of
A are obvious. Continuity
of
A
follows immediately from (1) and the definition of the corresponding topologies. Now we want to prove that
A
is surjective if we impose
some
further conditions.
4. LEMMA:
Foft
wb~et 06
E . A~~ume that i
i=1,2
let
let g be a any 6unc.tion -I..n
Ei
be a l.c..
C~O(nl'C~O(n2»'
-t..6 c.ont1.nuou.o. FolC. any
(j,k)
E
and let
~i
bean open
EI x E~ -1...6 a kIR-~pac.e 60ft any (j,k) EJN 2 .
a)
b)
~pac.e
]N2,
any
SPACES OF DIFFERENTIABL.E FUNCTIONS AND THE APPROXIMATION PROPERTY
UYl-t6 oJtm£.y Q
PROOF:
E
Q~
60Jt a.Yly
a) Observe that for any open subset
1. c. space
F
and any
f
E
C~o (n, F)
(x,y)
is continuous on
Kl of
K x Qj
->-
11 of a l.e. space E,any
f(j) (x) [y )
for any compact subset
111 and any compact subset
2
belongs to
Q
l
of
l
x
6~,C(K2
x
Q~))
= C(K ('
This proves the continuity of f J, for any (j,k) b)
E
l
k)
of
11 and
K2
sub-
, the function
in
112 and any com-
Pk«g(j)
(0)
[o])(k»
~) o
Q~
l
any
x K2 x
Qr
]N2.
The second assertion is a consequence of the following con-
siderations:
=: A ( t)
E
in E2 ,we get from lemma lob) that C(K
K
F. Hence for any compact
is linear and continuous for any compact set pact set Q
~e.t
the function
compact set Q in E and has values in set
c.ompac..t
Ei (i =1,2).
-tYl
i
y1
-tYl
301
+
B (t) •
3D2
MEISE
uniformly in
Yl E
By lemma 2.2
Qr
and
Y2 E
Q~
we get
It follows from a) that
f(j,k+l) is uniformly continuous in a neigh-
bourhood of the compact set
{Xl} x {x } x 2
Qr
x
Q~
,
hence we also have
lim A (t) t+o
uniformly in
5. THEOREM:
Yl E
Fote.
(Ei)~~
equaR.-4
Qr
and
Q~
R.et Ei be a qua-4.i-c.ompR.ete R..c..-4pac.e wlt.ic.1t
i =1,2
topoR.og.ic.aU.y and l a 0i be an open -4ub-4et 06
AMume. 6u.tc.tltete.molle. that Then the mapplng
Y2 E
E~
Er x
A: C~o WI x 02)
.i-4 a kIR --4pac.e +
note.
C~o WI ,C~o (n2»
any (j ,k)
Ei • E
]N2.
, A(f) : xl + f(x l ,·),
.i-4 a topoR.og.ic.aR. .i-4omote.ph.i-4m.
PROOF:
First let us show that
A is surjective. Let
g be any ele -
C~o WI ,C~o ( 2 ) ). By lemma 4 the function f: (Xl'X2 ) +g(xl ) (X2) 1s continuous on nl x 02' We shall prove f E C~o(Ql x 02)' Then it ment of
1s obvious that
A(f)
=
In order to prove
g, hence f
E
A is surjective.
C~o WI x 02) let us remark the following:
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
303
Let (j,k) E lN 2 and a surjection a : {]., ••• ,j} U{l', ••• ,k,} ... {l, ... ,j+k} be given. Then we define a continuous linear map 110' :(E l by
X
'+k '_k ... EI XEi E )] 2
( (e l , 0'( l) , ..• , e l , a (j) ) , (e 20 (1' ) , ... , e 2a (k ' ) } ) .
By lemma 4.a) the function Q
1
x Q
2
x
(E
i(j,k)
a
:=
f(j,k)
011
a
is
continuous on
x E )j+k
and i(j,k}(x x .) is (j+k) -linear for 2 a l' 2' x Q2' Because of the continuity of i~j,k} the map-
1
any (x ,x ) E Q 2 l l ping f~j,k) : Q x Q ... L~~k(El x E ,lR), 2 2 l
defined by
f(j,k) (x x) a l' 2
is continuous. Using the mappings duction that
f
belongs to
f(j,k) it is easy to prove by ina
00
C
co Wl
x
(2)' Let
us
show
that
f
is
Gateaux-differentiable: by
Define 0'2
¢ U {It} ... {l}
:
by
x = (x ,x ) E Q x Q 2 l 2 1
lim
0'2(1')
and
and
define
= 1. Thenwe get from 4.b) thatfor
h = (h ,h ) E El x E2 1 2
~ (f(x +th) - f(x) - f(l,O) (x)! thl - f(O,1) (x)! thl)
t ...o
0'1
0.
0'2
(f (1, 0) (x) + f~O,l) (x) )[h), and f E c~o W x Q2) l 0'1 2 by lemma 2.3. From this and lemmma 4.b) we get by induction that for
Hence
any
t
ft (x)[h)
=
E IN the function
be represented as a sum of k in
lN o with
j +k =
is in
f
t
t Cco W l x Q2) and that
f(t) can
f(j,k) where the sum runs over all j and a and
over certain
a.
This proves
that
:114
MEISE
f E C~O(nl
X
n 2 }. oo
A: COO co (nl x .n 2 ) +C co (nl'C'"co (n 2 »
Hence we have shown that
bijective. From the representation of (A-l(g» follows that A- l
is
(i) indicated above it
is continuous. Then A is a topological isomorphism
by proposition 3.
REMARK:
Results of the same type as in theorem 5 are also given
the lecture notes of Frohlicher and Bucher (15)
in
(with a different OOfi-
nition of differentiability) and in Colombeau [11), [12]. Itseems to be impossible to get the result on (DFM)-spaces given below by
bor-
nological methods. Concluding this section, let us combine theorem 5 and some
of
the results in section 3. Then we get
6. THEOREM:
Le.t El and E2 be. e.ithe.1t (F)-.&paee..&
ni
Ei
le.t
be. an ope.n .&ub.&e.t 06
oOlt
i =1,2.
011.
(OFM) -.&paee..& and
The.n we. have. the.
601-
lowing topologieal i.&omoltphi.&m.&
7. THEOREM:
Ei
FOIt
i=1,2, le.t
ni
be. art ope.n .&ub.&e.t 06 the.l.e . .&pac.e.
A.&.&ume. that e.ithe.lt a.nd E2
1)
El
2)
El and
E2
alte.
(F) -.&paee..&, one. 06 whic.h ha.& a.p.,
0It
alte. (DFM) -.&paee..&, one. 06 wh.i.eh v., a (OFSA) -.6paee..
The.n the. 60110wing hold.&
8. REMARK:
The dual of
C~o(n) forms a natural generalization of the
space of distributions with compact support to infinite
dimensions.
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
305
It is obvious that many of the results of this article can regarded as results on the dual of
also
C~O(n). E.g. theorem 3.10
be
is of
importance in connection with the theorem of Paley - lViener - Schwartz (in order to see this one has to extend several results valued functions on
T E C~o(n,~), as a
morphic function on the complexification of growth condition). Theorem 6 can be used x COO (E)'
co
complex
n, then (for certain l.c. spaces E) one can de-
fine the Fourier-Laplace transform of any
* : COOco (E)'
to
-+
E~,
to
holo-
satisfying a certain
define
a
convolution
COO co (E) '. The precise formulation of the results
just mentioned will be contained in a subsequent paper.
REFERENCES
[1)
A. ARHANGEL'SKII, Bicompact sets and the topology Soviet Math.
[2)
(Doklady) 4 (1963)"
of
spaces,
561 - 564.
R. ARON, Compact polynomials and compact differentiable
map-
pings between Banach spaces, in "Sem-ina..i)Le. (Ana.!y~e.)
P-i"-'tJ1.e.. Le.!ong Annee. 1974/75", Springer Lecture Notes Math. 524
(1976), p. 213 - 222. [3)
R. ARON, Approximation of differentiable functions on a
Banach
space, in "1 n6-inLte d-ime.n-6-iona.! holomoJtphy a.nd a.ppUca..tLOn-6'~ North-Holland Mathematics Studies (1977), p. 1 - 17. [4)
R.
ARON and M. SCHOTTENLOHER, Compact holomorphic mappings on Banach spaces and the approximation property, J. Functional Analysis 21 (1976), 7 - 30.
[5)
K.-D. BIERSTEDT and R. MEISE, Lokalkonvexe Unterraume in topologischen Vektorraumen und das €-Produkt,manuscripta math. 8 (1973),143-172.
[6)
K.-D. BIERSTEDT and R. MEISE, Bemerkung
uber
die
Approxima-
tionseigenschaft lokalkonvexer Funktionenraume, Math. Ann. 209 (1974), 99 -107.
3)6
MEISE
[7)
K.-D. BIERSTEDT and R. MEISE, Nuclearity and the Schwartz property in the theory of holomorphic functions on metrizable locally convex spaces, in "In6inite dimen~ional holomo~phy and applic.atio Mil, North-Holland Mathematics Studies (1977) , p.93-129.
[8]
J. L. BLASCO, Two problems on kIR-spaces, to appear Math. Sci. Hung.
[9)
P. L. BOLAND, An example of a nuclear space in infinite dimensional holomorphy, Ark. Mat. 15 (1977), 87 - 91.
[10)
F. BOMBAL GORDON and J. L. GONZALEZ LLAVONA, La propiedad de aproximacion en espacios de funciones diferenciables,~s ta Acad. Ci. Madrid 70 (1976), 727 - 741.
[11)
J. F. COLOMBEAU, 1973.
[12)
J. F. COLOMBEAU, Spaces of coo-mappings in infinitely many mensions and applications, preprint Bordeaux 1977.
[13)
S. DINEEN, Holomorphic functions on strong duals of FrechetMontel spaces, in "In6inite dimen~ional holomoltph!l and applic.ation~", North-Holland Mathematics Studies (1977) ,147-166.
[14)
K. FLORET and J. WLOKA, Ein6Uh-tung in die The.oltle d~ lokallwnvex.en Raume, Springer Lecture Notes in Math. 56 (1968).
[15)
A. FROLICHER and W. BUCHER, Calc.ulu~ in vec.to~ ~pac.e~ no~m, Springer Lecture Notes in Math. 30 (1966).
(16)
H. HOGBE-NLEND, Les espaces de Frechet-Schwartz et 1a propriete d'approximation, C.R. Acad. Sci. Paris A 275(1972),1073-1075.
[17]
J.
HORV~TH,
Vi66e~entiation
Topologic.al
et
bo~nologie,these,
vec.to~ ~pac.e~
and
in
Acta
Bordeaux
di~tltibution~
di-
without
I,~
ing, Mass, Addison Wesley 1965. [18)
H. H. KELLER, Vi66e~ential c.alc.ulu~ in loc.ally c.onvex. Springer Lecture Notes in Math. 417 (1974).
&pac.e&,
307
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
[19)
G. KOTHE, Topological vec~o~ ~pace~ I, Springer der Math. 159 (1969).
[20)
J. B. PROLLAandC. S. GUERREIRO,
Grundlehren
Anextension
of Nachbin's
theorem to differentiable functions on Banach spaces
with
the approximation property, Ark. Mat. 14 (1976), 251 - 258. [21)
H. H. SCHAEFER, Topological
[22)
L. SCHWARTZ, Theorie des distributions
vecto~ ~pace~,
a
Springer 1970.
valeurs
vectorielles
I, Ann. Inst. Fourier 7 (1957), 1-142.
a
[23)
M. DE WILDE, Reseaux dans les espaces lineaires semi-normes, Memoires Soc. Royale Sc. Liege, 5 e serie,18, 2 (1969).
[24)
S. YAMAMURO, Vi6 6e~ential calcullH in topological lineM Springer Lecture Notes in Math. 374 (1974).
~pacu,
This Page Intentionally Left Blank
Apppoximation TheopY and FUnctionaZ AnaZysis J.B. PpoUa (ed.) ©Nopth-HoZZand Puhlishi11{l COTIl{Jany, 1979
A LOOK AT APPROXIMATION THEORY
LEOPOLDO NACHBIN Instituto
de Matematica
Universidade Federal do Rio de Janeiro 20.000 Rio de Janeiro RJ ZC-32 Brazil Department
of Mathematics
University
of Rochester
Rochester NY 14627 USA
1. INTRODUCTION I would like to describe very briefly how I was led to
become
seriously interested in Approximation Theory, that is, to indicate the motivation that I had in my mind. This field has developed in Brazil in the past ten years or so, thanks also to the work of Silvio Machado,
Joao
Bosco
Prolla
and
Guido Zapata, as well as the research school that they formed. If I had to reduce bibliographical references to a bare
mini-
mum, in what concerns the work of the Brazilian school in Approximation Theory and its relationship to the research of other groups, would quote my monograph Element4 well as Prolla's monograph (1977)
06
App~ox~mat~on Theo~y
App~ox~mat~on
06
Veeto~
(see [34], (54]). However, the bibliography
(1967), as
Valued at
I
the
funetion4 end
is
up-to-date and complete with respect to the work by Machado, Prolla, Zapata and myself.
It is extremely incomplete otherwise.
emphasize the following aspects:
Let
me
310
NACHBIN
1)
I shall restrict myself here to the real valued case. The
vector valued case was treated in a desirable degree
of
generality
through vector fibrations by Machado [16 J and Prolla [40 J
(see also
[35 J, [36]). 2)
In the complex case, I point out the work by Machado on the
Bishop and Weierstrass-Stone theorems [18]. (See also [54]). 3)
We call attention to the work by Zapata on Mergelyan's
orem and quasi-analytic classes [65] 4)
~
(see also [54J).
Weighted approximation in the continuously
differentiable
case was studied by Zapata [63], [64J. 5)
A density theorem for polynomial algebras of
continuously
differentiable mappings in infinite dimensions and its
relationship
to the Banach-Grothendieck approximation property was investigated by Prolla and Guerreiro [531 (see also [38 J). 6)
Nonarchimedean Approximation Theory has
PrOlla [56],
been
studied
by
and Carneiro [ 7 1, [8].
2. APPROXIMATION OF CONTINUOUSLY DIFFERENTIABLE MAPPINGS In 1947, Marshall Stone carne from the University of Chicago to lecture at the Universidade Federal do Rio de Janeiro (known then as Universidade do Brasil) for three months.
He
offered
a
beautiful
course on "Rings of Continuous Functions". Among other things, talked about his celebrated paper mat~on Theo~em
A Gene~at~zed
We~e~~t~a~~
he
App~ox~
which he had just written. It was published next year
in volume 21 (1948) of Mathematics Magazine. This is a good
example
of an article that became famous in spite of the fact
:is
that
was
published in an obscure journal. Stone's course dealt with continuous functions, and was
going
to have a lasting influence on me. It was during and Shortly after it that, in 1948, I thought of and proved, but did not
publish
then,
A LOOI< AT APPROXIMATION THEORY
311
what I called the Weierstrass'-Stone theorem for modules [34).
I will
come back to this aspect in a brief while. The reason I did not publish right aw!ay that result for modules was this. It took
me
years
until 1960 -1961, while I visited Brandeis University for four IIDllths, !
to realize the interest for Approximation Theory of modules in place of algebras, and to get started in weighted approximation proper for continuous functions. In 1948, I went to the University of Chicago
for
a two
year
visit during 1948 -1950, at the invitation of Stone. While there, had an apportunity, in 1949, of presenting at Andre Weil's
seminar
the then recent article "On ideals of differentiable functions" Hassler
I
by
Whitney, just published in volume 70 (1948) of the American
Journal of Mathematics. After my lecture,
Irving Segal
asked
me:
how about a similar result for algeb:ras of continuously differentiable functions, along the lines of the Weierstrass-Stone theorem? In other words, the problem was to describe the closure of
a
subalgebra
continuously differentiable functions, or equivalently, to
of
describe
the closed subalgebras of continuously differentiable functions,
in
the spirit of the Weierstrass-Stone theorem. To the best of my knowledge, this problem has remained open so far; see below for the conjecture that I have in mind in this respect. Pressed by Segal's question, I studied immediately in 1949 [23) the noteworthy case of dense subalgebras, to obtain the following resuIt. Let E be a real m-differentiable (m=O,l, ..•
,co)
finite dimension. Denote by
all
Cm(E) the algebra of
m-differentiable real functions on E,
and all their differentials up to order
OOIC.
Tm
16m ~ 1, a .6uba.lgeb!C.a. A 06
-i6 and only -i6:
of
continuously
endowed with the topology
of uniform convergence on the compact subsets of
THEOREM 1:
manifold
Tm
E of such functions
m.
Cm(E) -i.6 de.n.6e
-in
312
NACHBIN
(Nl)
Fo~ eve~y
x E E,
(N2)
Fo~
x
eve~y
the~e ~~
E,y
E
E, x ;o! y, theJte
E
wc.h tha;t f (x) ;o! O.
f E A
~
!.>uch that
f E A
f(x) ;o! f(y).
Fo~
(N3)
eve~y
at x,
and
x E E
the~e ~~
at
f E A
~uc.h
df(x) (t) ;o!
ar(x)
tangent
eve~y
vecto~
to E
t;o! 0
that
o.
These conditions do not depend on m. The case
m=0
excluded
by the above result is covered by the Weierstrass-Stone theorem. Coming back to the question Segal asked me in 1949, bit by bit I was led to formulate the following conjecture. If it is true,
the
Whitney ideal theorem and the above density theorem are subsumed
by
it. For the sake of simplicity of terminology and notation only, let us assume that of
JRn
1 < m <
for some
00
and that
= 1,2, ••• ,
n
E=U
extension to arbitrary
being easy. If A is a subalgebra of relation
U/ A
defined by
equivalent when
CONJECTURE 2:
=
f(x)
16
f
E
A on
fey)
is a nonvoid open
U,
subset
E and to
Cm(U), consider the
equivalence
according to which x, Y E U
for all
m = 00
are
f E A.
Cm(U) and A ~!.> a !.>ubalgeb~a 06
Cm(U) then f
only ~6),
6o~
eve~y
c.la-6.-6 modulo
c.ompact
loag(x) - Oaf(x) oa
and
U/ A
I
<
E
-6ub~et eve~y
60~
any
K 06
U
c.onta~ned ~n ~ome
e: > 0, theILe
x E K
and
~~
g
any
E
A
paILt~al
equ~valence ~
uc.h
that
de~~vat~ve
06 o~de~ at mo~t equal to m. Notice that
f
belongs to the closure of A in
Cm(U) for
when the above condition holds true for every compact subset K of by definition, not just for those
K contained in
some
Tm
u,
equivalence
A LOOK AT APPROXIMATION THEORY
313
U I A. The above conjecture is an aspect of what I called
class modulo
lOl.'-a.U.zab"[lLty (see below too). If true, the above conjecture has a natural extension to modules in place of algebras. There is a more naive conjecture, which is easily seen false. We might indeed conjecture that every subalgebra which is closed for
m.
For
=
CO{E)
be
Cm{E) of sim-
E of functions and all their differen-
m
of fact, the statement that gebras of
of
A
Tm is also closed for the topology Tms
ple convergence at points of tials up to order
to
= 0,
this is indeed the case; as a matter
TO
and
C{E) is easily
T have to same closed subalos
seen
to
be. the. .6uba.ege.btta 06
cl
be
equivalent
to
the
Weierstrass-Stone theorem.
EXAMPLE 3:
LeA:
uI.'-h that
f{l/k)
.6
A
=
f{O)
nM
06 aU f E e l OR) 00, 2 aU k=l,2, ... and motte.ove.tt 2":k=l (l/n)/n =0. (lR)
A few years ago, I asked Jaime Lesmesthe question of extending the above Theorem 1 to infinite dimensions. I also did
raise
question during a lecture I gave at Madrid, where Jose Llavona
that got
interested in it. Recent work along this line was done by Lesmes [13) and Prolla [49], [53)
in Brazil, and by Llavona [14), [15]
in Spain.
We now summarize that aspect very succintly, along the lines of [38]. Let E, F be Hausdorff real locally convex spaces, E U a nonvoid open subset of
E and
the vector space of all mappings differentiable 1)
f
m
'1O,
F'IO,
= 1,2, ... ,00. We denote by ~(U;F)
f: U
~
F
that are continuously m-
in the following sense:
is finitely m-differentiable; that is, for every finite
dimensional vector subspace
S of
we assume that the restriction
f
E with
I (U
II
S 'I
°
and
U n S nonvoid,
S) is m-differentiable in the
classical sense. Thus we have the differentials
314
for
NACHBIN
k =0,1, ..• , k _< m, with values in the vector space £
all symmetric k-linear mappings of Ek to 2)
F.
The mapping
is continuous of every
k=O,l, ... , k.::. m. In particular, dkfCx) be-
longs to the vector subspace k-linear mappings of Ek to We endow
£sCkEiF) of
CmCU;F) with the topology
~~Cf)
X E K,
k = 0,1, ..• , k .::. m, i3
all
continuous symmetric
F.
of seminorms depending on the parameters
for
CkEiF) of
as
Tm defined by the k,
13, K, L
tEL} E
m
being a continuous seminorm on
being nonvoid compact subsets of
E
U,
family
F and K,L
respectively.
We shall use the notion of polynomial algebra; see the convention on page 63, [54].
THEOREM 4:
Le:t
and A be a polynomial .6ubatgeblla 06 CTICU;F).
m> 1
SuppOhe :tha:t :thelle i.6 a .6ub.6et G 06 :the vec:tOlt .6pace c.on:tbtuou.6 lineall endomOltph-Lhmh 06
E
E' ® E 06 a.U
w-Lth 6ini:te d-Lmen.6iona.e. .i.mage-6,
.6uch that: 1)
The -Ldenti:ty mapp-Lng
belo ng.6 to :the clo.6 ulle 06 G
IE
the compact-open :topology on the vec.toll .6pace a.e..e. c.ontinuou.6 l-Lneall endomoltphihmh 06 2)
Folt evelly
:tha:t
J
E
(f 0 J)
U
and evelly
f E A, it 60llow.6 that :the
Iv =
I V)
f
0
CJ
06
E.
G, evelly nonvoid open .6ub.6e:t V 06
J(V) C U
htlt-Lc.Uon
£CE; E)
6Oil
be.e.ong.6:to the
.6uch Ile-
c..e.Ohufte in
316
A LOOK AT APPROXIMATION THEORY
T
m
06
A
i v.
(Nl)
Fo~
eve~y
x E U,
(N2)
Fo~
eve~y
x E U, Y E U, X
that Fo~
(N3)
f(x) eve~y
~
the~e ~~
~ueh
f E A
~
y,
the~e
~~
f
E
~
0,
the~e ~~
f
E A
o.
~
that f(x)
~ueh
A
f(y) .
x E U, tEE,
t
that
a£
Tt(x)
If
df(x)
(t)
~
o.
E is finite dimensional, conditions 1) and 2) of Theorem 4
are satisfied by
G reduced to
IE. Hence Theorem 4 implies
Theorem
the
Banach-
1.
Condi tion 1) of Theorem 4 implies that Grothendieck approximation property, that is, closure of
E'
~
E
in
E
has
belongs
IE
to
£(E;E) for the compact-open topology.
the Thus
Theorem 4 leads to the following conjecture:
CONJECTURE
5:
FM eve~y g~ven
E, the
6oR.R.ow~ng
m
then
eond-<'t~on~
Me
eq~
v(lR.evtt: (Cl)
Fo~ (l~b~t~a~y
atgeb~a
A
Ovtty -<'6) A (C2)
E
ha~
the
U, F,
~~ devt~e ~vt ~at.L~ 6-<'e~
> 1,
Cm(U;F)
(Nl),
(N2),
eve~y
poR.ynomiat
OM Tm -<'6 (avtd
~
ub-
aR.w(ly~
(N3).
Banach-G~othendiec~ app~ox~mat-<'on
It is known that (Cl) implies (C2). The conjecture implies (Cl) is an attempt to improve Theorem 4.
p~ope~ty.
that
(C2)
316
NACHBIN
In the direction of research that I just mentioned,
there
is
more generally the question of studying Approximation Theory for algebras or modules of continuously differentiable vector valued mappings by using weights. This question however is still wide open,
in
spite of the available results. See the next section for the continuous case.
3. WEIGHTED APPROXIMATION FOR MODULES AND ALGEBRAS OF CONTINUOUS PONeTIONS Let me talk now about the Weiestrass-Stone theorem for modules, how it led me to the Bernstein approximation problem and what I then called the weighted approximation problem (or the Bernstein - Nachbin approximation problem, according to a more recent terminology by other authors) • Let E be a completely regular topological space, and C (E) denote the algebra of all continuous real functions on E endowed with the compact-open topology. There is the ideal theorem for If I
is an ideal in
C (E)
belongs to the closure of
and I
I-I (0) in
C(E) which reads
n
f
as
E I f- l (0), then
C (E) if and only if
f
follows. fEe (E)
vanishes
on
I-I (0) • More generally, there is the Weiestrass-Stone theorem for a subalgebra
A of
C(E)
which reads as follows. Let
E / A be the equiva-
xl - x 2 if xl' x2 E E and f(~) = f(x 2 ) A-l(O)= n Cl(O) which either is one fEA
lence relation on E defined by for every
f E A. Consider
of equivalence classes modulo belongs to the closure of
E / A or else is void. Then
A in
C (E) if and only if
on every equivalence class modulo E / A and
f
f
f E C (E)
is constant
vanishes on A-I (0) if
A-l(O) is nonvoid. In the ideal theorem, we have a module I over the algebra A= C(E).
A LOOK AT APPROXIMATION THEORV
317
In the Weierstrass-Stone theorem, we have a module
A over the alge-
A + lR generated by A and all constant real functions on
bra
we considered just a vector subspace module
W over the algebra
W of
C (E), we would
E. If
have
a
A of all constant real functions on E. In
the succession of these three cases, the algebra of multipliers varies from the largest to the smallest possibility containing the unit. More generally, let A be a subalgebra of
C (E) which we may
OCM
assume to contain the unit without loss of generality, and let W be a vector subspace of
C (E) which is a module over A so that Awe W.
