Fourier Series in Orthogonal Polynomials

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tourier ueries in Orthogonal Polynomials

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tourierueriesin Orthogonal L olynomials

Woiis Osilenker Moscow State Civil Engineering University

World Scientific Singapore • NewJersey•London* Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

FOURIER SERIES IN ORTHOGONAL POLYNOMIALS Copyright © 1999 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-3787-1

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

Contents 1

Preliminaries 1.1 Notations 1.2 Some Topics from Function Theory of a Real Variable 1.3 Some Topics from Functional Analysis 1.4 Interpolation Theorems. Estimates of the Hardy-Littlewood maximal function

1 1 3 24 33

2

Orthogonal polynomials and their properties 2.1 Orthogonal systems in a Hilbert space 2.2 Elementary properties of the orthogonal polynomials 2.3 Jacobi polynomials 2.4 Some estimates of the orthogonal polynomials

49 49 59 80 105

3

Convergence and summability of Fourier series in L^ 3.1 Fourier series in an abstract Hilbert space 3.2 Fundamental Theorems on a Convergence of the Fourier series in O.N.S 3.3 Behaviour of the partial sums of Fourier series in orthogonal polynomials 3.4 (C, l)-summability almost everywhere

121 121 124 134 143

4

Fourier orthogonal series in L£ (1 < r < oo) and C 155 4.1 On a divergent Fourier series of the continuous functions 155 4.2 Estimates of the Lebesgue A-function 161 4.3 Strong summability of Fourier polynomial series in L£ (r > 1) . . . 179 4.4 Linear methods of summability of Fourier polynomial series in L^(r > 1) and C 184

5

Fourier polynomial series in L^. Analogs of Fatou T h e o r e m s 5.1 On an almost everywhere divergence of orthogonal expansions . . . 5.2 On linear methods of <=vmmability at the Lebesgue points of the orthogonal polynomial series

v

197 197 201

vi

Contents

5.3 Poisson-Abel's summability of the orthogonal polynomial series . . 215 5.4 Analog of Fatou's Theorem for differentiated Fourier expansions . 223 6 The tion 6.1 6.2 6.3

representations of the trilinear kernels. Generalized TranslaOperator in orthogonal polynomials 231 The representations of the trilinear kernel 231 A Generalized Product Formula 241 A construction of the "double-humpbacked majorant". A Generalized Translation Operator in orthogonal polynomials 248

Notes

261

References

267

Author Index

280

Subject Index

283

Chapter 1

Preliminaries 1.1

Notations

We use Peano's decimal system of section's numbering. The numbering of the theorems and the lemmata, as well as that of the formulas starts, in each paragraph from the beginning and it stands as the first numeral in the number of the corresponding paragraph. References without numbers of the chapter relate to the chapter in question. In the case of a reference to another chapter the number of the chapter (a Roman figure) stands before the number of the formula (or of the Theorem, respectively). The numbering of the theorems and lemmata goes on continuously. We use Kronecker's symbol (Snm = 0 or 1, according as n / m, or n = m) and the symbols R 1 , Z, and N are used to denote the set of real numbers, integer numbers and positive integers respectively. As usual, Z+ = {0} U N. Closed intervals are denoted by [a, 6], open intervals by (a, b) and half-open intervals by (a, 6] or [a, b). We write for a real x sgnx =

{

1, i f x > 0 0, if x = 0 -1, ifx<0.

By 9ftz, Sz, \z\ we mean the real part, the imaginary part and the absolute value, respectively, of the complex number z. We denote by a + and [a] the positive part of a and the integral part of a, respectively.

1

2

Chapter 1.

Preliminaries

We use the usual o , 0 notation and also ~ in the following sense: if { a n } (n e Z + ) and {6 n } (n e Z+) are real sequences, then 6bnn = 0 ( a n ) ,

b&nn = o(an)

(ann > 0)

to state that bn/an is bounded, or tends to 0, respectively; and an ~ bn if and only if for some C\ and C 2 Ci < ^ < C 2 , bnn

n > 1.

Similar notations apply to functions and sequences of functions. We shall omit unnecessary parameters except where confusion can arise. As a matter of terminology a "constant" will always mean a quantity independent of n, symbol used exclusively to index orthogonal polynomials. Some constants will depend on certain parameters: C a , Cp, Co, C i , . . . These will sometimes be indicated, but not always. For later use, we denote f(x) = g{x), if the identity is true for all x f(x) := g(x) instead of: f(x) = g(x) by definition. We shall consider polynomials in x of the form m

U(x) = InImm((xx ) := ^2ckxkh

(cfc € R 1 , k = 0 , l , ......,,rma , me

Z+).

fc=0 A:=0

We call c m the leading coefficient of II(x). Here m is called the degree and if Cm ¥" 0> t n e precise degree of II(x). In what follows an arbitrary polynomials of degree at most equal to m will be denoted by irm. A trigonometrical polynomial in 0 of degree m has the form T{0) = do T(^) ao + H- ai ai cos cos0^++ 61 61sin sin00++ . . . . ++ aamm cos cosmO mO++ bbmm sin sin m0, ra0, with arbitrary real coefficients. Here m is again called the degree of T(0); m is the precise degree if \am\ + |6 m | > 0. According as all the bk or all the a^ vanish, T(9) is referred to as cosine or a sine polynomial. A function f(x) is called increasing (strictly increasing) if x\ < X2 implies f(xi) < f{x2)\ it is called nondecreasing if x\ < X2 implies f{x\) < /(X2). An analogous terminology will be used for decreasing functions. We shall call monotone functions increasing, nonincreasing, decreasing and nondecreasing functions. As usual, x(x) (x € E) is t n e characteristic function of a set E.

1.2. Some Topics from Function Theory of a Real Variable

3

We subdivide the interval [a, b] by means of the pointsXQ = a , x i , . . . , x n = b and then form the sum £ l / ( * i ++ ii)-/(*i)|. £l/(*i )-/(*i)|. i=0

The least upper bound of such sums for all possible subdivisions will be denoted n-l

Var V a (ar ,M6) f(x) /(*) = = Vf(f) := sup £

|/(x i + 1 ) - /(*,)! f{xt)\

i=0

and will be called the variation of the function f in the interval [a, 6]. If Var[a>5] /(#) < oo, we call / of finite (or bounded) variation on [a, b]. It is obvious that monotone functions are finite variation. If Var[a>5] f(x) < oo, then f(x) = /(a) + /i(a:) + /2(x), for all x G [a, 6], where /i(x) does not decrease and f2(x) does not increase. Let {An} (n € Z+) be an arbitrary sequence of numbers, then AAn = An - An+i

(n G Z+) Z+)

is the first difference of An, and +i + An+2 A2An = A(AAn) = An - 2An+l

(n €G Z+)

is the second difference of An. If A2An > 0 for all n

(A2A^n) < 0 for all n),

then {An} is called convex {concave).

1.2

Some Topics from Function Theory of a Real Variable

Classical Theorems of Function Theory. A certain class K of sets, defined in the space 8, is said to be completely additive, if the following conditions are valid: 1) the empty set belongs to K\ 2) if E belongs to K, then the complement of a set E (with respect to S) belongs totf; 3) if En 6 K for all n, then the set U^L^n also belongs to it.

4

Chapter 1. Preliminaries

The function fi(E) is called a measure, if it is defined and nonnegative for every E from AT, and the equation oo oo

M(u~=1

£„) = £>(£«) n=l

holds, where En (n = 1,2,...) are mutually disjoint sets from K. Number fi(E) is said to be a fi-measure of a set E. The Lebesgue measure of a set E we denote \E\. We call dfi(x) a mass distribution (or distribution). If a certain property A holds everywhere except on a set E of /x-measure zero, we shall say that this property holds /z-almost everywhere on E and use the abbreviation /Li-a.e. By the aid of the measure one can define the Lebesgue-Stieltjes integral

f/ fdfi. fdti.