The Weierstrass-Stone theorem for modules reads as follows. Introduce as before the equivalence relation E/A on to the closure of W in set K of £
>
E. Then
f
E
C (E) belongs
C(E) if,and only if, for every compact sub-
E contained in some equivalence class modulo
0, there is
g
E
W such that
Ig(x)
f (x) I <
E
E/A and every
for every
x
E
K.
This is an aspect of what later I called "localizability" (see belCM). It is known that a topological vector space sentation by continuous real functions, that is homeomorphic to a topological vector subspace and only if
W has some repre-
W is isomorphic and
W of some
logical vector space
Wand an algebra
a
plictly, we want to know when we can find
More
TEA
and
ia
A ... A
W ... W
so that im [T{x)]
x E W.
We have the following three results that I proved in never published. Let W be a topological vector space, and algebra of linear operators of tity operator of
ex-
A, W in some C(E) as above,
a surjective vector space isomorphism and homeomorphism
for every
W,
W, when does the pair A,
W have some representation by continuous real functions?
and a surjective algebra isomorphism
topo-
A of linear operators of
A contains the identity operator of
ia(T)im{x)
if
W is a Hausdorff locally convex space. Thus the follow-
ing representation theory question arises naturally. Given
where
C (E) ,
W. Assume that
1956, A
but
be an
A contains the iden-
Wand is commutative (without commutativity of
A
318
NACHBIN
we would below replace ideals by left ideals in For every ideal subspace of
J
in
A, let
W spanned by the
that a subset
X of J
for all ideals
T{x) with
T E J
x E W.
and
X is convex and X ==
A of codimension 1 in
A.
We
say
rl
We say
J
(X + JW)
that
W
is
A in case the A-convex neighborhoods of 0 in
locally convex under
form a basis of neighborhoods at the usual sense, of course. If of
J W denote the A-invariant vector
W is A-convex i f
in
A).
O.
W
This implies local convexity in
A is reduced to the scalar operators
W, then A-convexity and local convexi ty under
A reduce to
con-
vexity and local convexity in the usual sense. The above definitions are subsumed by
§ 3, [24 J •
A linear operator
neighborhoods such that
V of
T{V)
C
0
T on in
W is said to be "directed"
when
the > 0
W, for each of which there is It == It (V)
A V, form a basis of neighborhoods at 0; in equiva-
lent terms, when corresponding to every neighborhood
of
U
in
0
W
we may find another neighborhood V of 0 in Wand E > 0 such that co k k Uk=O T (E V) C U. More generally, the members of a collection C of linear operators on
Ware said to be "similarly directed"
neighborhoods
0 in
such that at
O.
V of
T{V)
C It
if
W, for each of which there is It = It (V ,T) > 0
V for every
TEe, form a basis of neighborhoods
Directedness of a linear operator implies its continuity. Both
directedness and similar directedness reduce to continuity when a normed space. These concepts arise only in treating
more
topological vector spaces. Thus the hypothesis in Theorem that the operators in isfied when
THEOREM 6: 6unction~
undelL
the
W is
general
6
below
A be similarly directed is automatically sat-
W is a normed space.
The
pai~
A, W ha~ ~ome ~ep~e~entat~on by cont~nuou~ ~eat
i6 and onty i6
W
i~
A, and the opelLatolLl.> il1
a
A
Hau.6do~66 aILe.
.6pace wh.i.ch
~
toea.Uy convex
.6im.LtalLty dilLected.
A LOOK AT APPROXIMATION THEORY
76 :the paL'!.
THEOREM 7:
Jteat 6unc.tionJ.> and
A,
undeJt
A, W haJ.> J.>ome JtepJteJ.>en:ta:tion by
S iJ.> a vec.toJt J.>ubJ.>pac.e 06
:then :the quo:tien:t paiJt
76 the paiJt
!teat 6unc.:tionJ.>,
76
A,
W
whic.h
:tain
:then
invaJtian:t
Jtep!teJ.>en:ta:tioI1
S i& c.to&ed in
A, W haJ.> &orne !tepJte& en:ta:tion
by
W.
c.ontinuou&
:then J.>pec.:t!tat J.>yn:theJ.>iJ.> hotd& in :the 60ttowing J.>en&e.
S iJ.> a c.to&ed pJtope!t vec.to!t J.>ub&pac.e 06
deJt
c.on:tinuouJ.>
W whic.h iJ.>
haJ.> J.>ome
A/S, W/S
by c.on:tinuouJ.> !teat 6unc.:tion& i6 and onty i6
THEOREM 8:
319
W whic.h i&
invaJtiantun-
S i& :the bt:te!tJ.>ec.:tioI1 06 att c.to&ed vec.to!t /.)ubJ.>pac.('h 06
a!te invaJtian:t undeJt
A, have c.odimen&ion one in
Wand c.on-.
S.
The passing to a quotient statement of Theorem 7 implies
spec-
tral synthesis in Theorem 8, which may be viewed as an abstract version of the Weierstrass-Stone theorem for modules. Let us also point
A is reduced to the scalar operators
W,
then
Theorem 8 becomes the following statement. Every closed proper
vec-
out that, when
tor subspace
S of a locally convex space
all closed vector subspaces of and contain
of
W is the intersection
of
W which have codimension one in
S. As it is classical, such a statement
is
W
equivalent
to the Hahn-Banach theorem. Thus Theorem 8 may be looked upon
as
a
generalization of both the Weierstrass-Stone theorem for modules and the Hahn-Banach theorem for locally convex spaces. We may then ask the following natural question. To what extent the condition of the operators in
A being similarly directed is cru-
cial for the validity of Theorem 6, or Theorem 7, or Theorem 8? Local convexity under
A is not superfluous.
In fact,
reduced to the scalars operators of
W, then it may
every closed proper vector subspace
S of
sll closed vector subspaces of and contain
letting be
A
false
W is the intersection
W which have condimension one in
S, in case W is not assumed to be locally convex.
be
that of
W The
320
NACHBIN
answer to the above natural question is no. The example that I found in 1957 led me to the classical Bernstein approximation problem, asI shall describe next.
EXAMPLE 9: tions on
Let W be the Frechet space of all continuous real funcJR
A = P (JR)
that are rapidly decreasing at infinity. Call
the algebra of all real polynomials on
JR. Every
a E C(lR)
that
is
slowly increasing at infinity gives rise to the continuous linear opera tor Thus
Ta : fEW
->-
which is directed i f and only a is bounded.
a fEW
A may be viewed' as a commutative algebra
operators of
of continuous linear
W containing the identity operator of
W, but each such
operator is directed if and only if the corresponding polynomial constant. It is clear that some
W is locally convex under
w E W vanishing nowhere
in lR such that
Aw
A.
There
of
JR
w E W
that is not a fundamental weight in the sense
B A P - 2 or B A P - 1 below). Then the closure
Aw
in W is a closed
proper vector subspace of W which is invariant under never vanishes in
is
is not dense in
W (this is easily seen to be equivalent to existence of some vanishing nowhere in
is
JR., it can be shown that Aw
any closed vector subspace of
A.
Since
w
is not contained
in
W which is invariant under
A, having
condimension one in W. Thus Theorem 8 does not hold in this case doo to lack of directedness. A fortiori Theorem 7 and Theorem
6
do not
BeJt~.te.i.n
a.pp!tox.-L-
hold in this case for the same reason. This counterexample leads us to the c..e.a..6.6-Lc.a..e. ma.~on
p4oblem, usually formulated in the following two forms, where
P (lRn )
is the algebra of all real polynomials on mn for n = 1,2, ..•. B AP - 1. Let
and
v: lRn
lR+ be an upper semi continuous "weight" n be the vector space of all fEe (lR ) such that vf n 0 at infinity, seminormed by II f II = sup{v(x). I f (x) I ;x EJR }. ->-
CVoo (lRn )
tends to
Assume that
v
v is rapidly decreasing at infinity, that is p(JRn) CCvoo(£).
A LOOK AT APPROXIMATION THEORY
n P(m )
When is
321
n Cv",(m )? We then say that
dense in
v
6uVlda-
is a
me.Vltal we.-ight. We shall denote by S"l n the set of all such fundamental weights in the sense of Bernstein. For technical reasons we also introduce the set Clearly
r n of all such
rn C S"ln
0
such that
v
k
E S"ln
k > O.
for all
This inclusion is proper. n Coo(m )
B AP - 2. Let ing to
v
be the Banach space of all
at infinity, normed by
the special case of
n Cv",(m )
it
is
n WE C(m )
is
IIfll= sup{jf(x)l; x E mn}i
when
v=1. Assume that P(m n ) w
rapidly decreasing at infinity, that is w a we...i.ght. When is
f E C(:nf)tend-
n
P(m ) w dense in
C (mn ),
C
and call
'"
C",(mn )? We then say
that
w
is a 6uVldame.Vltal we...i.ght. If
wE C(IR n )
is rapidly decreasing at infinity, then
fundamental weight in the sense of
~n
vanishes on
and
Iwl
B A P - 2 if and only if
is a fundamental weight in the
B A P - 1. However a fundamental weight vanish on that
n
lR
B AP - I
v
w is a w
never
sense
of
in the sense of B A P - I
rray
and may fail to be continuous.
It
is
in
is a better way of looking at the concept
men tal weights in the sense of Bernstein than
this sense of
funda-
B A P - 2.
The following are the simplest criteria for an upper semicontinuous function
v: m
->-
m+
to belong to
by
r I ' thus to
increasing degree of generality:
BOUNDED CASE: ANALYTIC CASE:
v
ha~
Th e.~e.
a bounde.d a~e.
C > 0
~uppo~t.
a.Vld
c
> 0 60lL wh..i.c.h, 6o~ anlj x E ~,
we. have
v (x)
QUASI-ANALYTIC CASE:
We. ha.ve.
< C • e -cl x'i •
~oo
m=l
I
+
00
whe.Jte,
322
m
NACHBIN
we.6 e.t
0, I , ... ,
In
BAP-I,
P(mn )
the subalgebra
Cvoo(m n ), and we have the weight
C(mn )
of
v in the definition of
Thus
weAflhted
I
n CVoo(IR ). In
P(mn )
B AP - 2, the subrnodule p(mn)w over the subalgebra is contained in
is contained in
of
CORn)
C00 (mn ), and we have the weight w in the definition
was led
app!Lox.[mat.[olll
to
the following general
formulation
of the
pll.obtem. The viewpoint thus adopted embraces the
Weierstrass - Stone theorem for modules, thus for algebras, Bernstein approximation problem. Actually, it is guided by
and the
the idea
of extending the classical Bernstein approximation problem in the same style that the Weierstrass - Stone theorem generalizes
the classical
Weierstrass theorem (see [34] for details). Let V be a set of upper semi continuous positive real functions on
a
completely regular topological space
d.[ll.ected in the sense that, if VI' v
such that
vI'::' A v and
v 2 < A v.
2
E.
v E V
and any
£
Each element of
CVoo(E).
V
is called
f E C(E) such that,
Each
... IIfliv = sup {vex) • If(x) Ii x
f
is
a for
> 0, the closed subset {xEE; v(x)'if(x)1 >d
is compact, will be denoted by seminorm
V
E V, there are A > 0 and v E V
we.[ght. The vector subspace of C(E) of all any
We assume that
natural topology on the we.[gh.ted llpace
E
v E V E}
C Voo (E)
determines a
on is defined
by
the
family of all such seminorms. Let
A
C
C (El be a subalgebra containing the unit, and W C CVoo(El
be a vector subspace. Assume that W is a module over
that
is
appll.ox~ma.t~on
pll.obtem consists of asking for a
description of the closure of W in
CVoo(E) under such circumstances.
AW C W.
The we.[gh.ted
A,
We say that
W is £.oca£..Lzab£.e undell. A .[n CVoo(E) when the following
A LOOK AT APPROXIMATION THEORY
condition holds true: if of
W in
CVoo (E)
f E CVoo(E), then
f
belongs to the closure
if (and always only if), for any
and any equivalence class v(x) • Iw(x) -
323
f(x) I <
E
modulo
X
E/A,
v E V, any
there is
w E W
E
> 0
such that
x E X. The I>:tltie:t weigh:ted appltoxi-
for any
ma:tion pILoblem consists of asking for necessary and sufficient conditions in order that We denote by
W be localizable under G (A) a subset of
A in
CVoo(E).
A which topologically generates
A as an algebra with unit, that is, such that the subalgebra generated by
G(A) and one is dense in
We also introduce a subset W as a module of by
G (W)
A,
is dense in
G(W) of
A for the topology of
of
A
C(E).
W which topologically generates
that is, the submodule over W for the topology of
A of
W
generated
C V00 (E) .
A basic result is then the following one.
Al>l>ume :tha:t, 60IL evelty
THEOREM 10:
w E G(W), :theILe il>
y E fl
v E V,
evelty
a E G(A) and eveILY
I>uch :tha:t
v(x). jw(x)i < y[a(x)]
n01t
any
x
E. Then
E
W il> locaV.zable undeIL
A -tv!
CVoo(E).
We may combine Theorem 1'0 with the indicated criteria for membership of
f l ' Let us consider explici tly the analytic case.
COROLLARY 11:
evelty
AI>.6ume :tha:t,
wE G(W), :thelte alte
nOlL
eveILY
C
0
>
and
v E V,
c
>
eveILy
x
E E.
Then
W
i.6 localiza.ble
undelL
and
.6ueh :tha:t
0
vex) • Iw(x) I < C • e-c'la(x)
nOll CUlY
a E G(A)
j
A
-ll'l
CVoo(E).
As a particular case of the above results for modules,
we have
324
NACHBIN
the following ones for algebras. For simplicity sake, assume that is strictly positive, that is, for every that
v(xl
o.
>
caLizabte .in f
E
Let
A be contained in
x E E
there is v E V suen
CV '" (El . We say that A is to-
CV",(El when the following condition
CV",(El, then
always only if)
holds
true:
if
eV",(E) if
(and
is constant on every equivalence class nodulo
E!A.
f f
V
belongs to the closure of
We denote by
G(A) a subset of
A in
A which topologically generates
A as an algebra with unit, that is such that the subalgebra of A generated by
G(A) and one is dense in
A for the topology of
CV",(E).
The particular case is then the following one.
A~~u.me
THEOREH 12: i~
y E r
~u.ch
l
that, 60Jt eveJtIj
x
E. Then
E
E
G(A), theJte
that
v(x)
60Jt any
and eveJtIj a
v E V
A
i~
~
y[a(x»)
toeat.izabte in
eV",(E).
We may combine Theorem 12 with the indicated criteria for membership of
rl. Let us consider explictly the analytic case.
A~~u.me
COROLLARY 13: aJte
and
C > 0
that, 60Jt eveJty v
c > 0
v(x)
60Jt anlj
x
E
E.
We quote
Then
A
~u.eh
[34], (37)
and eve.Jty a
that
< C • e- c • ia(x)
i~
E V
I
tocat.izabte.in
ev", (E)
•
for additional details.
E G(A), theJte
A LOOK AT APPROXIMATION THEORY
325
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L. NACHBIN, On the weighted polynomial approximation in a
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L. NACHBIN, Sur 1 'approximation polynomiale ponderee des tions reelles continues,
A~~~
GlLoupemen~ de Ma~he.ma~~Q~en~
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L~ne,
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(1963), (28]
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Ma~hema~~Q~an~,
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L. NACHBIN, Weighted approximation over topological spaces and the Bernstein
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L. NACHBIN,
Aproxima~ao
ponderada de
fun~oes
continuas por po-
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Fortaleza 1961 (1965), 146 -189, Instituto
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Matematica Pura e Aplicada, Brasil. [33]
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Funct~on Atgeb~a~
and
(Editor:
F. T. Birtel (1966), 330 - 333, Scott and Foresman, USA. [34]
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Uement~
0
6
app~o x~mat~on theo~y
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Concerning weighted
approximation, vector fib rations and algebras of
opera-
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[38 ]
L. NACHBIN, Sur la densi te des sous-algebres polynomiales
d' ap-
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Letong et
HeM~
Siwda
(Anaty~e),
1976/77,
Springer Verlag
Lecture Notes in Mathematics, to appear. [39]
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Vecto~
6~b~at~on~
and
atgeb~a~
06
ope~ato~~,
Publications du Seminaire d'Analyse Moderne, Universite de Sherbrooke (1968/69), Canada.
A LOOK AT APPROXIMATION THEORY
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de operadores,
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for
density
in
tensor products, Indagationes Jvlathematicae 33(1971),170-175. [42]
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89 (1971),
145 - 158. [43J
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Annalen 191 (1971),
283 - 289. [44 J
J. B. PROLLA, Weighted approximation of continuous
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Bulletin of the American Mathematical Society 77(1971), 1021 - 1024. [45]
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J. B. PROLLA and S. MACHADO, Weighted Grothendieck
subspaces,
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186
(1973), 247 - 258. [47J
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[48]
J. B. PROLLA, TJte.~
c.on6e.Jte.nc.~a~ ~obJte. te.oJt~a
de.
apJtax~mac.~on,
Publicaciones del Departamento de Ecuaciones Funcionales, Universidad de Sevilla (1974), Spain. [49]
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e.t
App.e.~c.at~on~
225 - 232, Hermann, Paris.
(f':diteur: L. Nachbin)
Fonc.-
(1975),
330
[51)
NACHBIN
J. B. PROLLA, Dense approximation for polynomial algebras, Bonner
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[54}
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App~ax~mat~on Theo~y and Funct~onai Analy~~~ (Editor: ~ B. Pro11a), Notas de Matematica (1979), North - Holland, to appear.
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[59}
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A LOOK AT APPROXIMATION THEORY
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(1975), 277 - 283, Hermann, Paris.
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Notas
de
This Page Intentionally Left Blank
Approximation Theory and Functional Analysis J.B. Prolla (ed.) ~North-HoZland Publishing Company, 1979
BANACH ALGEBRAS OVER VALUED FIELDS
LAWRENCE NARICI St. John's University Jamaica, New York, 11439, USA and EmvARD BECKENSTEIN St. John's Uni versi ty Staten Island, New York 10301, USA
ABSTRACT By "Gelfand theory" here is meant the study of the consequences of topologizing the maximal ideals of a Banach algebra.
The
theory
is most rich when the underlying field is that of the complex bers. If the underlying field is
R or some other valued
num-
field,
a
theory can still be developed however and that is discussed here. First the Gelfand theory for
com~lex
Banach algebras is reviewed
briefly;
then the analogous theory for the case when the field carries a nonarchimedean real-valued valuation is presented. In the course of the latter discussion, a Stone-Weierstrass theorem is needed. In the last part of the paper some versions of the Stone-Weierstrass tireoremwhich hold in algebras of continuous functions over fields with nonarchi medean valuation are discussed.
1. CLASSICAL GELFAND THEORY.
If
([ 1 I, [9 I )
G is an open subset of
<1:, the complex numbers, and
a topological vector space, a map
x: G
333
-+
X
X
is
is anaR.y:Uc in G if the
334
NARICI and BECKENSTEIN
difference quotient has a limit at each point in
G.
For the vector-valued version of Liouville's theorem to
hold,
the vector space must have a good supply of continuous linear functionals. The dual space f
E
vanishes at
X'
x'
must be totat in the sense that if every
x, then x must be
1.1. LIOUVILLE'S THEOREM:
16
X i.!> a
x:
O.
and
TVS
X'
{6
i.!\ total the.n
([ 1 1 , p.211).
x mU-6t be. c.on.6tant.
For the remainder of the results in this section we assume that X is a complex commutative Banach algebra with identity e
h is a Ite.guf.alt point 06
A complex number vertible. The set
p (x)
over the resolvent map
x E X
of regular points of
x
if
(II e II = 1)
•
x - he is in-
is an open set.t-iore-
rx: p (x) .... X, h .... (x - he) -1 is analytic,
([1],
p. 208). An important consequence of these results is:
1. 2. GELFAND - MAZUR THEOREM ([ 1
a (x) a 6
.6 pe.c.tltum
1,
p. 212):
x, tho.6 e. c.ompte.x numb e.It.6
(al
Fait e.ve.lty
x
h
601t whic.h
x - he
E
X, the.
i.6
not inve.lttible., i.6 not e.mpty. (b)
16
X
.(.-6
ve.I!.-6e..6)
the.n
PROOF:
(b) Since
Since
a divi.6ion atge.blta (all nonze.lto e.le.me.nt.6 have. in-
X .(..6 .(.-6omoltphic. and iMme.tlt.(.c. to
a (x) I-
¢, x - he
X is a division algebra,
is not invertible for sare A E
x - Ae
must be
0, i.e. x
= he.
The proof of part (a) depends heavily on the Liouville theorem. Consequently one would suspect that this result would
not
transfer
easily to Banach algebras over other fields, and indeed this is case. Even in real Banach algebras there may be elements with
the empty
spectrum. As long as the underlying field is
algebras
and locally m-convex algebras ([ 1 1 , p. 212 - 3). The only change that
BANACH AL.GEBRAS OVER VAL.UED FIEL.DS
335
occurs is that the "isometry" of part (b) is replaced by "homeomorphism". For a time i t was wondered ([ 6 ) logical division algebras other
th~n~.
if there were complex t0poWilliamson ([12) ,[1, p.214ll
showed that there were by providing an algebraically compatible ~
pology for the field
(t) of rational functions in
t
with
to-
complex
coefficients. An important consequence of (b) above is that maximal ideal
x
function
M of maximal ideals of which sends
Minto
it becomes possible to view tions mapping
M with
for any
X. We denote the coset (complex number) x + M by
M of
x (M). It now becomes possible to view on the space
a:
X/M is
~.
M into
X as a collection of functions X. We associate
x(M) • Once
x
M has been
X as a collection
of
E
X with the
topologized,
continuous func-
Among other things, even without
endowing
a topology, it now follows that
1.3
Q'
(x)
x(M) •
In algebras of continuous or analytic functions
([ 1 I, p. 202-3)
characterizations such as 1.3 are the rule for describing i. e., the spectrum of a function
is its range.
M with the weakest topology which will make each
We endow
x continuous
the maps
x
spectra,
and call this the Gel6and
~opology.
M
of then
becomes a compact Hausdorff space. A B*-algebra is a Banach algebra with involution satisfyingthe condition
/I x* x /I
= II
x /1
2
. The celebrated representation theorem
of
Gelfand and Naimark states:
1.4. REPRESENTATION OF B*-ALGEBRAS ([1
algeb.l!.a, 06
~hen
con~inuou~
maximal
ideaf~
X
I,
i~ i~ome~ftically i~omoftphic ~o ~he
complex-valued 06 X
wi~h ~up
16
p. 259f.):
~he
X i~ a
algebJta
6unc~ion~
an
noftm (and
pointwi~e ope.l!.ation~).
compact
~pace
C
B*-
(M ,
~)
M 06
336
NARICI and BECKENSTEIN
2. GELFAND THEORY OVER VALUED FIELDS Here we assume that
X is a commutative Banach
algebra
with
identi ty over a field F where the norm on X and the valuation on F each satisfy the strong
("nonarchimedeanlf)
triangle
inequality:
II x + y II 2. max (II x II, II y II). Among the consequences of this inequality are that i f
IIxll i"llyll, then
Ilx+ yll =max (1Ixll, Ily II) and that
every point in a sphere is a center. All norms and valuations are assumed real-valued. A detailed discussion of such normed spaces
and
algebras can be found in [10], such spaces being called nOYlMc.IUme.de.a.n
.6pac.e..6. The critical result
((1.
2»
that
each
element have
nonempty
spectrum fails to hold for nonarchimedean algebras. There may beelements with empty spectrum ([10], p. 105). The worst consequence this is that we cannot say that X/M is
x.
of
F for each maximal ideal of
X/M is merely a superfield of F. If we hypothesize
separately
that each element have nonempty spectrum then, exactly as in proofof (1,2) (b), division algebras are isometrically isomorphic to the
derlying field. We define a Gel6and
alge.b~a
to be
a
nonarchimedean
commutative Banach algebra X with identity such that X/M = F each maximal ideal
M of
un-
for
X.
Although we cannot show that each element hasnonempty spectrum in an arbitrary nonarchimedean Banach algebra, we can show for any x that
a(x) is closed and bounded, the proof being about the same
for the complex case ([ 10], p. 114). Thus if F is locally each element has compact spectrum. Also (cf. (1.3» true that
a(x) =
it is
as
compact, generally
x(M) n F.
In an attempt to duplicate the complex Gelfand theory, we wish to introduce a topology to the maximal ideals. Two main choices
are
available: Restrict consideration of what elements x are to be chosen or consider only certain maximal ideals. More specifically we consider ([ 10], p. 1l7f.):
BANACH ALGEBRAS OVER VALUED FIELDS
2.1. THE GELFAND SUBALGEBRA maximal ideal
M,
x (M)
F
E
2.2. THE GELFAND IDEALS for every
Those
337
such that for
X
X E
every
or
M: g
Those maximal ideals
M such that x(M) E F
x.
In the first case we retain all the
M'S; in the second,all the
x's. It now follows that (a) for each (maximal) ideal in
M E
M,
M
(Le., X 1M n X = F); g g
X g
n
X
is a Gelfand
g
(b) X=X
g
iff
M= M g
(X is a Gelfand algebra iff each maximal ideal is a Gelfand ideal or X coincides with its Gelfand subalgebra); gebra of
(c) X
g
is a closed subal-
X.
We may now consider the following topologies.
2.3. THE WEAK TOPOLOGY: weakest topology for
Define the weak Get6and
M such that each
X E X
~opotogy
to be the
is continuous.
g
induces the weak Gelfand topology on
Mg
2.4. THE STRONG TOPOLOGY:
!.>~ltong Get6and ~opotogy
Define the
This
to be
the weakest topology for Mg such that every xE X is continuous. This is clearly stronger than the weak Gelfand topology.
REMARKS:
(a) Strong topologies yield spaces with more structure. (b)
M is generally not big enough to yield information about g
the Gelfand ideals e.g., if
Mgg of
x E X ' then g
formities and
Mg and
X whereas
Xg are rich enough to help describe
a(x)
= x(M
gg
).