(2.1)

JE

Most important cases are the following: 1. n{E) is the Lebesgue measure; 2. let fi(x) be the additive measure assigning value fi{an) to the sets consisting of a single point x = a n , n = 1,2,... and vanishing for sets not containing any such point. In this case integral (2.1) may be defined as /(ajb)|i(ojb) k

and the series converges absolutely; 3. /x(x) is a certain function with a bounded variation in [a, b]. We get an additive function if to every [a,/3] C [a, b] corresponds the number F(/3) — F(a). In future, we shall consider the case, where fi(x) be a positive, bounded, monotone increasing function in the finite interval [a, b] C R 1 . The (real) function f(x) is called U^-integrable in [a, b] (r > 1) (or Lrdyiintegrable), if it is \i-measurable and moreover the condition \f(x)\rd»(x)
Ja Ja

is fulfilled. If //(#) is absolutely continuous and //(x) = w{x) (x 6 [a, 6]), then for any L^-integrable function r f/ \f(x)\ |/(s)rd»{x) dfi(x) == ff

Ja

Ja

\f(x)\ \f(x)\rrw(x)dx w(x)dx

1.2. Some Topics from Function Theory of a Real Variable

5

is valid. In this case we shall say that f(x) is an L^-integrable and we shall call w(x) to be a weight function (or weight). If, in particular, w(x) = 1 (for all x £ [a, &]), then we shall say - in accordance with usual terminology - that f(x) is Ll-integrable. Let [ ( [ \f{x)\r dfiix))1

ll/llr,M : = {

KJa

'

II I I

where

r

}

ess sup ess x€[a,6] sup |/(#)| |/(x)| x£[a,b)

ess sup f(x) := a<x
for 1 < r < oo, for for rr = — oo, oo,

inf

sup

Nc[a,b]

X£[a,b]\N

(2.2)

\f(x)\

for all sets of //-measure zero N C [a, b]. Denote r Lrdtl[a,b] = L Z£[a,b] ^[a,b] =={/, {/, / /i iss/ / /— —measurable measurableand and ||/|| ||/||r,/z oo}r,/x << oo}-

(2.3) (2.3)

We call simple any function taking only a finite number of values. The set of all simple functions will be denoted by S. It follows from the definition of the integral that the set S is dense in every L £ ( l < r < o o ) (we remind that /j,[a, b] is finite). Moreover, every function / £ I/J^[a, 6] (1 < r < oo) can be represented as point wise limit (almost everywhere) of nondecreasing sequence of simple functions. In what follows we sometimes need the formula the integration by parts: / f(x)dn(x) + f rtx)df(x) = /(6)p(6) - f(a)ft(a), JJa a

Ja

where a and b are finite, /x(x) is of bounded variation and f(x) is a continuous function. Second mean-value theorem of the integral calculus: Let /(#) > 0 be a nonincreasing function, and let g(x) be continuous, a < x < 6, a and b finite. Then j f[x)g{x) dx = f(a + 0) J g(x) dx Ja

(a < £ < b).

Ja

We remind (without proofs) some classical results of Function Theory. Theorem 2.1 (Lebesgue's dominated convergence Theorem). // the sequence {fn} of L^-integrable functions on the set E tends on E ^-almost everywhere to the function f, and furthermore there exists an L^-integrable function g such that \fn\ < g (n G Z+), then the function f is L^-integrable and the equation lim / fndfji= / fdfi ->°° JE JE

n

6

Chapter 1.

Preliminaries

is valid. T h e o r e m 2.2 (Fatou's L e m m a ) , If{fn} is a sequence of nonnegative functions L^-integrable on the set E and convergent /i-almost everywhere to the function f, then

liminf fndfi > /f / fdfi. lim inf / fndp,> d\x. n ->°° JE JJE E T h e o r e m 2.3 ( B . Levi's T h e o r e m ) . If {fn} is a sequence of increasing functions L^-integrable on the set E such that

IJ>"

\ j fndix\
then the limiting function is L^-integrable

n

monotonically

( n € ZZ+), +), (ne also, and the following

relation

lim / fndfi= fndfi = f/ fdfjL fdp, ^°° JE JE JE

is valid. In particular, if L^-integrable functions UQ(X), t i i ( x ) , . . . , un(x),... relation oo oo

£ Y^

satisfy the

.