Xg '
(c) These topologies are uni-
Mgg are complete. Thus
Mg or
Mgg is compact
iff they are totally bounded. The last remark helps to obtain the following compactness suIt.
re-
338
NARICI and 8ECKENSTEIN
2.5. COMPACTNESS ([ 10
I, p.
16
124):
-tJ.. loeally eompae.t .then
F
M
.6bLOngly eompae.t, .then ei.theJt F i.6 loeally eompae.t OJt .the 06 any elemen.t in
rl
Mg
X
g
.6pee.tJtum
Xg iJ.. nonemp.ty, eompac..t, and nowheJte den.6e.
2.6. DISCONNECTEDNESS ([101, p. 125):
Eaeh 06 .the Gel6and .topolog-te.6
i.6 O-dimenJ..ional an.d eaeh 06 .the .6paee.6
M , M , M " Xg in .the .6.tJtong g gg .topology -t.6 .to.tally di.6eonnee.ted and Hau.6doJt66.
2.7. SEPARATION ([ 10
I, p. 126):
The 6011ow-tng .6ta.tement.6
Me
eQu-i.va-
len.t. (a) The
(wea~)
Gel6and .topology on
(b) The 6une.t-ton.6 6Jtom (c) The
nune.t-to n.6 6Jtom
(d) The map M +M
nxg
M
i.6 Hau.6doJt66.
X .6epaJtate po-tnt.6. g Xg .6epaJtate poin.t.6 .6tJtongly.
-t.6
1 - l.
Maximal ideals must always be of codimension 1. Conversely, in
[ 5 I , Gleason proved that a linear subspace of codimension
1
in a
complex commutative Banach algebra with identity is a maximal
ideal
iff it consists of Singular elements. Hence, in a nonarchimedean Banach algebra, one might consider the question: If
M is a linear subspace
of codimension 1 consisting solely of singular elements, must Gel fand ideal? The fact that Gleason's
M bea
argument uses deep theorems
from complex variable theory gives Warning that
the
nonarchimedean
question could be difficult. In [2 I
the authors considered Gleason's question in the topo-
logical algebra (endowed with the compact-open topology) C(T,F) continuous function from a topological space field
T into
a
of
topological
F. It is shown there that Gleason's result is true if F is the
field of complex numbers, false if
F is the reals, and true if F is
an ultraregular field containing at least three points under any the following conditions.
of
BANACH AL.GEBRAS OVER VALUED FIEL.DS
339
1. F
is not algebraically closed.
2. F
possesses a sequence of distinct elenents converging to O.
3. F
is discretely valued.
4. The topology of
5. T
F is given by a valuation.
is ultranormal.
We say that a Gelfand algebra is
c-o~~c-~de~
~~
X
w.-Lth the
~egu!a~ ~66
hul!-~e~~e.e
the
-
if the functions
x
M strongly.
separate points and closed subsets of
2.8. REGULAR:
~egula~
Iwea~)
topology 011.
Ge!oa~d
M.
topology
o~
M
([10 I , :c;>. 135).
In the complex case, X is regular iff the hull-kernel to:c;>ology is Hausdorff and the :c;>roof relies heavily on the com:c;>actness of the Gelfand to:c;>ology. By
c~oosing
nonarchimedean algebras in which
is not com:c;>act, one obtains counterexamples to "if the topology is Hausdorff, then
each maximal ideal that
U
c W. If
X and let
Since II x (M) II.::. II x II
M}.
M
hull - kernel
X is regular".
U be the unit ball in
Let
M in
U = W, we call
X
W = {x I II x(M) II .::. 1 for every
for
M, it is clear
a V*-algebJta.As will be seen shortly, ( see
the V*-algebras are the nonarchimedean analogs of B*-algebras (2.10». I t is easy to verify ([10], p. 148)
that V*-algebras must be
semisimple.
2.9.
16
.-L~
T
p!ete then
T
a .-L~
O-d.-Lme~~.-Lonal
c-ompac-t
Hau~do~66 ~pac-e
homeomOJLph.-Lc- to the .6pac-e
C(T,F) ul1.deJt the map
M
0
6
and F
max~mal
.-L~
.-Ldea!~
c-om06
~
M = {x E C(T,F) I x{t) t the Gel6and topology. Al~o) C(T,F) ~~ a v*-algebJta ([10], :c;>. 154). In
add.-Lt.iol1., .-Lo
S
t
~~ O-d.-Lmel1.~~o~a.e,
meomoJtph'<'c- to T '<'66 C(S,F) .-L.6
c-ompac-t a~d Hau~do~66 the~ S ~ ho-
.-L~omo~ph.-Lc-
to
C(T,F).
As a first representation theorem we have 2.10. ([ lO),p. 164)
16
Xg .-L~ a V*-Ge!6and
algeb~a
and
Mg
.-L~
c-ompac.t
NARICI and BECKENSTEIN
340
~hen
Xg
low~
~ha~
x
i~ome~~lcally i~omo~phic
i~
i~ i~ome~~icallq i~omo~phic ~o
i6
C(M g ,F),
X i~ a v*-Gel6and algeb~a in which ~o
6~om
M
which
i~
i~
60l-
compac~ ~hen
C(M,F).
For the proof of (2.10) one needs a version of a Stone-Weierstarss theorem for algebras of continuous functions which take values in
a
nonarchimedean valued field. Such theorems are the subject of the next and last part of the paper.
3. STONE-WEIERSTRASS THEOREHS
F denotes a field with nonarchimedean valuation. Generalizing a result of Dieudonne ([ 4 1),Kaplansky ([ 7 1)
ob-
tained the following analog of the classical Stone-Weierstrass theorem.
3.1. KAPLANSKY-STONE-WEIERSTRASS THEOREM: i~
a
compac~ Hau~don66
~a~e~ poin~~
and
~pace
con~ain~
and Y a
([ 7
I,
~ubalgeb~a
con~~an~~ ~hen
i~
y
[10, p. 1621):
16
06 C(T,F) which
~epa
den~e
in
T
C(T,F).
An immediate consequence of this is
3.2.
and
([ 71, [10, p. 1631):
a
Y
~u.balgebna.
in6inity - which in
06
16
T
i~
a locally
Coo(T,F)-continu.ou~
~eyJMa~e~
point~
and
compac~ Hau.~dM66
6unction~
contain~
which
con~tanU
~pace
vani~h
at
then Y in deMe
C",(T,F). As has been observed by Nachbin ([8 I), it is not really neces-
sary to consider subalgebras
Y for Stone-Weierstrass type theorems:
sub-modules suffice. To quote just one of many possible illustrations of t:'lis
viewpoint ([ 3 I , for example) we have the following result of
Prolla's. 3.3.
([ 111, Cor.
2.5):
Let T be a compact
Hau~dM66
~pace,
X
a
BANACH ALGEBRAS OVER VALUED FIELDS
341
vwnaJtchime.de.an nOJtme.d cpace. ove.Jt F and l·t an A-cubmodule. 06 C(T,X), whe.Jt e. A ic a The.n
J.>
e.paJtating
J.>
ubalg e.bJta 06
W iJ.> de.nJ.>e. in C(T,X) iSn
C (T, F) •
bOlt e.ac.h
t in T, v/(t) ={w(t)lwEW}
REFERENCES
[11
E. BECKENSTEIN, L. NARICI and C. SUFFEL, Topological Alge.bJtQJ.>, North-Holland Publishing Co., Amsterdam, 1977.
[21
E. BECKENSTEIN, L. NARICI, C. SUFFEL and S. WARNER I
I~ximal
ideals in algebras of continuous functions, J. Anal. Math. 31(1977),293-297. [31
R. C. BUCK, Approximation properties of vector - valued
func-
tions, Pacific J. l4ath. 53(1974), 85 - 94. [41
J.
DIEUDONN~,
Sur les fonctions continues p-adiques, Bull.Sci.
Math. 68 (1944), [51
79 - 95.
A. GLEASON, A characterization of maximal ideals, J . Anal. Math., vol. 19(1967), 171-172.
[61
I. KAPLANSKY, Topological rings, Bull. Amer. Hath. Soc. 45(1948)
809-826. [71
I. KAPLANSKY, The Weierstrass theorem in fields with valuations, Proc. Amer. Math. Soc. 1(1950), 356 - 357.
[81
L. NACHBIN, AppJtoximation TheoJty,
van Nostrand, Princeton, 1967.
Reprinted by Krieger Publishing Co., 645 New York
Ave-
nue, Huntington, N. Y., 1976. [91
M. NAlMARK,.NoJtmed RingJ.> , Nordhoff, Groningen, TheNetherlands, 1964.
[101
L. NARICI, E. BECKENSTEIN and G. BACHIvlAN, Func-ti.onnl Ana.f.yJ.>.ic and Valuation The.oJty, Marcel Dekker, New York, 1971.
NARICI and BECKENSTEIN
342
[11)
J. B. PROLLA, Nonarchimedean function spaces. To appear Li~ea~ Space~
a~d Appnoximatio~
in :
(Proc. Conf.,Oberwolfach,
1977; Eds. P. L. Butzer and B. Sz.-Nagy), ISNM vol. 40, Birkhauser Verlag, Basel-Stuttgart, 1978. [12)
J. H. WILLIAMSON, On topologising the field C(t), Proc. Math. Soc. 5(1954), 729 - 734.
Amer.
Apppoximation Theopy and FunationaZ AnaZysis J.B. PpoZZa (ed.) ©North-HoZZand PuhZishing Company, 1979
APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS
PHILIPPE NOVERRAZ Universite de Nancy I Mathematiques 54037 NANCY CEDEX, France
If
U is an open and connected subset of
sional locally convex vector space (resp. [-oo,+oo[ )
a:,
E on
an infinite
an application
dimenf: U -+
a:
is said to be hoiomOltphic (resp. piuJU.6ubhaJl.monic)if
a)
f
is continuous (resp. upper semicontinuous)
b)
the restriction of
f
to any finite dimensional
subspace
is holomorphic (resp. plurisubharmonic). Let us denote by
H(U)
(resp. P(U), Pc(U»
the set of holomor-
phic (resp. plurisubharmonic, plurisubharmonic and continuous) functions on
U. If
In
a: n ,
1)
Any
2)
K is a compact ,subset of
{x
E
u,
{x
E
u, vex)
U, let us denote by
if(x)i < IfiK, \/f
< sup v, K
\/v
E
E
H(U)}
P(U)}.
n ~ 2, the following results are well known (3): v in
p(U) is the pointwise decreasing limit
of
COO
plurisubharmonic functions in a strictly smaller open
set
U'
of
If
U is pseudo-convex (ie
compact
U (ie
K of
d(U', C U) > 0).
U)
then
-
-
Kp (U) is compact in
Kp (U) 343
= Kpc (U)
.
U for any
344
NOVERRAZ
3)
If
U is pseudo-convex, then for
compact subset of al, ... ,a j
U there exist
fl, ... ,f
in
j
>
£
0 and K
H(U)
and
£,
If
K = KHeU ) is compact in a pseudo-convex open set U, then any holomorphic function in a neighborhood of K can be
approximate:'! uniformly on 5)
Pc (U),
positive numbers such that
Iv - sup a i log IfiliK < i 4)
v in
K by elements
of
H (U) .
are pseudo-convex, U c U' then KHeU ) ,=KH(U') for any compact subset of U if and only if H (U') is dense If
U and U'
in
H(U) for the compact open topology.
Properties 3), 4) and 5) have been generalised to larger classes of locally convex spaces with Schauder basis including
Banach
spaces (6). ~n, condition
We shall investigate conditions 1 and 2. In is obtained by regularisation (ie convolution) of
v by a Dirac
quence, so it is natural to consider some measure.
1) se-
For the sake
of
simplicity we shall consider here only (infinite dimensional) Banach spaces and Gaussian measures following Gross (5). It is well known that in a Banach space E there are no
sub-
stitute to the Lebesgue measure that means there does not exist
a
measure invariant by translations or rotations. A Gaussian measure
~
on E can be characterized as follows: there exists an Hilbert space H~
densely and continuously irnbeded in E such that
~
the cylindrical Gauss measure on the cylindrical sets of triplet (H
~
arises H~
from The
, i, E) is called an abstract Wiener space. The following
properties hold: 1)
Let be T in
£ (E,E), if T restricted to H" is in £(H ,H ) ,.. ~ ~ and is unitary then ~ is invariant by T (ie ~T-l = ~).
APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS
2)
Let be
11
x (A)
= 11 (x
346
+ A), A Borel in E, then 11 and 11 x are
either equivalent or orthogonal, they are equivalent if and only if
x belongs to
H
]J
We have the following Lemma:
16
LEMMA 1:
.i...6 a GaU.6.6.i..an mea..6uJte
]J
haJtmon.i..c. 6unc.t.i..on .i..n an open .6ub.6et
]J
60Jt r
[B (O,R)
n E and.i..6
06
U
E
I
1
vex) <
0
1
.i...6 a pluJt.i...6ub-
v
we ha.ve
v (x + y) 11 (dy)
lIyll~ r
-6mall enough.
Suppose that
PROOF: mapping
x
->-
e
i8
x
v
is bounded from above in the ball B(x,r), the
induces a
unitary mapping
T 8 on
Hll ,so
II
is
T8 - invariant, and we have
v(x+ye
i8
v(x+y) ll(dy) ,
) ll(dy)
but
vex) <
....!...2Tf
vex + ye
i8
)d8.
The result follows from
Fubini
theorem. A(v,x,r)
Let us note
Iii yli
v (x + y) II (ely),
A(x,v,r).i...6 apluJt.i...6ubhaJtmon.i..c. 6unc.t.i..on
PROPOSITION 1: 1)
06
.i...6 a c.onvex and .i..nc.Jtea.6.i..ng 6unc.t.i..on 06 log
ll(r)A 2)
1
Il(O,r)
vex) = lim
we have:
x
and
r.
A(x,v,r).
r=O 3)
A(x,v,r)
H
Let us recall that a function any
x in
E the function
entiable at
y = O.
II
-
d.i..66eJtent.i..able.
is H -differentiable if for II y ....
NOVEAAAZ
346
1) is a consequence of the fact that plurisubharmonic
PROOF:
func-
tions depending only from II x II are logarithmically convex. 2)
Since
v is upper semicontinuous, for any
vex + y) < vex) +
II y II -< r X,E:
for
E:
-
vex} < A(x,v,r} 2 vex} + 3)
E:
E:
> 0
we have
hence
•
Is a consequence of a result of Gross (5).
Let us notice that, unless
v is continuous, A(v,x,r)
in general a continuous function of
is
not
x.
As a consequence of 2) and 3) we have:
PROPOSITION 2:
A
6unetion v
plu~i~ubha~monie
WiH Umit 06 a Hquenee 06 in61nUe1.y
i~
loeally the point-
H-di66e~entiabte pt~ubltMmonie
6unc;tion~.
There is an other way to approximate bounded functions: let be a Gaussian measure of parameter ~t{11
x II
~ C4 > O} ... 0
function P tf (x) = f
t ...
if
IE f (x
+ y)
0
t > 0, then
then Gross
~t (dy)
P tf
tends to
PROOF:
For
f
E:
!f(x) - f(y)! 2
uniformly on
< 0, there is
and has proved that
(5)
f
<
IE
E.
n such that
!x -
yl
~ n
implies
~t(dy)
!f(x + y) - f(x)!
< E:).Jt(1I yll
< 2 £:
if
is uniformly continuous
E:.
!Ptf(x) - f(x)!
the
H~ -differentiable
is infinitely
is bounded and measurable. Moreover if
~t
if
< n)
+
211fll", ).Jt(lIy II
t < t
E:
> n)
APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS
If
f
347
is only continuous, then the convergence of
P tf
to
f
is uniform on every compact subset. It is also well known (1) that there exist Banach spaces such that the bounded and in the space of uniformly ceding result gives a
Cl
several
separable
functions are not
dense
continuous and bounded functions. Thepre-
uniform approximation by H-infinitely differen-
tiable function. For plurisubharmonic functions this kind of approximation gives more or less the same result as proposition 2. Now we shall state the following proposition:
PROPOSITION 3: E and let
v
Le:t
U be a p.6eudo-convex open .6ub.6e:t 06 a Banach .6pace
be a pluft-<-.6ubhaftmoyt-{,c 6unc:t-<-on on
u,
:then
-<-.6
v
:the
po-<-n.tw-<-.6e ,Um-<-t 06 a decftea.6-<-ng .6equenc.e 06 conunuOu..6('£n 6ac.:t Up.6c.hilz)
Le:t
COROLLARY:
U be a p.6eudo-convex open .6ub.6e:t 06 a Banach
.6pace
E, .then
60ft any compact .6ub.6e:t
K
U•
06
For the proof we shall follow an unpublised paper of
C. Herves
and M. Estevez (2). They first generalize in the Banach case an idea of (3):Let f be a lower semi-continuous function bounded from then for any integer
above,
k define
inf [kilx - yil + fey»). y I t is easy to show that
Moreover
f
f k - l ~ fk ~ f and
I fk (x)
- fk (y)
is the pointwise limit of the sequence
I
fk •
~ k II x - yll.
348
NOVERRAZ
If U is pseudo-convex and v is a plurisubharmonic function in U we take this approximation sequence of the function exp (- v)
in U and zero outside
U
{(z,w)
E
and if we consider the norm
f defined by
U. If we state
a:, I w I
U x
k II z II + I w
<
I
If (z) I }
on
E x
a: ,
we have
d«z,O), CU)
Since U is a pseudo-convex domain it follows that - log fk is plurisubharmonic in
U,
moreover
v = lim [-log fk I.
Proposition 3
is
k
proved.
PROOF OF THE COROLLARY: If
Xo '! Kp(U)
It is sufficient to prove that
there is
v in
P(U) such that
.
~c(U)
c
.
~(U)'
u(x o ) > ex > sup v. K
Proposition 3 implies that there is a decreasing sequence
(v ) n
in
Pc(U), hence:
K C {x E
u,
vex)
< ex}
U
{x
U, v (x) < ex}. n
E
n
Since K is compact and
vn+l
~
vn
there is an index
p
that
K C {x E U, v
p
(x)
< ex} •
We have
sup v K
hence
Xo does not belong to
P
KPc (U) . The corollary is proved .
such
APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS
349
REFERENCES
[1 I
R. BONIC and J. FRAMPTON, Smooth functions on Banach manifold, J. Math. Mech. 15(1966), 877 - 898.
[2 I
M. ESTEVEZ and C. HERVES, Sur une propriEhe de l' enveloppe plurisousharmonique dans les espaces normes, preprint.
[ 3 I
L. HORMANDER, Art .trttJtoduct.tort to complex artalY-6.t-6, Van Nostrand 1966.
[4 I
J. P. FERRIER and N. SIBONY, Approximation ponderee sur une sous-variete totalernent reelle de ~n, Ann. Inst.Fourier 26 (1976) I 101 - 115.
[5 I
H. H. KUO, Gau-6-6.tart mea-6uJte .trt Bartach -6pace-6, Springer Lecture Notes 464.
[6 I
Ph. NOVERRAZ, Approximation of holomorphic or plurisubharmonic functions in certain Banach spaces. PJtoc. Ort Irt6.trt.tte V.tmen-6.tonal HolomoJtphy, Springer Lecture Notes 364, p. 178-185.
[7 I
Ph. NOVERRAZ, P-6eu.do-cortvex.tte., cortvex.tte. polYrtom.tale e-tdomaJ.rte-6 d' holomoJtph.te, North-Holland Publishing Co., Amsterdam, 1972.
This Page Intentionally Left Blank
Appl'OJ:imation Theory and Functional Analysis J.B. Prolla (ed.) ~North-Holland Publishing Company, 1979
THE APPROXIMATION PROPERTY FOR CERTAIN SPACES OF HOLOMORPHIC MAPPINGS
OTILIA T. WIERMANN Instituto
PAQUES
de Matematica
Universidade Estadual
de Campinas
Campinas, SP, Brazil
§O. INTRODUCTION If
E and
F are locally convex complex Hausdorff spaces,
let
XS(E;F) be the vector space of all Silva-holomorphic mappings from E to
F.
(See Definition 1.13 below). In section 1, after the preliminary definitions, we study
the
XS(E;F) endowed with the topology of uniform convergence
on
space
strict (see Definition 1. 21) compact subsets of
E.
In section 2, we prove that for a quasi-complete space
E
the
following properties are equivalent: (a)
E has the S-approximation property (see Definition 1.31);
(b)
XS(E;~)
0 F
is
'S-dense in the space
locally convex space (e)
XS(E;F), for every
F;
Xs (E;C) with the topology,s' has the approximation
prop-
erty. For Banach spaces, Aron and Schottenloher [2 have some results about this for the space (X(E;~) ,TO)
E to
~,
I
1
(X(E;~),
and
Aron
TO)'
[1
1
(where
denotes the vector space of all holomorphic mappings from
endowed with the topology compact-open wish to thank Prof. Mario C. Matos for
351
'0)' his
guidance
and
362
PAQUES
encouragement during the preparation of this paper.
§l. SILVA-HOLOMORPHIC MAPPINGS. In this paper E and F are always locally Hausdorff spaces and
convex
U is a non-void open subset of
complex
E. BE will de-
note the set of all closed absolutely convex bounded subsets of If
B
E
BE'
EB
is the vector subspace of E generated
normed by the Minkowsky functional
PB
by
determined by B.
E.
Band
cs (E)
is
the set of non-trivial continuous semi norms on E.
1.1 DEFINITION:
n = 1,2, ••. ; £b(nE;F) will denote the
Let
space of all n-linear mappings from
vector
En = Ex .•• x E (n times)
F, which are bounded on bounded subsets of
En,
to
endowed with the 10-
cally convex topology generated by all semi norms of the form:
n
where
T E £b( ElF), Bl,.··,B n E BE
we denote II Til B
a (T (x
for all
B 1'···' n'
1"'"
x, E EB ~
i
x
n
' i
of ~
~b(
»
<
II T il· p (x) Bl, ••• ,Bn,a Bl 1
We will denote by
0
ElF) = £bs( ElF) = F
1.2 PROPOSITION:
Q
IIJ
£b(nE;F) are called Silva-bounded (S-bounded)
£b(nEIF ) of all such 0
= II T II B
fJ
= 1, •.• , n •
The elements of n-linea~ mapping~.
Q
8 E cs (F). If Bl = ••• =B =B, n . Notice that and
16
n
£bs( ElF) the vector
T that are symmetric. For and
II TII
B
n=O, we define
,13 = a(T), forevery T E£b(oE;F).
F i~ a c.omple;te loc.a.lly c.onvex ~pac.e.
i~ c.omple;te. Fo~ evelty
subspace
£b(~IF)
F, £bs(nE;F) i~ a c.lo~ed vec.;tolt ~ub~pac.e
06
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
1.3 DEFINITION:
For
Ts E £bS(nEiF )
T E £b(nEiF ), we
1
Sn
~
OES
The mapping
i6 and only i6
n
= 0,
n.
->
Ts E £bs( n EiF)
~
a con-
Fu~~he~mo~e
£bs(nEiF ).
T E £bs(nEiF ).
1.5 DEFINITION:
If
,xcr (n»'
T(X O (1)""
T: £b(n EiF )
~inuou~ p~ojec~ion 06 £b(nEiF ) on~o
Tx
syrrunetrization
n
is the syrrunetric group of degree
1.4 PROPOSITION:
we write
its
by
!iT where
define
363
n
Let
T E £b(nEiF ), x E E
to denote
we define
Tx
and
T(x, .•. ,x) where o
= T. A mapping
x
n E IN. If is repeated
P: E
->
n=1,2, ... , n
times.
is a Silva-bounded
F
n-homogen~_olU> polynomial if there is
T E £b (n EiF ) such that P(x) =TXn,
for every
corresponds to
T
P,
we
write
P = T. P ( EiF) denotes the vector space of all Silva-bounded b
n-'ho-
•
x E E. To denote that n
mogeneous polynomials from locally convex topology
E to
TS
F.
On
P
E
we consider
the
generated by all seminorms of the form:
sup {S(P(X»i
where
Pb(nEiF )
n
P ( EiF), b
and
x E E},
S E cs (F) •
Notice that
1.6 PROPOSITION:
The mapping
~o~ ~pace iMmMphi~m
and a
T E £bS(nE;F)
homeomo~phi~m
06
->
T E Pb(nEiF ), ~ a vec-
~he
6i~~~
o~o ~he ~econd
364
MOUES
.6pac.e. MOJteoveJt
(1) •
nn
11!
1.7 REMARK:
(Nachbin [9
is the best universal constant occurring
(1) •
1)•
16 F i.6 a c.omplete loc.ally c.ovwex .6pac.e,
1. 8 PROPOSITION:
i.6 c.omplete 60Jt all
P: E
~
F
for which there are
(k = 0, •.. , n) such that
P = P
o
+ ... + P
P
oJtom
P
n =0,1, ... ,
n
.
1.10 PROPOSITION:
(k = O, ... ,n)
P
P
and
p
1. 11 DEFINITION:
is a series in
A
0
k
F E
is
P ( k E;F) b
We will denote by P (E;F) b E to
F.
+ P n' with
+
n = 0,1, ••• ,
P
k
E
P ( k E;F) b
O. n ~
6oJtmal powe.IL .6 eJtie.6 6Jtom E to F ab 0 ut
x E E
a
Pb(E;F) , p 'I 0, theJte i.6 one and anly one
E
P
16
to
E
the vector space of all Silva-bounded polynomials from
way 06 wJt..i.ting
Pb(ll:E;F)
n=O,l, . . . .
A Silva-bounded polynomial
1. 9 DEFINITION:
mapping
in
~
E E
of the form co
k An (x _ ~}n, n=o where
n
An E .cbs ( E; F)
(n = 0,1, ••• ); or of the form
k
n=o where
P
-
n
An
E
n
P ( E;F) b
(n =0,1, ... ). An
coefficients of the power series.
and
P
n
are called
the
THE APPROXIMATION PROPERTY FOR SPACES OF HOL.OMORPHIC MAPPINGS
~ ek.[e~
16 the powek
1.12 LEMMA:
that 60k aii
~),
abou.t
~
E ~ ~u.c.h
E
S E cs(F), theke .[~ ~ome
and
B E BE
n~o P n (x -
355
that m
lim
S(~
m-+ oo
60k aii
Pn(x-~»
x E ~ + PBB, then S(Pn(t»
1.13 DEFINITION: A mapping f : U
(S-hoiomokph'[c.) on
U
0,
n=o
-+
=0,
OM
P
B
> 0,
~
if, corresponding to every
satisfying
~
~
fix)
+ PBB C U,
Pn(x -
tEE.
is said to be S.[iva-hoiomokph'[c.
F
Pm E Pb(~iF) (m=O,l, ... ) such that for all there is
n =0,1, ... , and
E U,
there are
S E cs(F) and such that
~),
n=o uniformly with respect to
~ + PBB. The sequence
S on
then unique at every point
sEE, by Lemma 1.12.
1
tions
We set the
is nota-
Pm' for p.1=O,l, ..••
if
Iii!
(P n) ~=o
The notation
fix)
~
m=o
1
"'iii!
indicates the Taylor series of
f
at
E,. J(S (U;F) denotes the
space of all S-holomorphic mappings from
1.14 REMARK:
U to
The above concept of holomorphy
F.
was
Sebastiao e Silva [16]. We will denote by J(U;F) all holomorphic mappings from this space see Nachbin [8] a mapping tinuous.
f E J(S(U;F) J(S(U;F)
U to
and
[9 J
vector
introduced
by
the vector space of
F. For some basic properties of and Noverraz [10]. Notice that
is holomorphic if and only if,
f
= J(U;F), for every open non-void subset
is U
conof
a
366
PAQUES
seminormed or a Silva space space
F.
In general,
borno1ogica1 space.
1.15 PROPOSITION: ~6
E and for every locally convex Hausdorff
j{'S(U/F) = j{'(U/F) i f
A mapp~ng
f: U ~ F U"
Le~
that
(1 -
A)
~
+
fix)
AX E U,
2
~
f E j{'S(U/F),
ho1omorphically
;i J
S-holomo~ph~c ~6
6M
A E
a:,
and only
eve~y
E U, X E U and
evefty
nM
~~
EB ,
B
1.16 COROLLARY:
a
(Matos [61).
~~ holomMph~c on
flu n E
is
E
I
A
f « l - AH + AX)
A-I
I 2..
p >
I
be .6uch
Then
p •
dA
IAI=p
1.17 COROLLARY:
and
p > 0
be .6uch ~hat
1
nT
Let f E JtS(U/F),
(Cauchy integral formula):
6n f(S)
(x)
~
+
AX E U,
6o~ eve~ A E IC ,
J
= _1_
2 1Ii
f
(~
+ AX) An + 1
I A
~EU, X
I.::.
E E
p.Then
d>'
IAI=p
60ft
n=O,l, ...
1.18 COROLLARY: ~ E
U
and
(Cauchy inequalities):
p > 0
.l p
60ft
~
+ pB C U. Then
~
be .6uch that
Let f E j{'s(U/F), SEcs(F), B E
sup {S(f(x»; X -
n
~
E pB}
n=O,l, ...
1.19 DEFINITION:
A mapping
f: U
holomoftph~c
if for every
¢ E F'
dual of
the function
¢
F)
0
f
+
F
(where
is said to be F'
wea~ly
denotes the
is silva-holomorphic.
S~lva-
topological
THE APPROXIMATION PROPERTY FOR SPACES OF HOI.OMORPH IC MAPPINGS
Let: F be ct .opctee w1.t:h t:he pllopellt:y thctt: 1.6
1. 20 PROPOSITION:
ct eompctct: .oub.oet 06
K 1..0
F, then the elo.oed ctb.oolutelif convex
r (K), 1..0 ct eompctct: .oub.oet 06
K,
357
F.
Then
S1.1va-holomollph1.e mctpp1.ng 1.6 and only 1.6
f : U -+ F f
1..0 S1.1va-holomOllph1.e.
The proof of this proposition follows from Proposition 1.lSand Nachbin [8 I .
1. 21 DEFINITION: A subset
if there is pact in
B E BE If
EB
K
E is said to be a .ot:Il1.et: eompctet set
of
such that
K
is contained in
E is normed, or Frechet (or
£ F l,
strict compact if and only if it is compact in We will denote b y , s of
EB
and
16
(Xs(U;F), 'sl
PROOF:
E.
the locally convex topology on
('0
F 1..0 a eomplet:e loeallif eonvex
BE BE'
U.
.opaee,
t:he.n
is complete, for
S E cs (F) •
be a Cauchy net in (Jes (U;F) ,'s) and
(falunEB)aEI
is a Cauchy net in (X(U') EB;F)"o)
is the compact - open topology). We know that
(X(U') EBi F ) "0)
F complete. Using this fact, it is easy to see that
there is
f E XS(U;F) such that (fa)aEI
(X (UiF),
's).
s
XS(U;F)
1..0 eomplete.
Let (fa) a E I
Then if
com-
K c: E is
then
uniform convergence on the strict compact subsets of
1.22 PROPOSITION:
is
converges
to
f
We now define the notion of Silva-holomorphic mapping of
on
com-
pact type, which will be needed in the next section.
1.23 DEFINITION:
For
linear mappings from of E,
and
E E
-+ I(J (
X
b E F, xl • b E F
I(J
E E'*, where
E to we
denotes the space
of
C, which are bounded on bounded
denote
the
S - bounded More
by
l(Ji E E'*, i=l, •.• ,n, n E IN
E'*
and
bE p', we denote
linear
subsets mapping
generally, the
all
for
S - bounded
358
PAOUES
n-linear mapping
by
The vecto::- subspace of
£b (nE;F)
generated
by all elements of
the
bE F, is denoted
by
form iplx ... xipn ·b, ipi E E*, i =l, ... ,n, and £bf(nE;F). We define the vector subspace be the closure of
£bf(nE;F) in
£b(nE;F), to
£b(nE;F). The topology on £bc(nE;F)
will always be the induced topology by complete space then
£bc(nE;F) of
n
£bc( E;F)
is
a
£b(nE;F). Hence, if complete
space.
We
F
is a define
£bfs(nE;F) =£bf(nE;F) n £bs(~;F) and £bcs(~;F) = £bc(~;F) n £bs(nE;F). For n = 0
we define all these spaces as
1.24 PROPOSITION:
1.25 DEFINITION:
n
£bfS(nE;F)
£bcs( E;F).
is said to be a S.Ltva-boundedn-Une.aJt
A E £b(nE;F)
mapp~ng 06 compact type if and only if Analogously, for
F.
A E £bc(nE;F) .
ip E E*, b E F, we denote the
n-homogeneous polynomial given by
X E
E
+
ip (x) n • b
Silva - bounded E
F
by
.pn • b E P ( n E;F). The vector subspace of Pb(nE;F) generated by all b elements of the form ipn • b, ip E E*, b E F is denoted by Pbf(~;F) • We define the vector subspace
Pbf ( n EiF) in
of
Pbc ( n EiF)
P
bc
a.
n ( EiF) of
Pb (nE iF) to be the closure
n
n
will
P ( EiF). Hence, if b
F
always
is a complet space
is a complete space.
1.26 PROPOSITION: ~nduc.e.6
bc
P ( EiF). The topology on b
be the induced topology by then
P
topolog~c.a.l
The natu~a.l mapp~ng and
•
n
T E £bS(nEiF ) +TEPb(EiF)
algebJt.a.~c. ~.6omoJt.ph~.6m
between
£bcs (nEiF ) and
(nEiF) •
1. 27 DEFINITION:
is said to be a
~va-bounded
n-homogeneouo
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
QompaQ~ ~ype if and only if
polynomial 06
1.28 DEFINITION:
Let
369
P E Pbc(nE;F).
Xsc(U;F) be the vector subspace of
of all Silva-holomorphic mappings f : U ->- F, such that for each 1 ~n n and n E lN, nT IS fix) E P ( E;F). An element f E Xsc(U;F) bc be called a Silva.- holomoJtph-LQ ma.pping a (\
QompaQ~ ~ype
06 U
A main tool of this paper is the notion of £-product by Schwartz [14]
which we want to review.
1.29 DEFINITION:
Given two locally convex Hausdorff spaces
F'c
F, we denote by
the dual of
.c £
(F
I
c'
E)
will
-Ln~o
F.
introduced
and
E
F endowed with the topology of uni-
form convergence on all balanced convex compact subsets of E £F =
x EU
F, and by
the space of all linear continuous maps from
FI
to
C
E, endowed with the topology of uniform convergence of all equicon tinuous subsets of seminorms
S £
£(F~,E),
U
S E cs(F) and
1. 30 DEFINITION:
the
a.pPJtox-Lma.~-Lon
E F',
0. E
lui
PJtopeJt~y,
EB
K of
E,
for all
x E K.
< £,
PJtopeJt~y
there is
and given
v EE', Ivl < o.},
if for every
for all
0.
E
E £F '" F (E.
is said to have
E cs(E), every
K of
E, there is T
£ E
> 0,
E' ® E,
x E K.
A locally convex Hausdorff space E is said to have
the S-a.pPJtox-Lma.tion set
s,
A locally convex Hausdorff space
o.(T(x) - x)
1.31 DEFINITION:
<
cs(E). We have that
and every balanced convex compact subset such that
£ (F' ,E) is generated by the £ c
defined by:
0.
sup {I (T(u),v) I;
(S £ 0.) (T)
T E
F'. The topology on
(S.a.p.), if given a strict compact sub-
B E BE
£ > 0, there is
such that
K
C
EB
and is compact in
T E E* 0 E, such that
Pa(T(X)- xl < £,
360
PAUUES
1.32 REMARK:
If
S.a.p., E' = E*, and all compact subsets
E has the
of
E are strict, then
E
is a normed space, or Frechet, of
E
has the approximation property. If (En) ~=O
sequence
property, then
E has the approximation property. Hence,
of Banach spaces
E has the S.a.p . .
£F, which has the S.a.p., then E is an inductive limit
Enflo in [3) S.a.p .•
Let E and F be locaU.y convex (ten~oJt
F
E
i~
E-topology) b)
A
pJtoduct 06
E
locaUy convex
and
Hau~doJt66 ~pace
all locally convex
60Jt all Banac.h E
i~
E ® F
Hau~dolt66 ~pace~
matIol'l pltopeJtty i6 al'ld ol'lly i6
16
E
and
F
I.>pace~
6
~pace~.
endowed
F,
~ub~pace
ha~
with 06
E e: F. (E
®e:
§2. THE APPROXIMATION PROPERTY FOR
~ub~et 06
E. Then
locally convex PROOF:
Let
~pace
Xsc(U;C)
601t
ha~
the apPJtou-
F.
E 0 F
E
il.> devt.6e il'l
Hau~doJt66
E £ F,
I.>pace~,
complete
E
®£
F
F denote~ the completion 06 E ®E F) •
Xs(U;(:).
S.a.p. and let
®F
F.
EEF,
We begin our study with an examination of the closure
Let E have the
E
in
and E olt F hal.> the appJtoximation pJtopeJtty, then
2.1 THEOREM:
E
the
F.
aJte locally c.onvex
il.> identical to
Then
the appJtoximation
den~e
be a qua~i-complete ~pace. Then
Let E
d)
Hau~doJt6
a topological vectOlt
pJtopeJtty i6 and only i6
c)
gives an example of
(Schwartz [14 I :
1. 33 PROPOSITION:
E ®
of a
En' which have the approximation
a Banach space which does not have the
a)
if
i~
Ts-den~e
of
U be a balanced
the
open
in
F.
K cUbe a stric compact set. By hypothesis there
is
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
B E
x E K. Let
such that
f E J(S(U;F),£ > 0
that there is
6 >
a.
is the complement of whenever
and is compact in
T E E* ® E
£ > 0, there is
every all
K C U n EB
such that
BE
x E K
and
6 x < distEB(K,CEB{U () E B
< 6.
Since
»,
X
fl
K,
E
U ()
there
{xl' ... ,x } C K. n r > a}).
and
y(x)
=
(B(a,r)
y: K
-+-
sup { 0 x, - PB (x - xi); i
x E K
and
B(x,a) C B(Xi,ox,)'
thus
Now for any
CE (U B
-
E ) B
()
fey) ) < £, continuous
<
£/2,
for
n
C
,u B(x, ,axi"~ l.=l l.
{x E E ; PB(x - a) < r, when B
= 1, •.. , n}
is continuous and y > O. Let
R
show
Define
l.
Then
< £, for
is
S{f(x) - fey»~
such that
xl
is
EB
is compact is EB ' K
a E EB
-
PB (T(x)
for
S E cs (F) . We first
and
(Proposition 1.15), then for each
for some set
so that
EB '
6 < dist (K,C (U " E » (where EB EB B U () EB in E B ) , such that S (f (x)
PB(x - y)
361
Y E B(x, 6),
X E K.
for
6 = inf {y{x); x
there
is
some
E
i
K}.
with
l.
S(f{x) - f{y»
Since
E has the
for all
S.a,p"
< S(f(x) - f{x
there is
x
E
K. Let
{gl"'"
n ~ cl>i (xl9 i ' i=l
T(x)
Let
E
» + S{f(x i ) - fey) l <
E* ® E
such that
E.
%(T(X) -xl < a,
x E K. By the above, we get that
S (f (T (x) for all
T
i
f (xl)
-
9 } n
< £,
be a basis in
T(E) and let
where
U = U () EB () T{E). Since o
f
is Silva-holomorphic,
f
can
be
considered as a holomorphic mapping from the finite dimensional balanced set
Uo
into
F,
362
PAQUES
~'"
f (z)
zPf
Ipj= 0
P
pE~n
E~
where (zl, ••• ,zn)
n
subsets of
Uo' Since
E~,
such that
is
M
S(f(x) -
T(K)
C
and convergence is uniform on compact un EB
and is compact in
x
there
uo'
n
~ zPf) < £, for all points [pl'::'M p
S(f(x) -
Th us, if
, fp E F
z =
~
i=l
CP.(x)gi
E
T(K).
1.
K,
E
~ CPp(x) f ) < S(f(x) - f(T(x») + S(f(T(x) ~ cpP(x)f}< 2£. Ipl.::.M p Ipi'::'M P -
Since
the proof is complete. Now, we give an extension of the previous theorem class of subsets of
2.2 DEFINITION: to be
U be a non-void open subset of
E if
Pb(Ei~)
said to be 6inLtely S - Rung e in space
Eo of
2.3 REMARK:
E,
If
U n Eo
is dense in
E. U
(JCs(U;~),
T )'
S
is
said
U
is
E if for each finite dimensional sub-
is S-Runge in
Eo
E is a Banach space, this definition coincides with
the Definition 2.1 of Aron-Schottenloher [2] . If open subset of
another
E.
Let
S-Runge in
to
EI
then
U is
a
U is finitely S-Runge and S-Runge
balanced in
E.
(Paques [ll]).
2.4 THEOREM:
Let E have the
S.a.p. and let
U
be a.n open
I.lubl.let
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
06
E which I!> 6in.£.tely S-Runge. Then 60~
JCS(UiF)
eve~y
I!>
JC (UiC) ® F
s
locally convex !>pace
363
TS - deMe .in
F.
For the proof of Theorem 2.4 i t will be needed
the
following
proposition, which has important corollaries.
Let: F be a !>pace !>at:.i-66ying t:he 60l10w.ing
2.5 PROPOSITION:
16
t:Ion:
vex. hull
open
K i!> a compact: !>ub!>et:
06
!>ub~et
06
F, t:hen t:he clo!>ed abMfu-tely con-
r(K),.i!> a compact: ~ub!>et:
K,
cond.i-
06
16
F.
U .i~ a vwn-vo.id
06 E, then
n E IN).
PROOF:
Let
for all to
T : JC (UiF) s
f E JC COiF) , s
->-
¢ E F' and
JCS(U;C). for each
be defined by (Tf) (¢) (x) =(¢
JCS(Ui(J:) cF
f E JCS(U;F)
x E U. Clearly, and
by
peg)
p
E
Tf : F~
We now show that the linear map ous. Indeed, let
¢
->-
JC (UiC)
s
fined by
q(¢)
= sup
p( (Tf) (¢»
for all
¢
E
Call it
K C U
g(x) E
sup{1 (¢
F'. Now
0
g
Let
f) (x) I i x
Tf E
A E JCS(UiC) cF (F~)'
L.
q
defined
JCS(UiC)
is a strict
compact f(K)
be the semi norm on
F
is a f
de-
{ I¢(t) l i t E L}. I t follows that
¢ E F'. Hence
Let now fine
F.
belongs
is continu-
set. By hypothesis, the closed absolutely convex hull of compact subset of
f) (x),
F'.
be a TS-continuous semi norm on
= sup {Ig(x) Ii x E K}, where
(Tf) (¢)
0
=F
£(F~i
E
K} < sup{I¢(t) Ii tEL} =q(¢)
JCS(Ui(J:».
= £(F~,
by the formula
JCs(U;C». For each g(x) (¢)
is weakly S-holomorphic, hence
=
(A¢) (x),
x E U, defor
S - holomorphic
all by
364
PAOUES
Proposi tion 1. 20. Clearly, Tg = A, and therefore T is onto JCS(UI(J:) e: F. On the other hand, T
is injective by the Hahn-Banach Theorem.
remains to show that
T
Let II(g)
= sup
is a homeomorphism.
8 E cs(F) and {Ig(x) II
It
K cUbe a strict compact
x E K},
subset.
Let
g E Jfs(UI
we have by the Hahn-Banach Theorem, that
sup {8 ( f (x) ); x E K} = sup { I cp ( f (x) ) :; cp E F I,
sup {
I( ¢
0
I cp I 2. 8,
X E K}
f , Y )I
(8 e: II) (Tf).
This completes the proof.
2.6 COROLLARIES OF THE PROPOSITION 2.5:
6ub-!>et 06 a)
16
U
F
OIL
i-!> a
non~void
open
E, we have:
16
F
,(,6
a complete .6pace and
(Je (UIC),
s
's>
ha.6 the
appILoximation PJtopeJtty, then
In paJttic.ulaJt i6 E ha.6 6inite dimen-!>ion and F i-!> a
com~
plete -!>pac.e, then
b)
16 F ha.6 the appJtoximation pJtopeJtty and c.ondit-ion 06 PJtopo.6i.t.[on 2.5, then in
0)
the
Jfs(U;e
Jf (U;F) • S
(Jfs(U;
i6,
-!>ati-!>6ie-!>
ha.6 the appJtoximation pJtopeJtty i6 and
JfS(Ui
-!>pace-!>
F.
i-!>
's-den-!>e in
JfS(U;F),
only
noJt ali Banach
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
The proof of a)
follows from Proposition 2.5 and
1.33 (d). The proof of b) 1.33 (b)i and c)
366
Proposition
follows from Proposition 2.5and
Proposition
follows from Proposition 2.5 and Proposition 1.33(c)
and Proposition 1.22.
PROOF OF THEOREM 2.4: and
Let
K CUbe a strict compact set, S E cs (F)
f E Xs (UiF). By hypothesis, there is
and is compact in satisfying
E
, so that given
B
PB(T(X) - x)
E
for all
< E,
>
B E BE
such that
0, there
where
(X(UoiF), TO)
is <
0,
F,
S(fl
with
u0
(y)
There are
Zj E F,
U
j = l , .•• ,m , with
is S-Runge in
o
Thus for
- fl(y»
Let
since
F.
<
£
I
E X(U it) 0
j
@
F,
Thus letting
h
it follows that
= f2
S(f(x) -hex»~
S(fl(y) - f
< S(f(x) -flu (T(x»
open ~ub~et E
PROOF:
E
Le.t
06
ha..6 th e
E.
2 (y»
I6
be a
and zJ' E F, j=l. ••
=
Zj) <£/2m ,
m
-
l:
'P
j =1
j
,m.
and with
0
Z.
J
E
< E, for all yET(K).
x E K,
+ S(fl u (T(x) -fl(T(X» 0
qua~.(-,,-omp.e.ete ~pa,,-e and
(XS(UiG:), TS)
is
y E T(K) •
for
oTlu E XS(Uill:) 0 F, we get for all
o
2.7 THEOREM:
2
there 0
E XS(T(E)iG:) f
Xs(T(E) iCI:)
flu
II
T(E), there are
has
(by Corollary 2.6
@£ F
(X(Uoi C ), TO)
""
is the completion of
F
fl E X(UoiG:) @
th e It
there
0 < distE (K,C E (U nEB» , such that i f PB(T(X) - x) B B S (f(x) fey) ) < £. Let U = U n U n T(E) . Since T(E) 0 B
finite dimension I
X E K,
-
then
a»
T E E* @ E
is
x E K.
As in the proof of Theorem 2.1, whenever 0 > 0,
K C un EB
+
U be. a non - vo.(d
ha~ the appltox.(mat.(on pltOpe.ltty,
S• a .p . •
We show that
E*S
(E*,T ) is a complemented S
subspace
of
366
PAQUES
(~S(U7C),
E~
TS)7 hence
we have that, if
E is a quasi-complete space, then
For
a E U, the mapping
for
f E
~S(U7()'
subset of in
E • B
(~S(Ui
Da:
E.
S,
is a continuous projection onto
Then there is
Let
E has the S.a.p .. -1 defined by Da(f) = Il f(a),
.... E
D2 = D . To show continuity, let a a
clear that
this,
has the approximation property. From
S E BE
ES' Indeed, it is
K be a strict
such that
compact
K C BS and is compact
a + <5K C U n EB • From Cauchy
<5 > 0, be such that
in -
equalities, (Corollary 1.18) i t follows that
sup xEK for all
f E
I '6 1 f
~S(U7
(a)
I
(xl
Then
Now, we show that
< -
Da
1
If
sup xEK
/)
(a
I,
+ ox)
is continuous.
E has the S.a.p .•
Since
E*
S
has the
proximation property, then for every balanced convex compact I
of
E
S'
for every strict compact subset K of
e: > 0, there is 'P E I. Since
g E (Esl g E (ES)
I
I
® E*, such that
1:
'Pi®x
i=l Since, for each for of
E. Let
for each
i
,
'PiE (E
s)',
i =1, .•• ,m, 'Pi : ES .... (
'P E E*, where Bi E BE
subset
and for
IIg('P) - 'PIIK <
£,
for every
xiEE*, i=l, •.. ,m.
-
is continuous I'P·~ ('P) I < ciP.~ ('P),
Pi ('P) = II 'P II Li ' for some strict compact subset Li be such that
i = 1, •.. ,m . Since
Li C ESi
and are compact in
B E BE such that K C EB and is compact in m L = r(i~l Li U K). Hence l'Pi ('P) I < c II 'PilL' where c D E BE
so that
BCD
and
Bi C D, for
Let
extensions of fore, for
'Pi'
for each
so that
i = 1, .•. ,m,
'Pi E
l.pi ('P)
i
I .::.
((ED)~)
I.
=1, ••• 1I.pIlL'
is a constant.
i = l, ... ,m, L
balanced convex compact set in the Banach space Banach Theorem, there are I
EBi'
K is a strict compact subset of E, there
is
For
every
® E,
m
g
E,
ap -
1
ED'
By
m , 'Pi: (ED) for
is a
the
Hahn-
I
linear
.p E (ED)
.... (
I.
There-
THE APPROXIMATION PROPERTY FOR SPACES OF HOL.OMORPHIC MAPPINGS
1/1:
Let
E~ -+
(ED) ~,
be defined by
is linear and continuous. Hence
1/1
~,
compact subset of (ED) wri te
ifl,
ID =
(V = {v E ED'
we get
K
where PD (v)
a},
<
= ID
equicontinuous.
Hence, we
can
a
for some
> 0). Hence
I~ =
voo =V.
Since
ED '
C
I
m ~
g
, for T E E*. ED is a balanced convex
V is a closed absolutely O-neighborhood in ED'
sup I g('P) (t) - 'P(t) tEK
where
1/I(T) = TI
1jJ(J)
and then
367
i=l
.
'Pi 0 x.l.
~
'Pi ('P D ) Xi (t)
i=l
£,
if
Hence,
-
m sup tEK
suplg('PD)(t)-'PD(t)1 < tEK
-
'PD (t) I <
E:
,
'PD E ID .
Therefore m
sup I'P D ( ~ 'Pixi(t) -t) 1< tEK i=l
for
€,
'PD E V
o
,
and then
m _ 1/£ •
(~
'Pixi(t) -
t) E V,
i=l that is, sup PD(g(t) - t) tEK Since, g E E* 0 E the
a
is independent of
E:,
i t follows that E has
S. a. p ..
2.8 DEFINITION:
Let
E be a locally convex complex Hausdorff space.
is said to have the S-hoiomoJLph..Lc- applLo x...Lmat..Lort plLOpe.lLty (S.H.a.p.)
E
if given K
and
< £0 •
C
EB
K C E, a strict compact set, there is and is compact in
such that
PB(g(x) - x)
EB
and given
< £, for all
£
B E BE
such that
> 0, there is g E J(S(E;(J:) S E
x E K.
368
PAQUES
It is clear that if
E has the
S. a. p., then
E has the S.H.a.p ..
For the converse it is needed that E be a quasi-complete space, that is, we have the following theorem, which contains the previous theorem for an open subset
which is finitely S-Runge.
be a qua.6i-c.omple.te .6pac.e and le..t
U be an open
2.9 THEOREM:
Le.t
.6ub.6e.t 06
whic.h i.6 6ini.tely S-Runge. Then .the 6ollowing c.onc.f..U;ioYl4
E,
E
U of E,
aJr.e equivalen.t:
ha.6 .the
a)
E
b)
FoJr. eveJr.y loc.ally c.onvex .6pac.e in
S.H.a.p .. @
c)
(JeS(Ui(C), TS) ha.6 .the, appJtoxima.tion pJtopeJt.ty.
d)
E
ha.6 .the,
S.a.p ..
The assumption of
only in
c)
b)
E to be a quasi-complete space is
needed
d) •
+
+
i.6 T s-den.6e
F
JeS(UiF).