\un(x)\ d}i(x) < 00, oo, / \Un(x)\dll(x)

JE

71=0

then the series Yl^Lo un{x) converges (absolutely) ^-almost everywhere. T h e o r e m 2.4 (Egorov's T h e o r e m ) . If a sequence of measurable functions fn is almost everywhere convergent on a set E, then for every S > 0 there exists a subset EQ C E, such that \E\EQ\ < 8 and such that the convergence of the sequence {fn} is uniform on E. T h e o r e m 2.5. A function of finite variation has a derivative almost everywhere. T h e o r e m 2.6. If f e L\, a(x) = f* f{t)dfi(t), then a'(x) = f(x)f/(x) holds for almost every x. T h e o r e m 2.7 (Fubini's T h e o r e m ) . / / the function f(x,y) is \i x fi-integrable on the rectangle P (a < x < b; c < y < d), then for ii-almost all x G [a, b], the function / ( # , y) is /i-integrable in the interval [c, d] as a function of the variable y, and the equation f(x,y)dn{x)dfjL(y)= / / f(x,y)dn{x)dfjL(y)

rb

rd

Ja rd

Jc rb

= I/ dfi(x) / = / Jc

holds. Inequalities.

d

Kv)

/ Ja

f(x,f(x,y)dfi(y) y) dii{y) /(x,f(x,y)dfi(x) y) dn(x)

1.2. Some Topics from Function Theory of a Real Variable

7

Lemma 2.8. The weighted geometric mean is always at most equal to the weighted arithmetic mean with the same weights, i.e. AaB0

(A > 0, B > 0, a > 0, 0 > 0, a + 0 = 1).

(2.4)

This is clear when A = 0. If A > 0, inequality (2.4) can be written in the form

A~(!)' S0 f(x) = l-/3 >O >(

((00<<xz << ll )) ..

(2.5)

Since the derivative f'{x) = (3(1 — x&~1) is negative for 0 < x < 1, and moreover / ( l ) = 0, inequality (2.5) and by the same taken inequality (2.4) are satisfied. Equality holds if, and only if, x = 1, i.e. when A = B. Theorem 2.9 (Holder's inequality). Ifr andr' are positive numbers satisfying the relation

i + i = l (r>l)

(2.6)

rr rr r and if f € L^a, b], g e L ^[a, b}} then f • g G L^a, b] and 1/r 1/r {ja { | jJf f(x)g(x)dn(x)\ f(x)g(x) dfi(x)\ <{j < {a J \f(x)\rd^x)} dn(x)}

J \g(x)\ \g(x)fr'dn(x)) d^x))^l,T

.

(2.7)

Proof. This inequality is obvious if one of the functions f(x) and g(x) is equal to zero /i-almost everywhere. Otherwise both integrals F= f[ \f(x)\rdfi(x), Ja

G= f

Ja

\g(x)\r'dfi(x)

are positive. On the basis of inequality (2.4), we see that

j:wrw^

< r \m\r ~Ja V rF

dfx(x)

\9(x)\r\ d/z(x) = T G ' )

- + — = 1, r T

the function under the integral sign in the left member of the above inequality is L*-integrable because it is at most equal to the (L*-integrable) function under the integral sign in the right member of the inequality. From this it follows directly that the product f(x)g(x) is L*-integrable and that inequality (2.7) holds.

8

Chapter 1. Preliminaries

In particular casep = q = 2, inequality (2.7) is called the Schwartz-Bunyakowskii inequality 1,2 2 2 \£f(x)g(x)d»(x)\ <{£\f(x)fd»(x)} [ J \f{x)\2 1/2 d^x)) {£\g(x)\ { dti(x)y j /2\g{x)\ . dfi(x)}1/2. \J f(x)g(x)dfi(x)\<

(2.8)