REMARK:
PROOF:
F, JeS(UiC!:)
c) is part (c) of Corollary 2.6, which is true for
open subset of
E.
c) ... d) is Theorem 2.7.
remains only to show that
a)
+
d) ... a)
is obvious.
b). This proof is analogous
proof of Theorem 2.1, substituting
g E JeS(Ei(C) @ E
for
any It
to the
T E E* @ E
(cf. Definition 2.8).
2.10 COROLLARY:
S.a.p •
Le.t E be a qua.6.i.-c.omple.te .6pac.e. Then
.i6 and only i6, 6oJr. eac.h
E
ha.6
.the
(Pb(nEi(C), TS) ha.6 .the.
ap-
S.a.p., it follows by Theorem 2.9, that
for
n
E
lN,
pJtox.i.ma.tion pJtopeJt.ty.
PROOF:
If
E has the
any open subset
U of E,
which is finitely S-Runge,
has the approximation property. Since for each
(Jes (U i(C),
T S)
n E lN, (Pb(nEiC),Ts)
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPH IC MAPPINGS
369
(Pb(~;«:)' TS)
is a complemented subspace of (;ICS (U;«:), T S), we have that has the approximation property.
Conversely, in particular, E * having the approximation property, E
has the S.a.p.
2.11 REMARK:
(as in the proof of Theorem 2.7).
By the previous Corollary, we have that
quasi-complete space and S-Runge, then if, for each
(;ICS(U;~),
n
E
IN,
U is an open subset of
E,
if
E
is
a
which is finitely
TS) has the approximation property, if and only ( Pb (nE ; C), TS) has the approximation property.
REFERENCES
[11
R. ARON, Tensor products of holomorphic functions, 35,
[21
Inda~Math.
(1973), 192 - 202.
R. ARON and M. SCHOTTENLOHER, Compact holomorphic mappings Banach spaces and the Approximation property, J. tional Analysis 21,
[31
on
Func-
(1976), 7 - 30.
P. ENFLO, A counterexample to the approximation property
in
Banach space, Acta Math. 130 (1973), 309 - 317. [41
A. GROTHENDIECK, P4oduit4 ten404iet4 topotogique4 et
e4pace4
nucieai4e4, Memoirs Amer. Math. Soc., 16 (1955). [51
c.
P. GUPTA, Malgrange theorem for nuclearly entire
functions
of bounded type on Banach space. Doctoral Dissertation, University of Rochester, 1966. Reproduced by Instituto de Matematica Pura e Aplicada, Rio de Janeiro, Brasil, Notas de Matematica, N9 37 (1968). [61
M. C. MATOS, Holomorphically borno1ogical spaces and
infinite
dimensional versions of Hartogs theorem, J. London Math. Soc.
(2) 17 (1978), to appear.
370
PAQUES
[7]
L. NACHBIN, Recent developments in infinite dimensional holomorphy, Bull. Amer. Math. Soc. 79 (1973), 625 - 640.
[8]
L. NACHBIN, A glimpse at infinite dimensional holomorphy,
In:
PJtocce.di.ng.6 on 1no.i.nLte. V.i.men.6.i.ona.t Ho.tomOJtphy, UY!.i.VeM.i.:ty 06 Kentucky 1973, (Edited by T. L. Hayden and T. J. Suffridge). Lecture Notes in Mathematics 364, SpringerVerlag Berlin-Heidelberg - New York 1974, pp. 69 - 79. [9]
L. NACHBIN, Topo.togy on Spae.e.6 06 HolomOlLph.i.e. Mappi.ng.6,Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 47, Springer -Verlag New York Inc. 1969.
[10 J
Ph. NOVERRAZ, P.6 e.udo - e.o nve.xLte., e.o nve.xLte. po .tynom.i.a..te et doma..i.ne.6 d'holomoJtphie en d.i.men.6ion in6.i.ni.e, Notas de Matematica 48, North-Holland, Amsterdam, 1973.
[11)
O. T. W. PAQUES, PJtoduto.6 ten.6oJt.i.a.i..6 de. 6un~oe.6 S.i..tva-ho.tomoJt6a.6 e. a pM PJt.i.edade de apJto x.i.ma~a.o, Doctoral Dissertation, Universidade Estadual de Campinas, Campinas, Brasil, 1977.
(12)
D. PISANELLI, Sur la LF-analitycite. In: Ana.tY.6e oone.t.i.one.t.te. et app.t.i.e.ation.6 (L. Nachbin, editor). Hermann, Paris, 1975, pp. 215 - 224.
(13)
J. B. PROLLA, AppJtox..i.mat.i.on 06 VectoJt Va.tued Function.6, Notas de Matematica 61, North-Holland, Amsterdam, 1977.
(14)
L. SCHWARTZ, Theorie des distributions a valeurs I, Ann. Inst. Fourier 7 (1957), 1 -141.
[15)
M. SCHOTTENLOHER, £-product and continuation of analytic mappings, In: Ana.ty.6e. Fonctionelle et App.ti.cat.i.on.6, (L. Nachbin, editor) Hermann, Paris, 1975, pp. 261 - 270.
[16]
J. S. SILVA, Conce.i.to.6 de 6un~a.o d.i.66eJtene..i.avel em e.6pa~o.6 .tode ca.tmente conveXO.6, Centro de Estudos Matemati:cos Lisboa, 1957.
vectorielles
Appro~mation
Theory and Functiona~ Analysis J.B. Prolla (ed.) ~North-Ho~land PubZishing Company. 1979
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
JOAO B. PROLLA Departamento de Matematica Universidade Estadua1
de Campinas
Campinas, SP, Brazil
1. INTRODUCTION Throughout this paper
X is a Hausdorff space such that Cb(X;l<}
(lK = IR or G:) separates the points of
X,
and
E is a non-zero locally
convex space. Our aim is to prove that certain function spaoes L have the approximation property as soon as
has the
E
C
C(X;E)
approximation
property. We show this for the class of all Nachbin spaces CVoo(X;E}. Such spaces include
C(X;E} with the compact-open topology;
with the strict topology; E =
~,
Co(X;E} with the uniform topology.
When
Bierstedt [1) , using the technique of E-products,had proved
that
CVoo(X;lK)
that
X is a completely regular
has the approximation property, under the hypothesis kIR -space, and that the family V of
weights is such that given a compact subset weight
Cb(X;E)
v E V
such that
vex)
~
1
for all
K C X, one can find
a
x E K.
The technique we use here was suggested by the paper
of
[5)
Gierz, who proved the analogue of Theorem 1 below for the case of
X
compact and bundles of Banach spaces. This technique of "locali za tion" of the approximation property was used by Bierstedt, in the case the partition by antisyrnrnetric sets (Bierstedt [2 ) ), but idea of representing the space of operators of
the
of main
L as another Nachbin
space of cross sections is due to Gierz. However our presentation is 371
PROllA
372
much simpler, in particular we do not use the concept of a C (X) -convex
locally
C (X) -module. In the Introduction to his paper, Gierz said
that his method could be applied to the vector fibrations in the sense of [8
J, and this led to our effort at simplifying
his
proof
and
adapting it to our context.
2. THE APPROXIMATION PROPERTY FOR NACHBIN SPACES vecto~
A
6~b~at~on
over a Hausdorff topological space
pair (X, (F x) x E X), where each lK
(where
F x is a vector space over
field
Fx' Le., f = (f(x»XEX.
on X is a function v on
we.~ght
norm over
the
is a
1< = lR or 11:). A c~o~~ -~ ect~on is then any element f of the
Cartesian product of the spaces A
X
x E X. A Nachbin
F x for each
L of cross-sections
f
X such that ~paee
LVco
vex) is a semi-
is a vector space
such that the mapping
X
EX" v(x)[ f(x»)
is upper semi continuous and null at infinity on X for each weight v d~~ec.ted ~et
belonging to a v
l
(i
' v
2
E V, there is some
= 1,2)
for all
x
E
V of weights (directed means that, given v E V
A
and
X); the space
>
~
AV(X)
with
the
0 such that vi(x)
L is then equipped
topology defined by the directed set of seminorms
f
..
II f IIv
and it is denoted by
sup {v(x)[ f(x»); x E X} ,
LV",
Since only the subspace we may assume that
L(x)
= Fx
L(x) = {f (x) ; f E L} C Fx is relevant, for each
The cartesian product of the spaces C(X;lK)-module, where
C(XilK)
x E X. Fx has the structure of a
denotes the ring of
all
continuous
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
lK-valued functions on ¢ E C (X; lK)
X,
if we define the product
and each cross-section
(¢ f) (x)
for all
x E
x.
If
WC L
373
f
¢f
for
each
by
¢(x) f(x)
is a vector subspace and
B C C(X;lK) is a
subalgebra, we say that W is a B-module, if BW={¢f;¢EB,f E w}cW. We recall that a locally convex space
E has the applLox.imat.ion
plLopelLty if the identity map e on E can be approximated, on every totally bounded set in
E,
by continuous linear maps of fi-
nite rank. This is equivalent to say that the space
E.
E'
$
E is dense in
ICE) with the topology of uniform convergence on
bounded sets of on
uniformly
E.
Let
p.
If, for each
rna tion property, then
P E cs (E)
I
let
Ep denote the space· E semi-
p E cs (E), the space
Ep has the approxi-
E has the approximation property.
Suppo-6e that, 601L ea.ch
THEOREM 1:
totally
cs (E) be the set of all continuous seminorms
For each semi norm
normed by
£c(E),
x E x, the -6pac.e
the topology de6.ined by the 6am.ily 06 -6em.inolLn-6 the apPlLox.imat.ion plLopelLty. Let
Fx equ.ippedw.ith
{v(x); v E v}
ha-6
B C Cb(X;lK) be a -6el6-adjo.int
and
-6epalLat.ing -6ubalgeblLa. Then any Nachb.in -6pace
wh.ich
LVoo
B-module ha-6 the applLox.imat.ion plLopelLty. The idea of the proof is to represent the space W = LV"" being
I
as a Nachbin space of cross-sect·ions over
X,
£ (W),
where
each
fiber
£(W;F x )' and then apply the solution of the Bernstein-Nachbin
approximation problem in the separating and self-adjoint bounded case. Before proving theorem 1 let us state some corollaries.
COROLLARY 1: Fx
Let X be a Hau-6dolL66 -6pace., and 601L each
be. a nOlLmed -6pace w.ith the applLox.imat.ion plLopelLty.
Cb(X;lK)
be
a -6el6-adjo.int and -6epalLat.ing -6ubalgeblLa.
x E X
Let
let B C
374
PROLLA
Let: L be a vect:oJt .6pace (Xi (F
x) x
CJtO.6.6 -.6 ect:ion.6
peltt:ainirlg
to
.6uch t:hat:
E X)
(1)
06
60Jt eveJty
f E L, the map
x .... IIf(x)1I i.6 upPeJt .6emicontinuoit6
and null at in6initYj i~
a B-modulej
(2)
L
(3)
L(x) = Fx
Then
noJt each
x E X.
L equipped with noJtm
f
1/
1/
= sup {I/ f (x) II i
X
ha.6 the
E x}
appJtoximation pJtopeJtty.
PROOF:
Consider the weight v on
for each 1/
f
1/
x E X.
= sup { 1/ f (x)
REMARK:
Then
LVoo
is
X defined by just
L
vex)
equipped
= norm with
the
Fx ' norm
II i x EX}.
From Corollary 1 it follows that all "continuous sums",
the sense of Godement [6] or [7],
in
of Banach spaces wi th the approxi-
mation property have the approximation property, if the X
of
is compact and if such a "continuous sum" is a
"base space"
C (Xi lK) - module. b
In particular, all "continuous sums" of Hilbert spaces and of C·-algebras, in the sense of Dixmier and Douady [3] tion property, if
have the approxima-
X is compact. Indeed, a "continuous sum"
sense of [3]
is a
COROLLARY 2:
Let X be a Hau.6dM66 .6pace .6uch that
in the
C(XilK)-module.
Cb(XilK) i.6 .6epa.-
Jtat:ing; let V be a diJtect:ed .6et 06 Jteal-valued, non-negative, uppeJt .6emicontinuou..6 6u.nction.6 on
Xi
and let E be a locally convex .6pace
wit:h the appJtoxima.tion pJtopeJtty. Then
CVoo(XiE) ha.6 the appJtoximation
pJto peltty. PROOF:
By definition, CVoo(XIE)
finity, for all
=
{f E C(X;E)
vanishes
at
in-
v E V}, equipped with the topology defined
by
the
i
v f
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
375
family of seminorms
sup {v(x) p(f(x»; x
II f II v ,p where
v E V Let
and
Lv denote
Lv(X)
=
0
by the seminorms
CVoo(XiE) equipped with the topology defined by
or
v E V
Lv(X) = E
is kept fixed. Then, for each x E X, equipped with the topology defined
{V(X)Pi p E cs(E)}. Hence in both cases, Lv(x) has
the approximation property. It remains to notice Cb(Xi~)-modules.
spaces are
X}
p E cs(E).
the above semi norms when either
E
property. Since
v E V
Therefore
Lv
has
was arbitrary, CV",(X;E)
that
all
the has
Nachbin
approximation the approxima-
tion property.
Let X and E be a.6 in COlLollaJI.y 2. Then
COROLLARY 3: (a)
C(XiE)
with the compact-open topology ha.6 .the applLoxima-
tion plLopelLty. Co(XiE) with the uni60lLm topology ha.6
(b)
the
applLoximation
plLO pelLty .
REMARK:
C(Xi~)
In (a) above, it is sufficient to assume that
is
separating.
COROLLARY 4:
(Fontenot [4 I)
.6pace, and let E
plLopeltty. Then
Let X be a locally compact
be a locally convex .6pace wi.th
Cb(XiE) with the .6tltict topology
the
applLoximation
B ha.6 the appltoxi-
ma.tion pltopelLty.
PROOF:
Apply Corollary 2, with
V
{v E C (X i lR);
o
Hau.6d0lL66
v > o}.
-
376
PROLLA
have the
PROOF:
app~aximatian p~ape~ty.
In Corollary 2, take
E
]I<.
3. PROOF OF THEOREM 1 Let
W = LV=
Let
Vo E V
and
x E X, where
for all
PROOF:
E > 0
be given.
T E leW) consider the map
For each
for
A C W be a totally bounded set.
and let
EX:W'" Fx
is the evaluation map, Le., Ex(f) =f(x),
fEW.
Just notice that
v (x) [ f (x)
For each and for each
x
1 -< /I
f
II v ,
v E V
consider the weight
sup {v(x)[ (U(x»
v E V.
v
on
T ==
(EX
0
T) x E
X defined by
(fl]; f E A}
U(x) E £(W;Fxl. Then
V(X)[ExOT]
for any
for any
T E £ (W), consider the cross-section
v(x)[U(xl]
for every
EX E £(W;F )' since
T E £(W).
sup {v(x)[ (T f)(x)]; f E A}
xi
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
x ~ V(X)[T(X)]
The map
STEP 2:
a~ .i.~6.i.~.i.t!J o~
X,
PROOF:
E
Let
Xo
6aJt each
377
i~ uppe~ ,~emico~~i~uou~ a~d
va~hu
T E £ (W) •
X and assume
Vex )[T(X ») < h. o 0
Choose
hOI
and
h'
such that
(1)
Let
6 = 2 (h" -
there exist
f
l
,f , ... ,f E A rn 2
i E {1,2, ... ,rn}
(2)
6 > O. Since
such that, given
T (A) is totally bounded, f E A,
there
is
such that
liT f - T fillv < 6 /4.
Since
x ->- v (x) [ (T f i) (x)
V , V "",V l 2 rn
for all
X. Let
U
I
= l,2, ••• ,rn).
= V l n V2
x E U
is upper sernicontinuous, there are such that
neighborhoods of
x E Vi (i
Let in
h'). Then
il
•..
and let
n Vrn . Then
U
f E A. Choose
is a neighborhood of
i E {1,2, .•• ,rn} such that
(2) is true. Then
vex) [ (T f) (x)] ::. vex) [ (T f) (x) - (T f ) (x) I + vex) [ (T f i ) (x) I i < liT f < 6/2
h"
T fillv + v(xo )[ (T f i ) (xo )] + 6/4
+ v(xo ) [ (T
f
i
) (x ) 1 o
- h' + v (xo) [ (T f i) (x o )
Xo
1•
PROllA
378
On the other hand, by (1), we have
v (x ) [ (T f.)
o
Hence
~
J -<
v(x0 ) [
v(x)[ (T f) (x)] < hOI
Therefore
v(X)[T(X)]
(T (x o ) J <
for all
< hOI < h,
h' •
f E A, and
for all
x E U.
xft U. x ~ v(x)(T(x)]
Let us now prove that the mapping
vanishes
at
infinity. Let
0 > 0
be given and define
KcS
{xE Xi v(x)[T{x}]
Ko
Since sup {II T f II v i Since
f
¢,if
sup{IITfll
> 6}.
;
v
may
assume
A} > o.
E
T{A) is totally bounded, there are
that, given
we
fEA}
f E A, there is
i E {l, ... ,m}
fl, ... ,fm E A
such
such that
(4)
m Let
u {tE Xi v(t)[(T f.)(t)] > 0/2}. i=l ~-
K =
Then K is compact, since each of the functions vanishes at infinity. Let now
v(x)[ (T f) (x)]
(5)
Choose
(6)
fi E A
K6
Ko >
and choose
f E A
such
(x)]
that
30 4
satisfying (4). Then
vex) [ (T f) (x)
Therefore Since
x E
x~V(X)[(Tfi)
1
< vex) [ (T f
0/2 < v(x)[ (T f ) (x)1 i
i
) (x)]
and so
is closed, this ends the proof.
+ 0/4.
x E K, i.e.,
Ko C K.
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
trW) over
= {T;
The above two steps show that the image
£
under the map T
£V 00
X,
is a l;jachbin space
pertaining to the vector fibration (Xi
we take as family
Fo~
STEP 3:
T
-+
eve~y
V of weights the family
V
=
379
T E £(W)}
of
of cross sections if
(£(WiFx»XEX)' {Vi
V
E V}
T E £(W), v E V,
sup liT fllv < sup V(x)[T(x»). fEA - xE X PROOF:
Let
f E A. Then
sup v(x) [ (T f) (x) I xEX sup v(x) [ (EX xEX
0
T) (f) I
"Til V
sup v(x)[T(x») xEX Let now
F
{T; TEW' 0W}.
Our aim is to prove that we can find
s up
fEA
II T f - f II Vo <
T E W' 0 W
E •
Hence, by Step 3, it is enough to prove that
i
where
=
such that
IIt-ill-
Vo
<E,
(Ex) X EX'
By the bounded case of the Bernstein-Nachbin approx:ilnation problem (Theorem 11, [8),
STEP 4:
F
STEP 5:
F04
i~
pg. 314) i t is enough to prove that
a B-module.
each
x E X, F(x) i~ den~e in
the topology de6ined by the
~emino~m~
£(W;F x )'
{v(x); v E V}.
equipped
wah
PROllA
~
PROOF:
To prove that
r
is a B-module, it is enough to prove that
cp
( 7)
for all
TE
T E W' ® Wand for all
F, i.e., for all
Mcp : W -+ W is defined by
T
acp (f)
cp f, for all
cp E Bj
and
fEW.
Now to prove (7), one has to prove that
(CP T)
(Mcp 0 T)A(X)
( 8)
for all
(x)
x E X. However,
(Hcp 0 T)A(X)
(¢
T)
(x)
¢(X)T(x)
¢(X)(E
And, for all
fEW
X
0 T).
one has
= ¢(x)
(T f) (x)
¢(x) (EX 0 T) (f).
This ends the proof of step 4. To prove step 5, we first notice that, since each with the topology defined by property, then
W' @ Fx
{v(x); v E V}
is dense in
with the topology of the seminorms
has
the
Fx equipped approximation
£C{W;F x )' a fortiori in £ (W;Fx) {v (x); v E V}.
THE APPROXIMATION PROPERlY FOR NACHBIN SPACES
Hence, all we have to prove is that for each
381
F(x) contains
W' 0 F x '
x E X.
Let then
T E W' 0 Fx
be a continuous linear operator of
fi-
nite rank, say
T
where
$i E W'
and
vi E Fx. Since
W{x)
F x ,choose
fi E W
such
that
for
1,2, ... ,n.
i
Define Then
n t $i 0 f.~ i=1
U
U E W' 0 Wi
so
UE
F. Now n
U(x)
Ex
0
(t
i=1
$i 0 f ) i
and therefore U(x) (f)
n t
T(f)
$1 (f) fi (x)
i=l for all
fEW.
REFERENCES
[1 I
K-D. BIERSTEDT, Gewichtete
R1iume
stetiger
vektorwertiger
Funktionen und das injektive Tensorprodukt. I, J. reine angew. Math. 259 (1973), 186 - 210. [2 I
K-D. BIERSTEDT, The approximation property for weighted
func-
tion spaces, Bonner Math. Schriften81 (1975), 3 - 25.
PROLLA
382
[3
1
J. DIXMIER and A. DOUADY, Champs continus d f espaces
et de C*-algebres, Bull. Soc. Math. France
hilbertiens 91 (1963),
227 - 284. [4
1
R. A. FONTENOT, Strict topologies for vector-valued functions, Can. J. Math. 26
[5
1
(1974), 841 - 853.
G. GIERZ, Representation of spaces of compact
operators
applications to the approximation property,
and
Preprint
Nr. 335, Techn. Hochschule Darmstadt, Feb. 1977. [6
1
R. GODEMENT, Theorie generale des sommes continues d I espaces de Banach, C.R. Acad. Sci. Paris 228 (1949) • 1321 - 1323.
[7
1
R. GODEMENT, Sur la theorie des representations unitaires, Ann. of Math. 53 (1951) , 68 -124.
[8
1
L. NACHBIN, S. MACHADO and J. B. PROLLA, Weighted approximation, vector fib rations and algebras of operators, J. pures et appl. 50 (1971), 299 - 323.
Math.
Approximation Theory and FUnational Analysis J.B. Prolla (ed.) ©North-Holland PubZishiYl{f Company, 1979
ON CARDINAL SPLINE SMOOTHING
I. J. SCHOENBERG Mathematics Research Center University
of Wisconsin
Madison, Wisconsin 53706, USA.
INTRODUCTION The present paper describes the methods, with some changes be mentioned below, whereby I solved the numerical problem
to
assigned
to me at the Ballistics Research Laboratories in Aberdeen, Maryland, during the second World War (see [4 1 ).
The problem was to smooth very
extended equidistant tables of drag functions
(or drag coefficients)
by approximating them by very smooth functions that were easily computable with their first and second derivatives. The first question to be answered was this: When maya "moving
aveJtage" opeJtat..i.on legLUmateR.y be c.alled a .6mooth..[ng 6oJtmula?
An
answer is given in Section 1 of Part I. That it is a reasonable
one
is shown in Section 2 of Part I. These ideas were later greatly generalized by Fritz John in his important work on parabolic differential equations (see [31).
The connection is briefly
mentioned
in
Section 2 of Part I. The second essential ingredient of our results of Part
II
is
the process of caJtd..[nal .6pl..[ne inteJtpolation (see [91). The required results are described in Section 3 of Part I. The changes from my war-time approach are developed completely in Part II. This is the new contribution of the paper, and it arises 383
SCHOENBERG
384
if we adapt the idea of E. T. Whittaker (see [101 and [7 ])
to
the
problem of smoothing a bi-infinite sequence of equidistant data. The result is the
Qa~d~nat ~pline ~moothing
p~oQe~~.
Let me thank Professor Ilio Galligani for asking me
to
visit
Rome in May of 1977, where I gave a short lO-hours course on Cardinal Spline Interpolation at his Istituto per le applicazioni del Calcolo "Mauro Picone".
PART I.
PRELIMINARIES
1. WHAT IS A SMOOTHING FORMULA?
The several smoothing (or gradua -
tion) methods discussed in [10) are all of the "moving average" type. By this we mean that we are given a sequence of real and
symmetric
weights 00
(l)
1,
which we apply to the data (x ) to produce their smoothed version (Yn)' n
by the formula
(2)
1:
v=-oo
an_\!x\!, (n
= 0,
±
1, ... ).
In other words, we convolute the sequences (an) and {xn }, an operation symbolically described by
(3)
By deriving a formula (2) according to a definite idea, such as a least squares formula, or Whittaker's formulae (see [ 10,
Chapter
XI J ), we may reasonably expect that it will transform a given quence
se-
n ) into a smoother sequence (Yn)' This point becomes doubtful, however, in case the formula (2) was otherwise obtained, e.g., (X
ON CARDINAL SPLINE SMOOTHING
385
by some approximating procedure that is not strictly
interpolatory.
An example of such a procedure will be our cardinal spline smoothing crnoo~hing
of Part II. A criterion for (2) to be regarded as a
60nmula
was proposed by the author in [4, pp. 50 - 54) and proceeds as follONS. To begin with, we assume the Laurant series
(4)
(r <
F (z)
Iz I
< r
-1
),
to converge in some ring containing the unit circle I z z = e
iu
=1. Setting
, we call the even function
a
¢(u)
(5)
the
chaJtaC~(ULic~ic
on
IzI
6unc~ion
o
+ 2a l cos u + 2a cos 2u + ... 2
of the formula (2). The regularity of F(z)
¢ (u)
= 1, implies that
is regular
in
a
certain
strip
11m u! < a, whence the valid Taylor expansion
(6)
¢(u)
E b
o
n
un
(I u I
< a).
This expansion allows us to express easily an important property (2): Its
JtepJtoduc~ive
poweJt, or degJtee 06
exac~ne¢¢.
We say that (2) has the degree of exactness is a natural number, provided that (2) reproduces Yn
xn
values
for all P (n) of
of
2m - 1, exactly,
where
m
i.
e.,
n, a sequence (x n ), if this sequence represents the a
polynomial of degree
2m - I,
but does
not have
this property for any higher degree. That the degree of exactness is always odd, follows from the symmetry condition (1). In terms of (6) we have the following easily established proposition:
The 60Jtrnuia i6
~he
(2)
hac
expan¢ion (6) i¢ 06
~he ~he
degJtee 06 60Jtrn
exac~ne¢¢
2m - I, i6 and only
386
SCHOENBERG
1 - Au
¢(u)
( 7)
2m
+ •••
(A
I
0).