Remark. If 1 < r < oo, then the number r' satisfying (2.6) is said a conjugate index. If r = 1, then r' = oo. In particular, if r = 2, then r' — 2. The following theorem is easily obtained from the Holder inequality. Theorem 2.10 (Minkowski's inequality). If r > 1 and f,g £ Lr^[a,b], then f + g e Lp[a,b] and

r /r 1/r 1/r r {£\g(x)\ dn(x)} 9(x)\ d^x)}dfi(x)y <{^ dn(x)} ++{^ {£ | 5 (x)rd^)} \g(x)\ dn(x)} -. {fij \f^ \Hx)+g(x)\ <{J \f(x)\ i/(x)r^(x)} +

a

r

1/r

r

1/r

l/r

r

l/r

1/r

C. (2.9) Proof. It is obvious for r = 1 or when the sum f(x) + g(x) is equal to 0 /z-almost everywhere. Otherwise, we write on the basis (2.7) /

J./aa

\f(x) + g{x)\ g(x)\rdn(x) < jf

fb\f(x)\\f(x)+g(x)rl1dn(x) dti(x)

Ja

,6

1 + gWr1 r-1d»(x) + / \g(x)\\f(x) \g(x)\\f(x)+g(x)\ \g(x)\\f(x)+g(x)r dn{x) d^x)

Ja

l/r'

1/r <{j \f(x)rdrfx)} \f(x) + g{x)\r rdrfx)} 1/r1,T{J r 1/r < { j£ " \f(x)\rrd»(x)} dn{x)} { { £ £\f(x) \f(x) + g(x)\g(x)\ dn(x)} dKx)}^' '

1/r'

| p ( x ) r r f M ( x1/r ) } 1 / r { ^ |/(ar)+p(x)rd/i(x)} \g(x)\r d»(x)} d(*(x)} 1{/ r { £jf* \f(x) + g(x)\rr d^x)}^'.

+ {J + {j

r Dividing both members of the last relation by•{/ {/ a 6\fl/+#^} + g\r dfi}11/^', ', we obtain (2.9). The above proof is justifiable if the functions / and g are bounded. If the functions / and g are not bounded, we write Minkowski's inequality for the truncated functions mf and mg, and pass to the limit. The following discrete versions of above-mentioned inequalities are valid: a) Holder's inequality OO

OO

| 2E >u ** ww * | ^< (: E ( ll>u * r ) fc=0 fc=0 fc=0

j/

OO

( £EM^ r ' ))

fc=0

k=0

j#

/

( r > l ; l / r + l / r ' = l).

(2.10)

b) Minkowski's inequality OO

+v

*i

OO

-. i

OO

rr

d>* *r) <(^w (E\*k+v *r)) + ((En En ) k\ ) <(Ei fc=0 fc=0 fc=0

r

w

*/

1 fr^ (r^)-)-

2n ((2.11) -)

1.2. Some Topics from Function Theory of a Real Variable

9

In particular, if r = r' = 2 we have

2 1/2 1/2 1/2 |f> (f>i22))1/2 V f e | < (f>i ) (f>i li>H^(f><) -. k=0 k=0

fc=0 fc=0

(2 12)

-

(2.12)

fc=0 fc=0

The inequality (2.12) is called the Cauchy-Bunyakowskii inequality. Theorem 2.11 (Bernstein's inequality). If P(x) = YH=ockXh (ck € R 1 , k = 0 , 1 , . . . , n; x e[a.b]) and \P(x)\<M then \P'(x)\< |P'(*)|<<

(x€[a,6]),

Mn M n ^/{x a)(b ^ / ( x - a ) ( 6 -- xx) )

(a < * < & ) .

Theorem 2.12 (Chebyshev's inequality). Let f € L£ (1 < r < oo), and Ze£ EA = { x € R 1 , |/(x)| > A}. Then

KEx)<(Vh*J. In fact, this assertion follows from the inequality

rp{E \rv(Exx)< )< ([ \f\ \f\rrd(i< dn< II/II^. ||/||^. JEx

The Lebesgue points. The point x is said to be a Lebesgue point (Zr-Lebesgue point) of the Lrintegrable (r > 1) function f(x) if the condition Umijp dt == 00 Mmjjjf+ \f(t)-f(x)\ |/(t)-/(x)| prA

(2.13) (2.13)

holds. Lemma 2.13. If f(x) is a function Lr-integrable (r > 1) over [a, 6], t/ien almost all the points x € [a, 6] are Lebesgue points. Proof. Let us arrange the rational numbers into a sequence {pn} and let us denote by En the set of those points i G [a, b] at which

lim um1J fr+ |/(t) \f{t)- -p„rPndt\r=dt\f(x) = \f(- Xp)n\-r P»r

h->0 h->o h Jxx does not hold. Obviously En is a set of measure zero, since the integral fa !/(*) ~~ P n | r ^ is almost everywhere differentiable, its derivative being \f{x) -

10

Chapter 1.