An example. Let (1) be the sequence a
- 1/16,
o
o
an
if
n > 2,
when the convolution (2) assumes the form
(8)
From (5) we obtain
(9 )
1 _
(5 + 4 cox u - cos 2u) /8
¢(u)
m = 2, A
and (7) is seen to hold with mula (8) has the
deg~ee
06
exactne~~
= 1/16.
1 16
It
u4
+ ...
follows that the for-
3.
May (8) be regarded as a smoothing formula? The general
cri-
terion is described by the following.
DEFINITION:
We ~ay
that (2)
cha~acte~-i~t-ic
6unct-ion
(10)
- 1
<: ¢(u)
-i~
~mootHng
6o~mu.ia.,
pltov-ided that
o
1
< u < 21T •
This condition evidently implies that the coe66icient
pea.lting in (7),
i~
po~itive.
a~
~ati~6ie~
¢(u)
<
a.
A. ap-
The characteristic function (9) is eas-
ily seen to satisfy (10), and so (8) is a smoothing formula according to our definition of the term. The criterion (10) raises the following kind of equation:
Do
the numerous smoothing formulae as given in [10 I satisfy our criterion
387
ON CARDINAL SPLINE SMOOTHING
(10)? See Greville's paper [1) where some of these questions are answered affirmatively. In [4, pp. 50 - 54) good arguments in support of (10) are presented. Even more convincing reasons
the criterion
were given in
[5,
Part I). In the next section we reproduce these arguments as pres ented in [6, pp. 200- 204).
2. THE BEHAVIOR OF THE ITERATES OF A MOVING AVERAGE FORMULA:
THE
PROBLEM OF DE FOREST. A conclusive argument in support of the necessi ty of our condition (1.10) is furnished by the solution of the following problem first stated and attacked
by
Erasmus
L. De
(1834-1888). If we subject the given sequence (xv> n times
Forest
in suc-
cession to the same tranforma tion (1.2), we obtain a linear trans formation
(n)
(1)
yv
E
k=-oo
which is the n-fold iterate of (1.2). Whati.6 thea.6!Jmptotie
06 the
coe66icient.6
06
(1) a.6
~
n
oo?
This question was
by De Forest and by G. B. Dantzig (for references see[S) when all coefficients of that
m=l
behaviolt answered
for the case
(1.2) are non - negative, hence necessarily
in (1.7). A general solution is as follows.
Let (1.2) be such that (1.17), (1.10), and (1.7), are satisfied, hence that
(2)
A
>
O. Let
1
27i"
oo
f
_00
e
-v
which is the normal frequency function
(3)
1 -- e
2 Iii
x2
4"
2m
cos vx dv,
388
if
SCHOENBERG
m=l, otherwise (m=2,3, •.• ) Gm(x) is an entire function
having
infinitely many zeros, all real.
1
1
1
(An) -2m G (v (An) -2m) + a Cn -2m) a.6 n .... m
(4)
where the ".U..t.tle
co ,
0".6 ymbol hold.6 1.1.1116 oltmly 6 alt all .i.l1teg elt.6
v.
For a proof see [5, Part I 1, where it is also shown by examples that (4) no longer holds if the equality sign is allowed in and that the coefficient n = 2k
a~n) diverges exponentially to +
tends to infinity through even values, if
(1.10) are reversed anywhere in the interval
the
(1.10), co,
as
inequalities
0 < u < 2rr •
The following discussion, while not directly related
to
our
subject of smoothing, will show the connection of the asymptotic relation (4) with the wider field of parabolic differential equations. Observe that (2) implies that 1 ( 5)
U(x,t)
t
-2m
1 G (x t m
2m)
1""
1
2iT
-co
e-
tv 2m + ixv
dv, (t > 0).
The function under the integral sign is immediately seen to satisfy for all v, the differential equation
(6)
(x E
R, t > 0),
which reduces to the familiar heat equation if also -plane
m=l.It follows that
U(x,t), defined by (5), is a solution of (6) in the upper half t > O. On the other hand, applying to (2) Fourier's inversion
formula and setting
v = 0, we find that
ON CARDINAL SPLINE SMOOTHING
These remarks imply the following:
Ix I
say, as
->
u(x,t)
. Lt. a -6oi.ut..i.on
t-
f(x).w C'-ontinuoUh a.nd
o(
I xl- 2 ),
the.n
00,
1
(8)
16
389
2m
ex.
1 G
2m } f(v)dv, m {(x - v)t-
06 the. d..i.66eJLe.nUai.
equat..i.on
(6)
(t > 0),
-6at..i.-6oy..i.ng the boundMy
C'-ond..i.t..i.on
lim
(9)
u(x,t)
f(x) .
t ... O+
This particular solution
u(x,t) may now also be
approximated
by the following numerical procedure: Draw in the (x,t) -plane
the
rectangular lattice of points
(vL'lx, n L'lt)
0, ± 1 , ...
(v
Define on it a lattice function
u
n
u
= 0,1,2,
v,n
..• ).
by starting with
f (v L'lx) ,
v,o
and computing the values along each horizontal line from those on the line below it, by means of the transformation (1.2). This amounts to iterating
(1.
evidently
2), and after n steps we obtain
(10)
For any given x and following: We.
6..i..IL~t
t
>
0, (10) will go over into (8) if we
C'-onne.C'-t the. me.-6h--6..i.ze.-6
L'lx a.nd
do the
L'lt by the. .lLe1.a...t<..on
390
SCHOENBERG
L'lt
(11)
A(L'lx) 2m.
16 the .LntegefL6 v and n aJr.e .ouch that
and
vL'lx .... x,
n L'I t .... t
a.o
L'lx .... O.
then
u v,n
->-
u(x,t).
This follows readily from (10) and (8), in view of the relation (8),
(4):
(10) differs
asymptotic
from a Cauchy-Riemann sum for too integral
by a quantity that tends to zero due to the uniformity in
v
of
the error term of (4). It is interesting to note that it does not matter which
for-
mula (1.2) we use in this construction, as long as it is of the degree of exactness
2m -1, Le., it satisfies (1. 7), and above all that it
satisfies the stabi Ii ty condition (1.10), the term "stabili ty"rreaning here stability on iteration. For the general theory of F. John,
of
which the equation (6) is a special example, see [3 I . In this section we dealt exclusively with formulae (1.2) which satisfy the symmetry relation. In [2 1 T. N. E. Greville dealt
with
the more difficult case of unsymmetric formulae.
3. CARDINAL SPLINE INTERPOLATION (see [9, Lectures 1 - 41). The prob-
lem of caJr.d.Lnal .LnteJr.polat.Lon is to find solutions
f(x)
of the in-
terpolation problem
(1)
f(v)
for all integers
v,
where (Yv) are the data. A formal solution is furnished by the series
ON CAROINAL SPLINE SMOOTHING
391
f (x)
( 2)
investigated in 1908 by de la Vallee Poussin, also later
by
E. T.
Whittaker, who called it the candinal ¢enie4. The difficulty with (2) sin'lTx
is the slow decay of the function
('IT x)
as
x
~
00.
solution of (1) is the pieeewi4e linean intenpolant
A much simpler 8 (x) 1
given by
( 3)
where
M {X) is the roof function defined by 2
M {xl =x+l 2
in
l-x
[-I,D], M (X) 2
in [0,11 andM:2{x) =0 i f lxi>l.
The purpose of candinal ¢pline intel!.polation is to bridge
the
gap
between the piecewise linear
8 (x) defined by (3), and the cardinal 1 series (2). It aims at retaining some of the sturdiness andruillplicity of (3), at the same time capturing some of the smoothness andsophistication of (2).
Let m be a. natul!.ai l1umbel!., a.nd let
(4)
{8
be the clau 06 cMdinai ¢plin.e4
{xl}
sex) 06 degnee
2m -1
de6ine.d
by
the two condition4:
(5)
(6)
The Jc.e..6.tJc.-<-ction
06
sex)
.to evelt?f unit intel!.val
(v ,v +1),
wheJte v i4 an integ en, ,U, a polynomial 06 degJte.e < 2m - 1.
SCHOENBERG
392
For
m~l
Sl
we find
to be identical with the class (3)
S2m-l
continuous piecewise linear functions. Observe that the class of polynomials of degrees not exceeding The role of the roof-function the so-called
contains
2m - 1.
M (X), of (3), is taken over by
2 M2m (x):
Qent~al B-~pfine
of
W~iting
= max
x+
(x,D),
it
may be de6ined by
1 6 2 m 2m-l (2m-I)! x+
( 7)
Clearly port
~
1
(2m -
1)
!
M2m (X) E S2m-l; we also find that
I:
(-1)
v
2m ~l (v) (x+m- v)+ .
v=O
M2m (X) > 0
in its
sup-
- m < x < m. The B-spline should be familiar in view of the. fun-
damental identity
rm
M
2m
The representation
a unique
(x)
dx ,
if we choose
which also shows that
admit~
(x) f (2m)
(3)
also generalizes, and
f (x) = x ev~y
S (x)
2m E
S2m-l
~ep~e~entation
s (x)
( 8)
whe~e
the c
v
aILe
Qon~tant~.
This is the so-called
~tandaILd
ILepILe~en
tation. The converse is clear: Every series (8) furnishes an element of
S2m-l
ments of
We now try to solve the interpolation problem (1) byele-
S2m-l'
In this direction there are two different kinds of
results. A. The data (Yv) sequence (Yv) is of
a~e
powe~
06 powelL gILowth (See [8 gILowth, and write
D.
We say thatthe
ON CARDINAL SPLINE SMOOTHING
(9)
393
(y) E PG,
provided that
O(lvI Y)
(10)
as
v
4
f(x)
E
PG,
±
for some
00,
Y
>
O.
Similarly, we write
(ll)
provided that
f (x) = 0 (I xl
Yl
) as
X
±
4
for some
00,
Below we exclude the trivial case when
potation
(13)
~equenQe
(Yv) i~
06
O.
powe~ g~owth,
(Yv)'
then the
inte~
p~obtem
(12)
ha~
16 the
~
m=l, since our prob-
lem is solved by (3) without any restriction on the
THEOREM 1:
Yl
6o~
S(v)
a unique
~otution
sex)
S (x)
E
~uQh
att
v,
that
S 2m-l n PG •
The assumption (9) of Theorem 1 is a rough one; it admits, e.g., all bounded sequences (Yv)' with
Y
o in (10). The second assump-
tion to which we now pass, is much more selective, and
takes
account the finer structure of the sequence; in fact it admits a narrow subclass of the sequences of
PG.
into only
As usual, with stronger as-
sumptions, stronger conclusions are possible: The interpolant will exhibit an important extremum property.
Sex)
SCHOENBERG
394
B. The
when
Qa~e
1:
[llmy [2 < v
00
(See [9, Lecture
6] ).
We
introduce the classes of sequences and functions as follows:
(14)
(15)
L~={f(x); f, ... ,f(m-l) are absolutely continUOUS,f(m~X)EL2(JR)}.
Of course
and
We may also describe
t2
ments of
L~
by
n
t2
are the familiar
and
L , respectively. 2
as the class of sequences obtained from elesuccessive summation. Similarly the elements
are obtained from those of
L2
by
n
of
successive integrations.
16
THEOREM 2.
then the intenpotation pnobtem
S(v)
(17)
ha~
a unique
~otution ~uQh
sex) E S2m-l n L~
(18)
Thi~
f(x)
(19)
and
that
i~
~otution
Sex)
ha~
an anbitnany 6unQtion
the 6ottowing extnemum pnopenty: ~uQh
f(x) E Lm 2
that
16
ON CARDINAL SPLINE SMOOTHING
(20)
f(v)
395
v ,
then
J~
(21)
unie!.>!.>
1f (m) (x) 12 dx >
Cx>
60Jt aii Jteai
x.
f (x) '" S (x)
m
In words: If (y) E £2'
2 1S (m) (x) 1 dx,
then the spline interpolant
S(x) mini-
mizes the integral
(22)
C"
I (f)
1f (m) (x) 12 dx
among all sufficiently smooth interpolants of If
v, where P (x) E
and therefore
Sex)
= P(x)
1T
m _ ' then P(x) ES2m-l nL , m l 2
by the unicity of the solution in Theorem
I(S) = O. In the general case of (y) E i~
2. However, here therefore say that
we may
S(X) is among all interpolants of (Yv)' the
that "is most nearly" a polynomial of degree P(v), where
If
(Yv)'
p(x) E S2m-l n PG, and so
P(x) E S(x)
=
1T
- , 2m l
< m - 1.
but
p(x) is the unique solution of The-
orem 1. Theorem 2 does not apply here because (Yv) ~ i~ . There no interpolant
teJtPoiat~on
f(x) such that
I(f) <
pJtobiem!.>? To answer this question let us for the
(Yv) E i
l
,
hence a fortiori
This insures the continuity of the periodic function
(24)
T(u)
is
00
assume that
(23)
one
moment
396
SCHOENBERG
which we call the
genenat~ng
6unct~on
of the sequence (Yv)' Here and
below we denote the relationship between a sequence and its generating function symbolically by writing
(25)
We also require the generating function of the sequence (M
(V», which
2m
is
(26)
This is a cosine polynomial of order
m - 1, because
I
(7), and we find that
x
I
> m. It is readily evaluated by
¢2(u) =1, ¢4(u) =
~(2+COS
u), ¢6(u)
=6~(33+26
M (x) 2m
o
if
cosu+cos2u), . . . .
It also has the property that
(27)
1
for all
u.
It follows that its reciprocal has an expansion
(28)
with real coefficients
1
w ' w_v = wv ' that decay exponentially. v
Let
us find the standard representation
(29)
s (x)
of the solution of the interpolation problem (17), which requires that
397
ON CARDINAL SPLINE SMOOTHING
( 30)
v.
for all
Furthermore let
( 31)
l: ij u j cj e ,
C(u)
or
->- C(u),
be the as yet unknown generating function of the (c.). Since the conJ
volution of two sequences has a generating function that is the product of the generating functions of the two sequences, we see by (24), (26), and (31), that the relations (30) are equivalent to the relation
T(u),
(32)
or
T (u)
C(u)
¢2m(u)
Now (28) shows that (c ) v
( 33)
v.
for all
l: YJ' W • j v-J
The6e ane the eoe66ieient6 06 the intenpotating 6ptine (29).
EXAMPLES:
1.
16
m =1, then ¢2(u) =1, hence
Section
51)
/3.
v
2. If we choose
(34 )
16
Wv =O(v
'I
0), and
m =2, we find (See [9, Lecture 4,
that
W
shows that
v.
for all
we obtain
Wo =1,
W
AI v
Yv
I,
= 0v
-2 +
where
,where
°
L
2m- l (x)
l: W M2 (x v v m
-.26795.
o (v 'I
0
Therefore the spline
v
/3
v)
O),then (33)
SCHOENBERG
398
is the solution of the interpolation problem
L
(35)
_ (v) 2m l
6u~dame~tat
The function (34) is the ~otut~o~
S(x) of the
(36)
ge~e~at
v.
for all
p~obtem
S(X) \)=-00
of the process, and the
6u~et~o~
(17)
~~
g~ve~
Yv L 2m- l (x -
by
v).
This cardinal interpolation formula bridges the gap between the linear interpolant (3) and the cardinal series (2). In fact, notice that if m =1 then (36) reduces to (3), while we have
( 37)
lim S2m-l (x)
m-+ oo
Also every derivative
sin
1TX
1TX
S (k) (x) converges to the corresponding deriva2m-l
tive of the right side of (37), uniformly for all real
x.
In our discussion we have assumed that (23) holds.
the
~atat~o~~
(33),
(29),
PART II.
a~d
a~e
vat~d
6o~
both
Theo~em~
1
a~d
THE CARDINAL SMOOTHING SPLINE
1. STATEMENT OF THE PROBLEM:
(1)
(36)
However,
We assume now that
1: I y
v
I <
co
and restrict ourselves to real-valued data and functions.
We
also
recall the definitions (3.14) and (3.15) of Part I, of the classes t and
In view of the inclusion relations
m 2
ON CARDINAL SPLINE SMOOTHING
399
(2)
(See [9, p. 104]), we observe that assumptions of Theorem 2 for all
We aJte given
THE PROBLEM:
implies that
(1)
(y)
satisfiesthe
m.
m and a -6moothing pafLametefL
<
€
O.
Among
aii 6unction-6
f(x)
( 3)
E L m,
hence
2
f(m) (x)
E
L2
-
y) 2
(IR),
we wi-6h to 6ind the -6oiution 06 the pJtobiem
J(f)
(4)
LEMMA 1:
(f (v)
In -60iving the minimum pJtobiem (4) we may JtuWc;tthe choice to the eiement-6 06
06 admi-6-6ibie 6unction-6
f(x)
(5)
S2m-l :I
PROOF:
minimum.
If
f(x)
is such that
apply Theorem 2 to the sequence
(6)
be such that
L~
J(f)
<
€
f (v)
f
for all
(s(m))2 dx
then
(f(v)), and let
s(x) E S2m-l n
s (v)
00,
L~
v. But then
+ E(f(v) - y)2,
(f(v)
-
y)
E
i
2
•
We
400
SCHOENBERG
and so
J(f) ,
in view of the extremum property of Theorem 2. Therefore, for any f(x), produces a value
Let
U~
f(x), the spline ~
6ind the
the~e6o~e
(7)
(8)
J(s)
~olution
8 (x)
minimum.
J(8)
Cx> (8 (m)
(9)
(x) ) 2 dx
whe~e
(10)
PROOF: From (7) we find that
r""
(5 (m) (x) ) 2 dx
of
sex) that interpolates
J(f).
Here we need another
(11)
sex) as expressed by (3.21)
ON CARDINAL SPLINE SMOOTHING
401
where (Y r ) is the even sequence defined by
I:
(12)
M(m) (x) M(m) (x - r)dx ,
where, to simplify notations we dropped the subscript
2m of M (x) . 2m
Integrations by parts show that
(_l)m-l
(13)
Observe that
M(2m-l) (x)
C'" M' (x)M(2m-l) (x -
r)dx .
is a step function assuming in consecutive
unit intervals the values
(14)
. .. , 0, 0, 1, -
( 2m; 1),
( 2~-1), ... , - 1, 0, 0, ...
This sequence has the generating function
(15)
except for a shift factor
e i uk which we disregard. Now (13) indicates
that (Y ) is the convolution of the sequence (14) with the sequence r V+ l
I
M' (x)dx
+
M (v
1)
-
M ( v)
->
-iu
(e
- 1) ¢ 2m (u) •
v
However, in (13) the sequence 2:
v
a b - • If v v r
(Y ) appears as a sum of the form r we pass from (a) to the reversed sequence (a_), we
obtain a genuine convolution
2:
v
a
b . -v v-r
Let us therefore
reverse
the first sequence (14). As we obtain the generating function of the reversed sequence by changing
u into
- u in its original generating
function, we find the generating function of (Y r ) to be (up to a shift factor e iuk ) the product
402
SCHOENBERG
(-1) m(l - e -iu) 2m r6
2m
(u)
Since (Y r ) is an even sequence, its generating function must be even, and therefore
establishing (10).
2. SOLUTION OF THE PROBLEM:
(I)
J(S)
= e:
l:
j,v
From (8),
(9), and (7) we find that
y. c.C +l: )-v ) v v
Let us minimize this function of the (c ). To obtain the normal equak tions, we differentiate J(S) obtaining
e: l: Y j _k c j + l: { l: c. M (v - j ) - y v} M (v - k) = 0 (k E j
v
j
If we sum within the double-sum only with respect to
(2)
where
2') •
]
v, we obtain
ON CARDINAL SPLINE SMOOTHING
403
( 3)
The normal equations thus become
(k E 'I),
or
(4)
(k E 'I).
However, by (3) and (1.10) we find
(5)
From
(M
2m
(v))
->-
¢2m (u) ,
and writing
(6)
(c)
->- C(u),
(y)
->-
T(u),
we find the normal equations (4) to be equivalent to the relation
T (u) ¢2m (u) ,
whence
(7)
This establishes
->-
C(u)
T(u) ¢2m(u) + E(2 sin ~)2m 2
SCHOENBE RG
404
THEOREM 3:
1~ te~mh
06 the
expa~hio~
1
(8)
¢2m (u) +
whe~e
Qoe66iQie~th
the
6iQie~th
(c.) ]
06 the
E
(2 sin ~ ) 2m
06 the
( ) WvEe
ivu I
WV(E)
holutio~
S(x)
(9)
v=-oo
mi~imum p~oblem,
L: c. M2 (x j ] m
j)
a~e
(10)
l:
j=_oo
Qa~di~al hmoothi~g
We call the solution (9) the
3.
A 6ew
p~ope~tieh
A. in (2.8)
06 the
hpli~e.
s(x) =S(X;E).
Qa~di~al hmoothi~g hpli~e
We have assumed above that
E
> O. However I
if we set
=0
it becomes
I
1
(1)
and a comparison with the expansion (3.28) of Part I, = W
v
B.
the
Qa~di~al hpli~e
S(X) 06
What i.6 the e66eQt 06 the
o~igi~al
sequence
v: Thih .6 ho Wh that
for all
i~te~polati~g
o~
E
(S(v))
data I
(Yv)?
S(x;O) Theo~em
= S(X)
shows ~eduQe.6
that
to the
2.
hmoothi~g .6pli~e
S(x)
= S(x;
E)
This we answer by detennining the "sm:x:>thed"
to compare it with (y ). By (2.9) and (2.10)we find v
(S (v))
ON CARDINAL SPLINE SMOOTHING
406
and therefore, by (2.7),
(S(v»
(2)
+
T(U)
C(u) ¢2m(u) 1
+E
(2 sin .J!) 2m 2
In terms of the expansion
1
( 3)
1 +
E
e
(2 sin .J!) 2m _---.,,....--,-..2:;-_ ¢2m (u)
a~i~e~
(2) shows that the sequence (S(V;E» ~moothing
n~om
ivu
the data (y ) by the v
60~mufa
(4)
S(V;E)
Observe that by (2.8) and (3) the coefficients by
a v (£l
= l: M2m (v j
j)
0V(E)
are expressed
W j (E) •
Is (4) a smoothing formula according to our definition of Part I, Section l? That it is one we see if we inspect its characteristic function
(5)
1
K(u;£l
1 +
E
(2 sin .J!) 2m _ _.,...-......,.....::;2,....-_ ¢2m(u)
for it is evident that
o
(6)
C.
The
< K(u;£l < K(O;£l
~moothing
powe~
1
06 the
for
60~mufa
o
< u < 211 •
(4)
inc~ea~e~
with
E. In [4, Definition 2, p. 53] we gave good reasons
infor
SCHOENBERG
406
the following definition: Of two different smoothing formulae having the characteristic functions
¢(u) and
¢(u), we say that the second
has greater smoothing power, provided that
i¢(u)i < i¢(u)i
(7)
However, if
for all
u,excludingequalityforall
u.
0 < £ < £, it is clear by (5) that
o
< K(U;E)
< K(u;£)
o
if
< u < 27f ,
and the criterion (7l is satisfied. D.
The degJtee 06 exactl1eM 06 the -6moothing 60Jtmula (4) .i.-6=2m-1.
This follows from (1. 7) of Part I, because (5l shows that we have the expansion in powers of
(8)
K(u;
E.
u
£l
1 - £u
2m
+ ...
If we drop our assumption (1.1), and assume only that (Yv'
-L-6 06 poweJt gJtowth, thel1 OlVL COI1-6.tJz.uction 06 the -6moothil1g -6pUl1e 8 (xl = 8 (x; E)
by the 60Jtmutae (2.8), (2.10), and (2.9), Jtema-Ln-6 appl-Lcabte.Of course, its earlier connection with the funtional holds. In fact we will find that sumably, it is still true that our
J(8) =
J(8), of (1.8), no longer 00
for all splines
8(x;£l minimizes
8. Pre-
J(8l, provided
that (Yv) satisfies the condition
of Theorem 2. However, this I was not able to establish. In any case I recommend the cardinal smoothing spline (8 (x; Ell, which represents the modification, found more than 30 years later,of may war-time approach to the problem of cardinal smoothing.
ON CAROINALSPLINE SMOOTHING
407
REFERENCES
[lJ
T. N. E. GREVILLE, On stability of linear smoothing
formulas,
SIAM J. Num. Analysis, 3(1966), pp. 157-170. [2J
T. N. E. GREVILLE, On a problem of E. L. De Forest in iterated smoothing, SIAM J. Math. AnaL, 5(1974), pp. 376 -398.
[3]
FRITZ JOHN, On integration of parabolic equations by
difference
methods, Corrun. on Pure and Appl. Math., 5 (1952) ,pp.155 - 211. [ 4]
I. J. SCHOENBERG, Contributions to the problem of approximation of equidistant data by analytic functions, Quart. of Appl. Math., 4 (1946), Part A, pp. 45 - 99, Part B, pp. 112 -141.
[5]
I. J. SCHOENBERG, Some analytical aspects of the problems
of
smoothing, Courant Anniversary vol ume "SWcUe-6 and E6.6ay.6 ", New York, 1948, pp. 351 - 370. [6]
I. J. SCHOENBERG, On smoothing operations and their generating functions, Bull. Amer. Math. Soc., 59(1953), pp. 199-230.
[7]
I. J. SCHOENBERG, Spline functions and the problem of graduation, Proc. Nat. Acad. Sci. 52 (1964), pp. 947 - 950.
[8]
I. J. SCHOENBERG, Cardinal interpolation and spline
functions
II. Interpolation of data of power growth, J. Approx. Theory, 6(1972), pp. 404 - 420. [ 9]
I. J. SCHOENBERG,
Ca~dinaf
.6pfine
inte~pofation,
Reg.
Conf.
Monogr. NQ 12, 125 pages, SIAM, Philadelphia, 1973. [10]
E. T. WHITTAKER and G. ROBINSON, The eafeufu.6 Blackie and Son, London, 1924.
Department of Mathematics United States Military Academy West Point, New York 10996
06
ob.6e~vation.6,
This Page Intentionally Left Blank
Approximation Theory and Functional Analysis J.B. Prolla (ed.) ©North-HoUand Publishing Company, 1979
A CHARACTERIZATION OF ECHELON KOTHE-SCHWARTZ SPACES
M. VALDIVIA Facul tad Pas eo
de Ciencias al Har, 13
Valencia
(Spain)
In [1 I , A. Grothendieck asks if each quasi-barrelled (OF) -space is bornological. We gave an answer to this question in [5 I structing a class of quasi-barrelled (DF)-spaces which bornological nor barrelled. In this paper, in
by
are
con-
neither
the context
of
Kothe's echelon spaces which are Montel, we characterize the
the
spaces
of Schwartz using certain non-bornological barrelled spaces.
As
a
barrelled
consequence, we prove the exis tence of non - bornologi cal (DF)-spaces.
K
of
denote
by
The vector spaces we use here are defined on the field the realor complex numbers. If \l
(E ,F)
the Mackey topology on
(E,F) E. If
is a dual pair, we
E is a topological vector space,
E' is its topological dual. In the sequel and
Ax
A will be an echelon space
its a-dual. Let us suppose that the steps defining
a(n)
(ai n ), a~n), ... ,a~n), ... ), n=1,2, ...
are all positive, they form an increasing sequence index
p,
there exists and index
q such that
each be
n-th whose value for this
is the space generated by 409
for
p
is one. Generally, we follow the terminology of [2 I ~
and,
a(q) '" O. Let
the sequence such that all its terms vanish except
of spaces. In particular,
A
the
kind vectors
410
VALDIVIA
En'
"x[
Here
n = 1, 2,
\.l (" x,
we always consider
a subspace of
,,) I .
Let
P
{In: n = 1,2, ..• } be a partition of the set
tural numbers, such that
In of
F n In
N such that, if
E J, then
j
F E F
is finite, n=1,2, . . . . Let
the set of all the filters on for some
N of
In is infinite, n = 1,2, . • . . Let
filter of all the subsets of tary in
as
I{)
N finer than
F be the
the complemen{F. : j
E
)
F so that, if
M n In f. 121, n =1,2, ..•
na-
be
J}
M
E
F. )
It follows immedi -
ately that, with the relation of inclusion, this set is inductive ordered. Using Zorn's lemma, let
PROPOSITION 1:
Fo.lr. eac.h
U be a maximal element.
n E N, :the .lr.e-6:t.lr.-Lc.:t-Lon 06
U :to
In
-L-6 an
ul :t.lr.a 6LU e.lr. •
PROOF:
Let Al and A2 be two non-empty subsets of
intersects all the elements of
belongs to
Al
U l
and
that
U [U
U and then
{Ip : pEN, P ". n} I
This completes the proof. n = AI· U c N, we denote by "x(U) the sectional subspace of
" x (U) ={a.=(al ,a 2 , ..• ,an' ... ): a. E " x , an =0, } \in E U . U belong to U it follows that U n U belongs to U l 2 2
"x[ \.l ( " x, ,,) I defined by If
such
A n I
U and
For each
n
Al n A2 = 121. Therefore, one of these sets, say Al
Al u A2 = In' and
A
I
and
and, therefore,
L
U
E U}
A CHARACTERIZATION OF ECHELON KOTHE·SCHWARTZ SPACES
is a subspace of gyof
>. x
containing
L is the one induced by
16
PROPOSITION 2:
Let us suppose that the topolo-
]J(>'x,>.).
A i.6 a MOVlte.-i'. .6pac.e. aVld
ab.6oftb.6 the. bOUVlde.d .6ub.6e.t.6 06
PROOF:
I{).
411
I{)
T i.6 a baftfte.-i'. in
~ >,x(N ~ I
n
),
60ft e.ac.h
L, Lt
n E N.
Let us suppose that there exists in
normal subset
B which is not absorbed by
struct a sequence (y ) in q
T. We now inductively ron-
B in the following way:
that we have already obtained the elements
Let
us
suppose
in
Yl'Y2, •.. ,yq
B such
that
~
a
r EN(p)
where
N(l), N(2) , ... ,N(q) N (1)
joints, such that
are finite subsets of
=
In' mutually dis-
In which does not lie in l-:l(p -1), being
N(l) U N(2) U ... U N(r). The space
I{)
n >.x(N ~ I
n
) is the topological direct sum of
I{)
Let
Bl be the projection of
B2 be the projection of
B onto
B onto
normal set it follows that
E2
El
n >,x(N ~ (I
according to
according to
E
l
B
l
. Since
B is not absorbed by
can find an element
E
~ M(q))).
n
2
, and
. Since
Bl U B2 C B. Moreover, Bl + B2
is a bounded subset of the finite-dimensional space sorbs
,
contains the first elerrent of I , and N (I ), p> 1, n p
contains the first element of M(r)
E K, P = 1,2, .•• , q
r
yq+l E B2 C B
T,
neither
such that
y q+ 1 ¢ (q + 1) T.
E B
2
l
B is ~
B.
let a Bl
, hence T ab. Therefore, we
412
VALDIVIA
can be written in the form
The element
l:
a
rEN(q+l)
where
£
r
N (q + 1) is a finite subset of
r
In' disjoint from
each
set
N(l), N(2), ... ,N(q) and that it contains the first element of In which is not contained in partition of
I
n
M(q). The sets of the sequence (N(q)) define
. Let
U{N(2q-l):q
U {N (2q)
Since the restriction of an
U E U
such that
U on
U () I
n
q
In
1,2, ... }
1,2, ... } .
is an ul trafil ter, there
coincides with
say. Therefore, Y2Q E "x(U) , q = 1, 2, ...
PI or
The space
relled, because is a sectional subspace of sorbs the set
"xl
exists
P 2 ' U n I n =P l , "x(U)
jJ (" x,,,)
is bar-
li hence T ab-
{Y ,Y , ... ,Y ... } and it contradicts 2q 2 4
Y2q ¢
( 2q) T, q = 1 , 2 , . .. .
is
Since the normal hull of every bounded subset of bounded, i t follows that
PROPOSITION 3:
16 "
1.6
T absorbs everyboundedsubsetof
£t
Mon.tel
bounded .6ub.6e.t 06
.6p£tc.e
£tb.6oltb.6
eveltlj
PROOF:
Let us suppose that there is in
set
B not absorbed by
is not in
a
£tnd
T i.6 £t b£t!l.ltel in
L, i.t
T. Let us choose in
a bounded and normal subB an element
Yl
T. By a recurrent process, let us define a sequence
which (Y ) q
A CHARACTERIZATION OF ECHELON KOTHE-SCHWARTZ SPACES
in
B.
If
are already defined, such that
k r E N(p)
where
arEK, p=1,2, ... ,q,
N(l), N(2), ... N(q) are finite subsets of
N(2) C I
U
{H
U I
nl+l
N(q) C In
Let
413
q-l
+1 U I
: p = 1,2, ... ,q}.
p
n +2 l
U
nq_l +
Then
N so that
2 U ... U I
H
nq
q
is the topological direct sum
of
Let
n "x(N - K )
Bl be the projection of
projection of
B onto
and
q
E2
B onto
El
according to
according to E
l
.
Since
B
E2 is
and B2
the
normal,
Bl U B2 C B. Moreover, Bl + B2 J B. From the previous proposition, it follows easily that T absorbs B . Since l an element
B is not absorbed by Y + l E B2 C B q
such that
Y + q l
Then
T, neither
¢
(q + l)T.
B
2
, hence we
can
find
VALDIVIA
414
being
N(q + 1) a finite subset of
p
is a partition of number
nq + l ,
Nand
Let
M
=U
C
nq
I n +1 q
Since
n + 2 u ••• q
In' Hence
T absorbs the set
bounded subset of
U
natural
In q+l
N - M
AX(N - M) is barrelled and
tradicts the fact of
a
such that
I
{N(q) :q =l,2, •.• }. Then
it follows that
Let
: n
n
U I
set with finite complement in
. Since
q
N(q + 1) is finite, we can find
larger than
N (q + 1)
{I
1'1 which not intersects K
intersects each
N - M
E
n
in a
FeU.
Yq E AX(N - M), q =1,2, ... ,
{Yl'Y 2 ""'Y q " " }
Yq ¢ qT, q = 1,2,. ••
I
. Clearly,
and it
con-
T absorbs each
'tl.
q.e.d.
A[ Jl (A, Ax) j a Montel space which is not Schwartz. 'Iherefore,
there exists a positive integer
k such that, if
subset of all the natural numbers
(
n so that
M is the
ordered
a(k) f 0, the sequence n
(k»)
:l
p
)
nE M
does not converges towards zero, p=k+l, k+2,,,., [2,p.422j. Let us define an increasing sequence in M
such that
lim i-+oo
A CHARACTERIZATION OF ECHELON KOTHE·SCHWARTZ SPACES
415
A[~(A,AX)] is a Montel space we can select a subsequence (m ) i of (qi) so that, for a particular number kl > k + 1
Since
o
lim i-+ co
Let
11 be the set
[2, p. 421] .
{m ,m , ... }. Obviously, M 2
1
finite set. Let us suppose that we have constructed I
so that
I
If
I
P numbers
n
p
I
n M
p
¢,
r
p
11
is an
subsets
inof
is an infinite set and
~
r,
p, r
l,2, ... ,q .
{r ,r , ... ,r , ... }, suppose also that there are two natural l 2 i k + p, i so that k > p p a(k) r.
ark) r.
1.
lim
a(k+p) r.
i-+ oo
cp
~
0,
lim i .... co
1.
1.
0,
i > i
(k ) a p r.
P
1.
Let
H
q
= U {I
p
:p =1,2, ... ,q}. If we arrange the terms of
H nM as q
a sequence
we obtain, for
u >
that
p=1,2, .•. ,q,
lim
0,
i-+ oo
From (1) and the condition of space, it follows that
i
> i
P
(1) •
A[~(A'AX)] not being a Schwartz
416
VALDIVIA
M~H
q
is an infinite set and the sequence
does not converges to zero. Therefore, we can select (t ) of (si) and a positive integer i
kq+l > k + q + 1
(k) at. cq +l
(k+q+l) at.
i-+oo
"I 0,
l.
lim
(kq+l) at i
i-rco
l.
Let element of ti tion
subsequence so that
(k) at.
l.
lim
a
o•
be the set {t , t , ... , ti ' ... } together with the first 2 l N which does not lie in
P = {In: n = 1,2, ... } of
H . In this way we obtain a parq
N such that
In is infinite,
whose
properties will be used in the sequel.
THEOREM 1: i~ il1
>.. x[ fl
16 (A x,
the Mantel
~paQe
>..)] a den~ e ~ ub-6 pace
G
which i-6 baltlt elled a.nd non bOlt-
l101og,[ca.l.
PROOF:
Using the number
construct
the space
and the subspace
k and the parti tion
L as we did at the beginning of
and the vector
a (k). We will prove that
nological. Let
T be a barrel in
[3, p. 324], hence
<{J.
this
<{J
L
G is barrelled and non bar-
G. By Proposition 3, T absorbs every
On the other hand,
T n
we
paper
>"X[fl(>"X,>..)] which is the linear hull of
G of
bounded subset of
P obtained above,
<{J
is a bornological
is a neighbourhood of the origin
space in
<{J.
A CHARACTERIZATION OF ECHELON KOTHE·SCHWARTZ SPACES
Since
If!
is dense in
G, the closure of
T
and it is a neighbourhood of the origin in relIed. Since
L is a subspace of
larger than one, to see that
II
417
If! in G is contained in T
G. Therefore, G
G whose codimension in
a
limi t in the sense of Mackey, of a sequence lying in Let us suppose that I Bin»
ber
G
is not
G is not bornological, according
result of Mackey [41, it is suffices to prove that
to
is bar-
is a sequence of
lk
to a
) is not the
L.
L which congerges
a lk ) in the sense of Mackey. Consequently, there is a natural num-
p such that I Bin»
pology of the norm
converges to
II • II
deduced from the uni t ball
I'b q
{B
We can find a p05i tive integer
>
Given
Bin)
Ib
~ c 2
r
p
.:::. a~k+P) , q = I, 2, ... } .
I
50 that
,
m > r, mEl
P
In) (n) In) ,b , ••• ,b , ... ), we have that 2 l q
q E N, a lk +p ) q
II a Ik) - B In) II
We can find
U E U such that
U
not finite, we can obtain a posi tive integer r, such that
AXlk + p) for the toa
a (k) in
5 in
I
p
,
~ O} .
Ii
larger
I
is
P
than
bin) = O. Then 5
II a Ik) _ BIn) II >
a lk +p )
I > -2
c
p
5
and we obtain a contradiction. Therefore,
G is not bornological.
VALDIVIA
418
Le.t E be. a FJtec.he.t-S c.hwaJttz
PROPOSITION 4:
.6u.b.6pac.e. 06
PROOF:
E' [ 11 (E' ,E»), the.n
pac.e.. 16 F .L6 a baJtJteLte.d
.6
F -<-.6 boJtnolog-<-c.al.
Let (An) be an increasing fundamental sequence of compact sub-
sets of
E' [11(E' ,E»). Let us suppose first that F is dense in E'[I1(E' ,E»).
Let
be the closure of
Bn
n F
A
n
closure of the normed space
FA
E' [ 11 (E' , E) I. Le t
in
n F
be the
in the Banach space
E'A
n the topology induced by the one associated to (DF) -space,
q and
Then A
p
EA' Since
r
such that
Ap
is a compact set in
F
C
r
Bq
p
in
and
continuous linear form v on
E'
and therefore
un of vn in
we can
afirm
proved obtaining the closure
F
with
F n Fn
Banach
F n F n is dense
can be extended
F n' Evidently, there exists vn in
that
to
Fn' n =1,2, . . . . Then
F is bornological. The general case
F
of
F in
E' [11 (E' ,E»)
and
F.J..
It
the orthogonal subspace of
E to
F.
A Monte.l .6pac.e.
follows from Theorem 1 and Proposition 4.
a
a linear v u is
proving
is the dual the Mackey of the Frechet-Schwartz space E / F
THEOREM 2:
PROOF:
F. Since
of
E~.
E'[I1(E' ,E») and, consequently, its restriction
F is continuous. Hence,
tha t
u to
which coincides with
is continuous on to
a
in-
Bq is a compact set in
u be a bounded linear form on
F n' the restriction
form
is
there are two positive
E'[I1(E',E») is the inductive limit of the sequence (Fn) spaces. Let
F
n
(Bn) is a fundamental sequence of canpact sets in E' [11 (E' ,E») ,
[2, p. 402). Given a positive integer tegers
with
n
1,
A CHARACTERIZATION OF ECHELON KOTHE-SCHWARTZ SPACES
419
REFERENCES
[1
I
A. GROTHENDIECK, Sur les spaces (F) et (OF), Summa Brasil.
3,
(1954), 57 -123. [2
I
G. KOTHE, Topological vector spaces
r. Berlin-Heidelberg - New
York. Springer: 1969. [3
I
T. KOMURA and Y. KOMURA, Sur les espaces parfaits de suites et leurs generalisations. J. Math. Soc. Japan 15 (1963), 319-338.
[4
I
G. MACKEY, On infinite dimensional linear spaces, Proc. Acad. Sci. USA 29
[5
I
Nat.
(1943), 216 - 221.
M. VALDIVIA, A class of quasi-barrelled (DF) -spaces which not bornological, Math.
z.
136 (1974), 249 - 251.
are
This Page Intentionally Left Blank
Approximation Theory and Functional Analysis J.B. Prolla (ed.) ©North-Holland Publishing Company, 1979
tHE RATIONAL APPROXIMATION OF REAL· FUNCTIONS
DANIEL WULBERT Mathematics Department University of California La Jolla, California 92093, USA
I.
INTRODUCTION This paper is closely related to the classical theory
uniform approximation of continuous functions by quotients nomials. That is, let
f
of best of poly-
be a continuous real function on [0,11
and
let (1.1)
q
~
p / q
where
0
on
[0,1]
and
irreducible}
Pn denotes the real polynomials of degree less than or
equal
n. It is classically known that there is an best approximation to
f
E
Rm n
which
is
a
[see for example Walsh, 19351 . That is,
11 f - rll
(1. 2)
r
dist (f , R~) .
Furthermore the approximation is characterized by f -
~
hav-
ing the zero function as a best approximation from the linear space (1. 3)
where
N
max { aq + m, Clp + n}. Hence
421
r
is a best approximation to
422
WULBERT
f if and only if
f - r
N + 2.
has an extremal alternation of length are
(Achieser [1930]). It follows that best approximations
always
unique. In this setting however the best approximation operator not generally continuous. In fact, it is continuous at if
f
has a normal point
either 3p
m or
f
is
if and only
p/q as a best approximation, that is,
3q = n (Werner
if
[1965]). The complex rational func-
tions, R~(CC), are defined similarly with
Pm and
P n
with
replaced
Pm(CC) and Pn(CC), the polynomials with complex coefficients. A complex function defined on [0,1] still has exactly one best Pn (CC). However approximation from R~ (CC) is not as
approximation from well understood.
It is still true that a best approximations exists
in R~(CC), but Walsh [1931] has constructed an example to show thatif the domain of the functions is a particular "crescent moon"
shaped
region of the complex plane (instead of the interval [0,1] as
in our
setting), then there is a complex function with more than one
best
approximation in Rm(CC). More recently E. Saff and R. Varga made n
surprising observation that in fact (x - 1/2)2 has nonunique approximations from
the best
Ri(CC) [1976].
If the function being approximated is the real function f, the.n its best approximation in
Pn(CC) is also real, and the
reduced to the theory of approximation from
problem
P . But for the ration
nal functions the Saff-Varga example shows that the analogous
reduc-
tion is not valid. It appears natural to consider approximations f from
is
to
Re R~(CC) - the real parts of R~(CC) functions. This paper is an exposition of part of such a study.
The
de-
tailed paper will appear elsewhere. As it turns out the theory of approximation from approxima tion from
Re R~ is, an intriguing mix of the regularity Rm wi th the pathology of that from n
are also some applications to approximation from
R~(CC).
of
R~(cr). There
THE RATIONAL APPROXIMATION OF REAL FUNCTIONS
423
II. EXISTENCE OF BEST APPROXIMATIONS In the classical settings, the existence of a best approximation is easy to establish. The idea is that a minimizing sequence r (i. e.
II r. - f II
+
~
i dist (f, Rn)) has a subsequence with converging m
merators and denominators. Cancelling common zeros of the limit
for f nunu-
merator and denominator produces a best approximation to f. Here the problem is that the limit function may not be in For example, for each
£
> 0
+
(0) (1) (2.1)
r
Re R~(~).
E Re R~(~).
£
So, (2.2)
xm+n
But, one easily shows that
is not of the form
pq + st 2+ 2 , P , t E Pm , s , q E P n q s
(2.3)
Hence the theory is actually about approximation from the closure of Re R~(~) (denoted herefter by R~). The first problem, then,
is
to
characterize R~.
Rm
Clearly,
n
{..E. q
(2.4)
C Qm+n , where
n
pEP \!
'
q E P 211 ; q > 0 on
:R, and q > 0 on [O,l]}.
In fact:
16
n < m + 1, then.
1.
PROPOSI,",ION:
2.
PROPOSITION: Foft a.Le.
a.ll
fEC[O,l]
m, a.n.d
~+n .
a.dm,.[tl.> bel.> t a.ppftO x,.[ma.t,.[o n.1.> to
424
WULBERT
III. CHARACTERIZATION AND UNICITY OF APPROXIMATIONS FROM
R~, the idea
As in the characterization of approximation from
is to change the problem to that of approximation from a more computable set. We will first state a special case so that the general case
a
will appear less absurd. Suppose that
1)
m
~
E
no common factors and that the degrees of
a
,
that a and
and
b
b
have
are such
that
2n + aa < ab + m. Let H(a,b) = {h
(3.1)
where
3.
M
ab + m
~~
a
and
~
PROPOSITION:
ze~o
PM: sgn h(x) = - sgn a(x)
E
~~ a be~~ app~ox~ma~~on ~o
Now in general suppose
b .::: 0).
a
and
~
a
f
- 1)
E
~
•
6~om
E
Z (b)}
f
~6 and onty
b have no common quadratic factors a
f.
~6
H (a,b) .
From the definition of
However it may be possible that
real zeros. Let
x
Z(f) denotes the zero set of a function
be~~ app~ox~ma~~on ~o
we may assume that
for
and b
have some
F be the greatest monic common divisor of
a
~ (i.e. common and b.
Put (3.2)
a IF
and
b
o
b IF .
Now put (3.3)
For
M
ba
E
Q.m n
max { abo + m, aa
+ 2n}
we now define
Z (b ) n JR (3.4)
o
Z(a,b)
{
if
2n + aa < ab + m
IZ':olOlRlU'.lUl-.lif
2n+'a"b+m
THE RATIONAL APPROXIMATION OF REAL FUNCTIONS
425
Por convience we will write f(oo)
(3.5)
for
lim f(x) x+oo
f (_00) for
and
lim f (x) , x~-oo
when these limits exists. Now define: H(a,b)
(3.6)
{Ph: h E PM : sgn h (x)
for
4. COMMENT:
x
E
- sgn a
o
(x)
Z(a ,b)}.
With the above notation proposition 3 above
is
still
valid. Our interest in proposition 3 is that one can compute the number of possible sign changes of members of H (a,b)
and
use
this
to
derive an extremal alternation type of characterization for approxi-
~.
mations from
However the result separates into many cases de-
pending on the number and parity of the pOints in and in
Z(a,b) () [1, 00).
Rather than presenting
Z (a,b)
the
() (- 00 ,0]
complicated
statement of the alternation theorem, we will give some of the consequences.
5.
COROLLARY:
6.
COROLLARY:
and Z(b) () m
Be6t
=
~6
app~ox~mat~on6
6~om
a~e
un~que.
Supp06e a, and b have no c.ommon 6ac.toM, m + db > 2n + Cla a ¢ . Then 1) b ~6 a be-6t app~ox~mat~on to f E C [
°,
a.nd only
in
f
-
a
b
2 + max {m + ab, 2n + da}.
7.
COROLLARY:
A c.on-6tant
6unc.t~on ~6
a be6t
app~oximat~on,
to
a
426
WULBERT
Qontinuou~
an
8.
6~om
6unQtion,
ext~emat atte~nation
16
COROLLARY:
r E ~
IV.
n
06 tength
a Qontinuou-6 6unQtion f
and
(i)
r i-6 a
(ii)
-r
be~t app~oximation
i-6 not a
APPROXIMATION FROH
06
f
be~t app~oximation
but to
f - 2r.
Rm(
cases
dist (f,R~)
dist (f,R~(
=
this phenomenon only occurs in a restricted setting, there are applications to the theory for approximation from
Let
p~oximation 6~om
f E CIO, IJ
R~(
II f whe~e
ing
COROLLARY: a~e
11. a
min {n -
N
Let
and
~ E R~. 16
~ i-6 a be~t
then ~ II~ dist(f,R~)
ab, m - aaL
f E C I 0 , 1 J; the 60Uow-
m > n, p E Pm-n ' and
(i)
p i-6 a be-6t
app~oximation
to
f
6~om
R~(
(ii)
p i-6 a be-6t
app~oximation
to
f
6~om
Rm. n
a
ap-
equivatent:
COROLLARY: i~
ap-
R~(
proximations from
THEOREM:
some
R~(
we can easily produce real functions which have nonunique best
10.
an
-6uQh that
In some special
9.
ha~
6unQtion
2 + max {m, 2n}.
the~e i~
n > 1
e~~o~
i6 and onty i6 the
Qm
Let
m > nand
be~t app~oximation
6~om
fEe I 0 , 1 J. A
Qon~tant
f
- a
6unQtion ha~
an
THE RATIONAL APPROXIMATION OF REAL FUNCTIONS
extkernaf
12.
afteknat~on
EXAMPLE:
427
06 fength m + n + 2.
(Saff - Varga)
For every
m > n > 1
there
.'l.re
con-
tinuous real functions which have nonunique best approximations from
R~(a:)
•
13.
COROLLARY:
Let n<m+l. Let ba
(Saff - Varga)
be the be¢t
appkox~rnat~on
an extkernal
alteknat~on
to
-
f
E
Ef1(Z(a) n Z(b) = tj») n
nkorn
06 fength at fea¢t
2 + m + mi n {n -
db, m -
da } .
REFERENCES
[ 1]
N. I. ACHIESER, On extremal properties of certain rational functions. Doklady Akad Nauk SSSR (1930), 495-499 (Russian).
[ 2]
E. W. CHENEY,
IntkoduQt~on
to
Appkox~rnat~on
TheokY.
HcGraw
Hill, New York 1966. [3]
E. W. CHENEY, Approximation by generalized rational functions, PkoQeed~ng¢
Syrnpo¢~urn
on the
Appkox~rnat~on
06 FUYlQUOM,
General Hotors, Elsevier Publishing Co., Amsterdam 1964, 101 - 110.
[ 4]
E. W. CHENEY
and H. L. LOEB, Generalized rational functions,
SIAM Journal [ 5]
C. J. DE LA
VALL~E
Numerical Anal. 1 (1964), 11 - 25. POUSSIN,
Sur les polynomes d'approximation
et la representation approchee d'un angle, Acad. Royale de Belgique, Bull de la Classe des sciences 12(1910).
[ 61
A. A. GOLDSTEIN, Rational approximations on finite point sets, Syrnpo¢~urn on the Appkox~rnat~on 06 FunQt~on¢, General Hators, Elsevier Publishing Co., Amsterdam 1964.
[7)
A. N. KOLMOGOROFF, A remark concerning the polynomials of P.L. Tschbycheff which deviate the least from a given function (Russian) Uspekhi Math. Nauk 3 (1948), 216 - 221.
[81
G. MAINARDUS
and R. S. VARGA, Chebyshev rational approximation
to certain en tire functions in
[0, + 00) ,J:Approx. Theory
3(1970), 300 - 309.
[ 9)
J. A. ROULIER tion of
[10)
and G. D. TAYLOR, Rational Chebyshev approxima[0, +00 ),J.Approx. Theory 11(1974), 208-215.