Preliminaries

pn\r (Theorem 2.6). E = U^LiEn is therefore also a set of measure zero. We now show that all the points of the set [a, b] \ E are Lebesgue points of the function f(x). For a given e > 0 we determine pn so that \f(x) — pn\ < e/2. Then since xeE, for sufficiently small \h\by Minkowski's inequality (2.9)

x)rdt uU [[+h+hmm ~ /"( /(x)rdt ))

<<

m

pn]rdt

+h l/(<) pn]rdt UU/*IT ~ ~ )) +h +h + ({j* ({J* \f(x)-Pn\ \m-Pn\rrdtj dt\

<

€ € e e + = 2 + 22 =

£

i.e. x is actually a Lebesgue point of f{x). It is clear that every point of continuity of the function f(x) is a Lebesgue point. Let /JL(X) be a function, nondecreasing in [a, b]. For eachte\p(a),it(b)] £ € [/i(a),/x(6)] there exists at least one point x = /i _ 1 (*) f° r which H(x /z(x -— 0) < t < fi(x p,(x + 0) holds. The singular values of t for which the function //'(£) defined in this way is not unique, are characterized by the fact that /JL(X) takes the same value at every point of a subinterval [/i~ 1 (t),/I~ 1 (t)]. Two intervals of this kind, belonging to different singular values of £, have no common points. Since a set of intervals on a straight line, disjoint by pairs, can be denumerable at most, the set of the singular points is also denumerable, consequently it is a set of measure zero in the sense of Lebesgue. Thus p>~l(t) is defined at almost every point of t G [//(a), /i(6)]; the same holds for / [ M - 1 ( * ) ] if f{x) is defined at every point. The Lebesgue-Stieltjes integral of a function f(x) is defined by the formula rb

/ f(x)dfi(x)= f(x)d»(x)= //

y./x(6)

f[(i-\t)}dt; f[n-Ht)\dt;

/ € L* is true if and only if f[/J'~1(t)} is integrable in [/x(a),/x(fe)] in the sense of Lebesgue; f € L7^ holds if and only if |/[M - 1 (*)]| r is integrable in [//(a),//(&)] in the sense of Lebesgue. R e m a r k s . 1. We assume f(t) to be equal to the characteristic function of a set of //-measure zero. Then by Theorem 2.6 /Qx f(t) dfi(t) = 0; it follows from this that every set of //-measure zero is the union of a set of measure zero in the sense of Lebesgue with a set at the points of which p!(x) = 0. 2. If g{T) is the characteristic function of a set A of measure zero in the sense of Lebesgue, then (T = n(t))

JJgMt)]diJL{t) g[fi(t)}dfi(t) = Jg(T)dT = 0, and g[p>(t)] vanishes therefore almost everywhere, with respect to the /i-measure g[li(t)] being nothing else than the characteristic function of the image set /JL~1(A).

1.2. Some Topics horn Function Theory of a Real Variable

11

Lemma 2.14. Let F € ££ (r > 1), then FeL^(r> lim —r± —-^ /TX ^°Sx dii(t)Jx

h

\F(t) \F{t) - F(x)\ F{x)\rr dn{t) dfi(t) = =0 V W "

(2.14)

holds fi-almost everywhere. Proof. In Lemma 2.13 we put / ( r ) = F[/x _1 (r)]. By the transformation T = fi(t) (2.14) goes over into (2.13) and the set of measure zero in the sense of Lebesgue of the exceptional T-points is transformed by t = /x _1 (T) into a set of /z-measure zero of t-points (see remark 2). FeL^(r>l), Theorem 2.15. Let F e Z£ (r > 1), then llim i mi i j^f ++

r \F(t) -\F(t)-F(x)\ F(x)\r d»(t) =0 d»(t)=0

(2.15)

holds for almost every x. Proof. We divide the interval I over which / | F | r d/z is to be carried out into three subsets: let M\ be the set of the points x with /JL'(X) ^ 0, Mi the set at the points of which //(a?) = 0 and Ms the set of the points x for which //(#) does not exist. 1. By Lemma 2.14 and the Remark 1, (2.14) holds - in the sense of Lebesgue - almost everywhere for x G M\. In view of Theorem 2.5 lim lim

n(x HJx u(x ++ h) h) -— - fi(x) fi(x) u(x) ,, ; = = /a a/ (Ix) Ix) < 00 oo #) < h

holds almost everywhere on J D Mi, consequently (2.15) is satisfied almost everywhere on Mi (see remark 2). 2. F(x) is finite at almost all points and by Theorem 2.6 Hm \1 j f +

\F(t)\r d»(t) = J ; j f \F(t)\r d»{t) dp(t) = |F(z)|V(z) |F(*)|V(*) = 0

holds almost everywhere on Mi\ thus almost everywhere I

0
rx+h

\F(t)\F(t)-F(x)\ - F(x)\r rdp(t) dfi(t)

fX+fl

rx+h

< 2r lim - / \F(t)\r dfi + 2r\F(x)\r lim / dfi(t) r i*->oh,Jx r < 2 lim - / \F(t)\r dfi + 2r\F(x)\>*-*>Jx lim / dfi(t) r r x + h X) r r = 2 \F(x)\ lim rt l ^ = 2 \F{x)\ n'{x) = 0, = 2 r |F(x)| r lim M» + fe)-M») = 2r\F(x)\r»'(x) = 0, whence (2.15) is valid almost everywhere on M2, too. whence (2.15) is valid 2.5, almost M2,oftoo. 3. By Theorem theeverywhere set M3 is on a set measure zero in the sense of Lebesgue. 3. By Theorem 2.5, the set M3 is a set of measure zero in the sense of ^From 1,2 and 3 follows the assertion of Theorem 2.15. Lebesgue. ^From 1,2 and 3 follows the assertion of Theorem 2.15.

Chapter 1. Preliminaries

12 A p p r o x i m a t i o n by polynomials.

T h e o r e m 2.16 ( T h e o r e m of Weierstrass). A junction, continuous in a finite closed interval, can be approximated with a preassigned accuracy by polynomials. A function of a real variable which is continuous and has the period 2TT, can be approximated by trigonometric polynomials. For information concerning this theorem we refer to Zygmund, v.I, [1959]. Let u{5) be the modulus of continuity of a given function / ( x ) , continuous in the finite interval I = (a, 6): a;(/;S) := m a x | / ( x i ) - f(x 7, |xi | m -- xx22\\ << 6. 6. u(5) = Lj(f;6) / ( x2)\ eJ,xJ,2£x2 £ 7, 2 ) | for xx G If for a certain a > 0, u(S) < C5 C6a, where C doesn't depend on < 6,S, then we shall say that the function / satisfies Lipschitz's condition of order a in the interval (a, b) and denote / G A a . Only the case 0 < a < 1 is interesting, since if / G A a , a > 1, then cj(*)/tf u(6)/6 tends to zero as 6 -> 0, and d'(x) existst and d\x) = = for all x, i.e. / = const. We shall say that the function satisfies Dini-Lipschitz condition and write / 6 DL(7), D L ( 7 ) , if

«(/,*)<

Ini J

{5 ((5 -+ -> 0). (*-+0).

For an arbitrary a € (0,1], 7 > 1: DL(j) DL{-y) D A a . The following two theorems are due to Jackson D. T h e o r e m 2.17. If f(x) is a In-periodic continuous function with the continuity modulus CJ(/, S), then there exists a trigonometrical polynomial Tn(x) of order equal u(f, 5), at most to n with the aworoximation approximation nronertii property max \f(x)-T | / ( x ) -nT (x)\=0L r nn ( x ) | = o L ( // ;;; M M - J| . o<x<27r \ n II \ Proof. Let us consider the integrals 3 3

r atf.. F

si innnn((tt//22 ) \ 44 ^ ^..^ s/ftann(t/2)Y ,.