E. B. SAFF and R. S. VARGA, Nonuniqueness of best complex
ra-
tional approximations to real functions on real intervals (1976), preprint. [11)
J. L. WALSH, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc.
Colloquium
Publications 20, Providence R.I., 1935. [12)
J. L. WALSH, On the over convergence of sequences of functions, Amer. J. Math. 54(1932) I 559 - 570.
[13J
J. L. WALSH, The existence of rational functions of best
rational
ap-
proximation, Trans. Amer. Math. Soc. 33(1931), 477-502. [14J
H. WERNER, On the local behavior of the rational
Tschbyscheff
operator, Bull. Amer. Mat. Soc. 70(1964), 554 - 555. [15J
D. E. WULBERT, The rational approximation of Amer. J. !1ath., to appear.
real
functions,
Approximation Theory and FunctionaL AnaLYBiB J.B. FroZZa (ed.) ©Nm'th-HoZland Publishing Company, 1979
FUNDAMENTAL SEMI NORMS
GUIDO ZAPATA* Instituto de Matematica Uni versidade Federal
do Rio de Janeiro
Rio de Janeiro,
Brazil
1. INTRODUCTION Here we will consider a general problem of polynomial approximation in euclidean n-dimensional space. The subject
of
polynomial
approximation was iniciated in 1885 with the first version
of
the
Weierstrass theorem for uniform approximation on compact sets of euclidean space. The non-uniform approximation problem
on
the
whole
space was iniciated with the Bernstein paper of 1924 [2]
and
con-
tinued to be developed in the so called Bernstein problem. Classically this problem has been studied from the point of view
of
continuous
and integrable functions in the natural context of weighted
spaces.
See, for instance [1], [4], [7], [9] and [12] for related developnents and additional references. More recently, Bernstein's
problem
has
also been considered in the context of weighted spaces of differentiable functions and distributions. See, for instance [10] and [13]. In this approach we use the unifying
notion
of
fundamental
seminorm in considering a polynomial approximation problem whichoontains all the above mentioned cases of Bernstein's problem. Further, this approach puts in focus the seminorm point of view in approximation
*
The author was partially supported by FINEP, Brazil.
429
430
ZAPATA
theory which has been undertaken for instance in [3
1•
The main results are a characterization of fundamental
semi-
norms on the real line (Theorem 1), a quasi-analytic criterion
for
such semi norms (Theorem 2) and a tensor product criterion for fundamental seminorms in the general case (Theorem 3) • We finish by listing some interesting open problems,
some
of
them unsolved even in the classical case.
2. PRELIMINARIES
In the following, n, m INU{oo}
t
E
and k will denote elements
respectively. We put k
lRn ,
is the algebra of all complex valued m-times continu-
.
{f
E
sup on
{I
f (x)
e~(lRn)
I,
x
E
lRn. If
T.
E
em(lRn) I support of
f
on
e lkl (lRn ) ,
f
then
akf
n lR }. Also for
m
E
sub-
is compact}
lR n we let
II f II
denote
:IN we define
the
the
norm
by
The topology defined by the family of norms denoted by
f
We will consider also the following
For a bounded complex function
11m
n and if
n
ously differentiable functions on
II
= kl + •.• + k
:IN* ,
. . . • tn .
em (lR n )
number
Ik I
in
II
11m' m
E
:IN, will
Unless explicitly stated, this is the topology
be
to be
FUNDAMENTAL SEMINOAMS
C~ (lR n ) .
considered on
REMARK 1:
t .... 1/(1 + t
The functions
C~(lR) • Hence for all
to
431
p E P(lRn )
(1 + x2)m E COO (lRn ) n 0
p/(l + xf)m
that the set of all products
2
t .... t/(l + t
) and
m > m
0
g E C~(lRn)
gp,
I
)
belong
mo E :IN such that
there exists
for any
2
. From this follONS
p E P(lR n )
contains
the sum of any two such products and also any polynomial.
The Be.fLMte-tn .6pac.e on
DEFINITION 1: B
when
lRn,
n = 1 I is the complex vector space
a be a seminorm on
closure of
of
all
Bn or simply
products
gp,
p E P(lRn ).
g E c~(lRnj, Let
denoted by
Bn
A
I
C
Bn. Then
A in the seminormed space (Bn/a)
DEFINITION 2:
A semi norm
a
ii. a
will denote the
:= Bn,a
Bn is po.tynom-ta..t.ty c.ompa..t-tb.te if the
on
module operations (g/f) E COO (lR n ) o
x B
fEB
n are continuous. SPC(lR )
n,a
when
Let
a
.... fEB
n,a
will denote the (directed) set of all poly-
nomially compatible seminorms on
EXAMPLE 1:
.... gf E B n,C!.
n,C!.
Bn.
be an -tnc.Jt.ea6-tng .6em-tnoJt.m on
I f I 2. I g I • Then
a E SPC (lR) .
Since
Igfl 2. IIg III f I then
I f I
I fl·
a(gf) 2.
Bn
that is a(f) 2.a(g) 00 n In fact, let g E Co(JR ), f E Bn.
IIg lIa(f). Also
ali) = a(f) sinoe
It is clear that finite positive linear combinations of increasing seminorms are also increasing seminorms. EXAMPLE 2:
Let
m E :IN * ,
ak ,
i k I
<
m,
be
a
family of increasing
ZAPATA
432
semi norms on Bn such that ki.s.ki,
ak' < constant. a
when
k
k < k', that is
i=l, .•• ,n.Let
a(f)
a E SPC(lRn ). In fact, let
Then
formula,
the fact that
a
g E C~(lRn), fEB . Using Leibnitz's n
is increasing, and the condition
k
on
the
family, i t follows that
a(gf) < constant· IIg IIma(f). Also, i t is clear that
Hence
REMARK 2:
There exist semi norms
n a E SPC(JR )
a(f) =a(f).
which are not of
types described in Examples 1 or 2. For instance, this
is
the
the case
for the semi norm defined by
a(f)
If(O)1 +sup{lf'(x)l, xE[O,l)}, fEB.
g E C~(IRJ
Then for all
and
fEB
we have
a (fl.
L e.t
PROPOSITION 1:
PROOF:
00 n go E Co(lR ),
Let
g E COO (lR n ) .... gp
a
be such that Then
9 g
m o
B
0
n,a
n
C~ (lR ).
in in
B
a
n SPC(lR )
E
be given.
Then
when II x II < m and
Hence
-
8 g po E
m o
mapping
the
is continuous. For m = 1,2, ••• let
0 -< em-< 1, em(x) = 1
.... go
DEFINITION 3:
B
E
n Po E P(lR)
n,a
emE COOc (lRn )
II akem II -<.1.,1< m - \kl -<m. n
C~ (IR )
for all
m and
n,a is 6u.l'lda.mel'lta..e. when
P(lRnl
is
dense in
FUNDAMENTAL SEMINORMS
Bn,a . We say also that n-dimen-6~anal
433
is a Beltn-6t:ein -6eminoltm an lRn. Beltn-6t:ein'-6
ct
pltobtem consists in describing Bernstein seminorms
on
mn. REMARK 3:
All the cases of Bernstein approximation problem mentioned
at the Instroduction consist in asking for necessary and
sufficient
a E SPC(JR n )
conditions in order that some convenient semi norm
be
fundamental. Here is a useful result.
PROPOSITION 2:
Let: (E,S) be a complex -6eminoltmed -6pace 06 6unct:ion-6 and
A-6-6ume t:hat: t:helte exi-6t-6 ~ndu.ced
in E and t:he
mElNU{oo}
a = siB
SPC(~n).
E
n
-6u.ch that
t:opology on
iimit t:opology. Then
PROOF:
p(JRn)
in the inductive limit topology [11],
dense in E. slBn
From this it follows that
PORn)
is
then
it
dense is
is dense in
E,
also since
is fundamental.
EXAMPLE 3: on lRn
Let
u be
such that
m
upper-semicontinuous nonnegative
u(t) t k
vanishes at in finity for all
u is a weight: on
say that
lRn ). Let
of all complex continuous functions at infinity, seminormed by ing seminorm. For 0 <
em-<
S(f
-
-
C~(lRn)
Necessity is obvious. Conversely, since
C~(lRn)
in
i-6 deMe .<.n E i6 and onty i6 a i-6 6u.ndamental.
1,
8 f) m
...
0
when
if
1 m
-+-
00 •
f
on
lR
n
S (f) = II u f II. Then
m = 1,2, ...
8 (x) m
n = C U oo (JR)
E
let
8
II xII < m. Also
m
S(f) <
-
k
E
lN n . (We
be the vector space
such that uf vanishes S IBn
E C ( JRn )
c
Then for
function
any
is an increasbe
such
f
E
lIuli II f II for all
f
that n C u oo (lR )
E
n Cc(lR ) •
Thus the conditions of Proposition 2 are satisfied. When PORn) dense in C uoo(lR n ), we say that u is a 6u.ndament:at weight.
is
434
ZAPATA
EXAMPLE 4:
Let
is j.l-integrable for all 13 = .cP-seminorm. Then E and
m. n such that
j.l be a positive Borel measure on k E lNn, 13IBn
1:5. P < +
Cc(mn ) is dense in
is increasing. Also
S(f):5. j.l(mn)l/Pllfll
f E Cc(mn ).
for all
E = .c P ( j.l ) ,
Let
00.
tk
Thus
the
condi-
tions of Proposition 2 are satisfied.
EXAMPLE 5: such that
m
Let
Uk :5. constant
f E Cm(mn)
of all
I kl :5. m, 13 (f)
Uk '
. Uk'
k' < k. Let
if
lIukdkfll. Then
m-+
0, when
->-
13 (f) :5. ( f E C~(mn).
13IB n
for all
k,
described
is of the type
is as in the proof of oo ,
n
E be the vector space
Ukd f vanishes at infinity for all
em' m =1,2, ..•
13(f - emf)
for all
Ikl :5. m, be a family of weights on m.
k
such that
Ikl~m
=
in Example 2. I f 1, then
IN*,
E
Proposition
fEE. Also
max II ~ II) II film Ik,:5.m
Thus the seminormed space (E,13)
satisfies
the
hypothesis of Proposition 2.
EXAMPLE 6:
Let
m and
~,
notes Lebesgue measure on Given
p,
1 < P < + "", let
tributions 13(f)
=
f
on
R
n
ik
I :5. m, be as in Example 5. If
m.n , let
d]Jk = Uk d A
such that
scribed in Example 2. Furthermore
I k I < m.
for all
akf
E
.cP(]Jk)
13i Bn Cm(mn) c
for all k, Ik
is also
of
is dense in
the E.
of this fact is similar to that used in proving density Also
de-
E be the vector space of all complex dis-
Ikl;m (JlakfIPd]Jk)l/p. Then
spaces [11].
d A
in
i
:5. m,
type deThe proof Sobolev
(3 (f)
Once again the hypothesis of Proposition 2 are satisfied.
Let
PROPOSITION 3: 1)
16
i3
E
a.
be a 6uI'ldamel'ltal ~em'£l'loJtm on
n SPC(m )
6undamenta.l.
.{.4 4uc.h that
Rn.
i3 < constant -0., then
i3 '£.6
FUNOAMENTALSEMINORMS
Let
2)
lR \ { 0 }, Xo
E
IR n and
E
S(f) = a(fo<;?) OM att
16
PROOF:
c
f
435
.then
Bn'
' X E lRn. o S i...o 6undamentaL
1) is an immediate conseguence of Proposition 1. In the case
n P(IR)
of 2), observe that
0
"'n
0
"'n = Cc(IR).
3. MAIN RESULTS In the characterization of dense subalgebras in spaces of differentiable functions the following is a crucial result.
LEMMA I
6uncti..on~
~ati..~6yi..ng
2)
Fait any
3)
Folt any x, v
x
gl" .. ,gl
E
A
and
h
n
x t- y, thelte
.ouch that
i..~
g E A
~uch
g(y) •
IR n thelte
E
Then given a.ny
IR
E
g(x) t-
that
A c Cm(lR n ), m > 1, be a .oet 06 Iteat
the 60llowi..ng condition.o:
Fan a.ny X, y
1)
PROOF:
Let
(Nachbin's Lenuna):
g
i~
IRn, v t- O, thelte
E
Cm(]Rn) a.nd
f
E
E
Cm(]Rl), h(O) = 0,
K
~uch
A
E
C
i~
.that
g E A ~uch
]Rn compact
t-
g(x)
O.
that ~(x) t-O. thelte
.ouch that
See [ 8 1 •
DEFINITION 4:
A set
A C Cm(]Rn),
m > I, satisfies conditions
(N)
i~ a ~el6-adjoint
~ub
if 1), 2) and 3) above are true.
LEMMA 2:
Le.t
a
E
SPC(lR n ).
16
A C COO (]Rn)
o
algeblta that hati...o6i..eh condi..ti..onh (N), then A i..h den.oe in
PROOF:
From Proposition 1 it is enough to show that
B n,a
436
ZAPATA
Further, since the topology defined by T,
we need only show that the closure of
A in the
is weaker than
latter
topology
n
c~ (IR ) •
contains If
C~
C is a subset of
cmn) we will denote by C its closure
in the topology T. Assume also that if
n c~ (IR )
on
Cl
gl, ...
,ge E C
are real and
C is a subalgebra. In this case,
hE cooCill-)
is such that
h(O)
-
then
G = {gl (x), ••• ,g-e (x), x E mn} is bounded in
the Weierstrass approximation theorem for differentiable we can approximate stant term, since
h on h(O)
G
= O.
i = 1, .•. , t. Hence
k E mn,
in the topology Let
by polynomials
pEP (IRt)
Furthermore, akg,
functions,
wi thout
is bounded
1-
lRi,using
p (g l' ... , 9 t ) approximates
f E Coo(lRn )
c
f 'I 0
be given. Assume
and
Al
h (g l' ... , 9 t)
let
H
is a subalgebra of
satisfies conditions (N). In particular, for any
Also
be its
x E H
Since
A and also
there exists
hE C"'(lR) such thath~O, h(g(x» >0
g(x) '10. Choose
such that
h(O) = O. From the above remark, it follows that hog
is posi ti ve on a neighbourhood of
ness, there exist Let
all
T.
is a self-adjoint algebra, then
g E Al
con-
for
support. Let Al be the set of real parts of functions in A.
and
n
h(gl, ... ,g-e) E C. In fact, it is clear that h(gl, ... ,g-e) E CoOR).
Since the set
A
= 0, 00
hlEC""(lR)
bourhood of
O.
gl E Al
and
be such that If
r >0
hl=l
x.
(r,+co),
,
K.
Hence
on a neigh-
Hand
fl E Al
has compact support, say
K. Then from Nachbin's Lemma, there eXl.st gl, ... ,gi f = h(gl, ... ,gi) on
on H, gl(O) =0.
hl=O
on
1
fl
since this is a closed subalgebra. Also, fl
such that
gl~r
such that on
hog E A . l Hence by compact-
E~,
eo
i
hE C (lR), h(O} =0
f = fl 'h(gl, ... ,gt)
mn and from the remark on subalgebras it follows that
f
E
on
A, since
h(gl, ••• ,g-e) E A. Now the proof is complete.
DEFINITION 5: defined by
For
z E
Q;\
IR, let
9 z be the complex function on
lR
FUNDAMENTAL SEMINORMS
1 x -
Cl
Let
Cl
'
x
E
lR.
gz E C~ (JR) .
It is clear that
LEMMA 3:
Z
437
r6
E SPC (JR).
gz E P (lR) a
6o!L
z E C \ JR, theYl
,6ome
6uYldameYltal.
~,6
PROOF:
Let
In fact, for
E = P (lR) m=O
true for some
U •
We claim that
g';P (JR)
C
E
for all
mE IN •
this is evident. Assume that the proposition
mElN.
Let
pEP(lR).Since
is
q=gz(p-p(Z»EP(lR),
then from the assumption it follows that
g~+ 1 (p
Now the mapping
Since
-
p ( z) )
f E Ba ~ g~f E Ba
is continuous hence
E is a complex vector space we have
g~+l(p _ p(z» + p(Z)g~+l E E.
So the claim is proved. Further, the mapping
f E Bu ~
h E '~s se If -a d'Jo~n , tsi P (JR) ' con t ~nuous, ence nce for all
•;s. So
is
E BCl
(-)l t EE gz =gz
l E IN ; whence
m, n
for all
From this it follows that the complex algebra gz
f
is contained in
tions (N) since
E.
{gz}
Also
E
IN .
A generated by gz and
A is self-adjoint and satisfies condi-
satisfies conditions (N).
From
Lemma
2
it
438
ZAPATA
follows that
P(lR)
Let
LEMMA 4:
is dense in
a E SPC(IR),
Ba' that is, a is fundamental.
z, z' E
a: \
IR. Then thelte eX-i..6t.6 a pO.6ilive
pEP (lR) .
PROOF:
Let
pEP (JR).
Since
g zp
it follows
From the definition of
a(gf) .::. CllgII
a, there exists
m
cdf),
for all
m
k~O
C > 0
and m E IN such that
g E C~(lR),
fEB.
k! r k+l
To finish, it is enough to observe that the number C ,=1+ iz-z' i CC z,z z does not depend on
THEOREM 1:
Let
p.
a E SPC(IR).
In
thelte e.x..i...6t-l>
z
E
a: \
lR
.6aeh :tha.t
a i.6
6andamen..ta..e
the .6et in eomplex. plane
i.6 unbounded, then a i.6 6anda.mental. Convelt.6ely i6 then
PROOF:
Po. (z) i~ unbounded
Assume that
60ft al!
z E C \ IR.
Pa(z) is unbounded. Let
p E P(lR)
be such that
FUNDAMENTAL SEMINORMS
~
a(gi P )
q
then
1
and
P (m)
E
p(z)
439
O. If
,
gzp
and
= p(z)
q - gz
By choosing a constant Cz,i > 0
.
as in Lemma 4 it follows that
c Z,l..
Since
P a (z)
is unbounded, then
gz
P (IR) a
E
and from Lemma 3
a
is
fundamental. Conversely assume that n
IN*
E
that
be given. Since
a(gz - p)
~ ~
a(gzq) = na(gz - p) ma 4 it follows that
Pn
Then
E
. ~
g
z
Let
a
is fundamental. Let
E
p(m)a,
there exists
q
= n(l
(x - z)p).
1. If
Ci,z
a (gl.. q)
< C.
P (m) , a ( g i Pn)
~
is
a
a:: \ m
E
pEP(JR) Then
q
E
and such
P (JR) and
positive constant as is !em=~. C.
To finish we let
1.,Z
1
z
1.,Z
n
and
Hence
-C-.-
Pa(z)
is
1.,Z
unbounded.
Let
THEOREM 2 (quasi-analytic criterion):
a
1
~
SPC(IR).
E
+
16
00
n=l
PROOF: on
Let
P (lR).
T be a continuous linear form on Let
B a
such that T vanishes such
that
on D. In fact assuming this,
from
D denote the set of complex
numbers
z
Imz < 1. Define h(z)
It is enough to prove that
T(gz)'
h =0
zED.
440
ZAPATA
- - a.
Hahn-Banach theorem it follows that a.
is
zED, n E IN.
If
n > I
~ z
T vanishes on
is also true for
Ih (z) I
n =
P (m)
o.
zED. Then
then n-l _x_ _ zn
-
it follows that
h (z)
Hence
zED,
for all
<
From the definition of
Since
"g z "m-
(*)
I h ( z) I < C II T II
for all
for all
< (m + 1) !
(m
C > 0
a, there exist
a.(gf) ~clIglim a(f)
+ l) !
and
g E C~(m),
zED
n E IN.
m E lN such that
fEB.
we have that
zED, n E IN.
for all
Let
(z - zo)gzgz
<
iz-zolliTIl
is holomorphic on
o
• Hence
a.(gzg~) ~ Iz- z o IIITII(C(m+1)!)3 cx (1). o
From this it follows that h
for all
fundamental from Lemma 3. Let
Since
gz E P (m)
h
is holomorphic on
D. Since (*) is true,
D and 2:
n=l
1
+ "",
then Denjoy conditions in Watson's problem are satisfied, vanishes on
D ([ 6
1 ). Now the proof is complete.
hence
h
FUNDAMENTAL SEMI NORMS
t:. be the -6et 06 a!! -6em.(.rwftm-6
Let
COROLLARY 1:
441
thefte afte po-6Ltive c.OYl-6tant-6
C
NI m
and
C
I
E
0. E
IN
SPC(lR)
60ft wJUch
(a..t.t depending on 0.)
-6uch that < C (c n log n • • • . • log
m
60ft aU
n) n
nand
n > N
lognn = log (logm _ 1 n)
m > 1.
t:.
Then
06
a d.(.ftec.ted -6 et
.(.-6
6undamenta! -6 eminoftm-6.
This is a direct consequence of Theorem 2 observing that the
PROOF:
"moments" of any two such seminorms have
a common estimate of the same
type.
Let
THEOREM 3: 0.
1
' ... Io.
then
n
E
n
0.
E SPC(lR ).
SPC (lR)
-6 uch that
16
thefte eX.(.-6t 6undamenta!
-<--6 6undamental.
0.
PROOF:
Let
1I:B
1I(C~(lR)
x •.•
xC~(lR»
since each
0.1
o.i
x ••• xB
o.n
be defined by 1I(f , ... ,f ) =f 0 ••• 0f , 1 l n n
.... B n,o.
--0.
--0.
C1l(P(lR) 1 x ••• xP(lR) n)
is fundamental and
11
c 1I(P(lR) x ••• xP(lR)o.
is continuous. Hence i f
the complex subalgebra generated by
11
(C~ (lR) x ... x C~ (lR»
viz
A
- - - 0.
then
~em~no!tm-6
A C P (lRn )
Since
A
is a self-adjoint subalgebra of
A
is
442
ZAPATA
and also satisfies conditions (N), from Lemma 2 it follows B • Hence n,a
is dense in
that
A
is fundamental.
4. OPEN PROBLEMS 1.
Give integral criteria like those in [ 7 I ing fundamental semi norms on
2.
Under what conditions on
3.
If
a
SPC(m)
4.
If
a E SPC(m)
E
5.
E
IC
we have
6.
p(IR)a.
/p(z) /
~
p
E
p(m),
the
a(p) 2.1
and
ce c1z / ? a
E
SPC(m) is it true that E
P (IR),
a (p)
space of entire functions on
>
a is
~ 1}
is
IC?
Give a characterization of fundamental seminorms on
n 7.
?
is not fundamental, are there positive con-
fundamental i f and only i f the set {p in
00
is not fundamental, describe
Under what conditions on
unbounded
is it true that a is
~ 1 = + i=l~a(xn)
stants c, C such that for all z
IR.
a E Spc(m)
fundamental if and only if
for characteriz-
IRn,
2.
Is the set of all fundamental seminorms on Same on
lR
directed?
lR n ?
REFERENCES
[ 1)
N. AKIEZER, On the weighted approximation of continuous tions by polynomials on the entire number axis,
funcAmer.
Math. Soc. Translations, Series 2, vol. 22 (1962), 95 -138. (2)
S. BERNSTEIN, Le probleme de l'approximation des fonctions continues sur tout l'axe reel et l'une de ses applications, Bull. Soc. Math. France 52 (1924), 399 - 410.
FUNDAMENTAL SEMINORMS
[3]
443
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INDEX
A
algebraic convolution integrals
71
almost simple
214
approximation, non-archimedean
121
approximation on product sets
46
approximation property
37,
approximation, rational
421
approximation, restricted range
226
approximation, simultaneous
227
B b - differentiable
161
Bernstein problem
433
Bernstein seminorm
433
Bernstein space
431
Birkhoff condition
192
Birkhoff interpolation problem
189
Birkhoff's kernel
222
C
cardinal series
391
cardinal spline interpolation
390
coalescence of matrices
198
coefficient of collision
200
compactly - regular
291
445
280,
373
446
INDEX
condition (L)
167
cross - section
372
D Dedekind completion
64
degree of exactness
385
differentiability type
164
differentiability type, compact
165
E
echelon Kothe-Schwartz spaces
409
e: - product
37, 269
F
Fejer - Korovkin kernel
78,
formal power series
354
fundamental seminorm
432
fundamental weight
433
fusion lemma
143
G Gaussian matrix
231
Gelfand theory
336
generating function
396
I
increasing semi norm
431
interchange number
202
interpolation matrix
189
interpolation matrix, poised
189
interpolation matrix, regular
189
79,
88
INDEX
447
K
Korovkin approximation
19
Korovkin closure
20
Korovkin space
20
Korovkin's theorem
63
L
level functions
199
M
meromorphic uniform approximation
139
N
Nachbin space
372
non-archimedean spaces
121
o order regularity
189
P
plurisubharmonic function
343
poids de Bernstein
237
point regulier
238
Polya condition
192
Polya functions
191
polynomially compatible semi norm
431
power growth
392
property (B)
168
pseudodifferential operator
13
448
INDEX
Q
q - regular
229
quasi - analytic cri terion
439
R
rational approximation
421
regular interpolation matrix
189
relative Korovkin approximation
28
relative Korovkin closure
28
restricted range approximation
226
Rogosinski summation method
103
Rolle set
209
S
S-approximation property (S.a.P.)
359
seminorm, Bernstein
433
seminorm, fundamental
432
seminorm, increasing
431
seminorm, polynomially compatible
431
sheaf of F-morphic functions
40
shift
203
S-holomorphic approximation property (S.H. a.p.)
367
Silva-bounded n-homogeneous polynomial
353
Silva-bounded n-linear map
352
Silva-bounded polynomial
354
Silva-ho1omorphic
355
Silva-holomorphic, weakly
356
simple
213
singular integral of de la Vallee Poussin
99
singular integral of Fejer
98
singular integral of Landau-Stieltjes
93
INDEX
singular integral of Weierstrass
449
96
smoothing formula
386
S - Runge
362
strict compact
357
supported sequence
194
V V*-algebra
339
vector fibration
372
very compact
275
W
weakly Silva-holomorphic
355
weight
372,
weight, fundamental
433
433
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