== Jn{x) > = 2xn(2n2 + +1)1)J_LT"wX /( *+ +l{x+t) >*> {'^m') \~^m~) 2<xn(2n* {^tJW) 2^^TT)J_J

MX) Jn(X

f o r n = 1 , 2 . . . . Since /sinn(*/2)\2 /sinn(t/2)V S=i^ fc cc [( sin(i/2) sinft/2) j) == nn + 22]^L( (nn- -f c* )0ccoo ss ff cctt

n#fJ = " + 22>~ )

(2.16) (2-16)

is a cosine polynomial of order ( n - 1 ) , the kernel of the integral Jn(x) and therefore Jn(x) itself too is a trigonometrical polynomial of order (2n - 2). Using (2.16), we

1.2. Some Topics from Function Theory of a Real Variable

13

obtain

£ ( W ) 4 *-£["♦'£<-*>«-*]'* n-l

\2n22 + 4 ^ ( n - fc) k)22]l = = -ir[2n 2/3(n- - l)n(2n l)n(2n- - 1)]. 1)]. = 7T [2n 7r[2n2 2++2/3(n k=i fc=i

So

J_ r / s i n n ( t / 2 ) \ 4 2TT / _. W„ V sin(i/2) sin(*/2) J

_ n(2n 2 + 1) ~ " 3

and thus

l/{+ +f) f{x)] 4 dt 5$?)' * 'J« <*> - '<«>" * s s ^ n y f"<* *■> -"'<*>' ( ?(W£) IJ»<*> - /wi * S ^ T T ) r

follows. Let us cover the interval [x-\t\,x+\t\] [x — \t\,x + \t\] by the minimal number N of congruent intervals of length < 1/n, having at most end points in common. Since the total length of the intervals is Nn'1 > 2\t\ and at the same time Nn'1 < 2\t\ + n _ 1 , N satisfies the inequality 2n|t|
|/(x /(x)| < ) < Nu>(f; 1/n) < (2n|t| + l)u,(/; 1/n). \f(x +1) - f(x)\ < J2 W Vkif) N«>(f\ Vn) fc=i

Thus we obtain the estimate .r / x

*/ M ./ x. ^^ M 6w(/;l/n) 6w(/;l/n) /"* / /ssinn(t/2)> si innnn( (t t/ /22) )\ \4 4 ^^ / ; i / n ) /"* 2 7r(2n + 1) , sin(ty2)

/ r/ r

, ,, ,x x

Let us divide the integral on the right-side into two parts, viz. into those extended over [0,7r/n] and [7r/n,7r]. Then it follows that

rr 7o

/sinn(t/2)\\ 44 __ ^4 4r7r" 7" /sinn(t/2 V sin(t/2) J ~ Jo

44

r d^ t 3

2 222

J*/n t

Hence | Jnn(x)-f{x)\<24u>(f;l/n) (x)-/(x)|<24u;(/;l/n) \J as J n (x) is a trigonometrical polynomial of order 2n—2, we have already proved our statement for even values of n. For n = 2m — 1 the best degree of approximation of the function f(x) by trigonometrical polynomials of order equal at most to 2m — 1

14

Chapter 1.

Preliminaries

is not greater than that by trigonometrical polynomials of order equal at most to 2m — 2; for these, however, the degree of approximation possesses the upper bound 1/n) < 24w(/; 2/(2m - 1)) < 48a;(/; 48u;(/; 1/n),

24CJ(/;

and this proves our theorem. Theorem 2.18. Let f(x) be a function, continuous in the finite interval [a,b] with the continuity modulus u(f;8). Then there exists a sequence of polynomials Pn(x) of degree equal at most n, for which max \f(x) 1/n)]. | / ( i ) --PPBn((x)\ x ) | = 0[w(/; 0[w(/;l/n)].

a<x
Proof. We may restrict ourselves to the case [a, b] = [—1,1], since the linear transformation u — (2x — b — a)/(b — a) carries the interval [a, b] over [—1,1] and f(u) has then the continuity modulus u;(/; 8(b — a)/2) = 0[cj(f] 8)]. Therefore, let us introduce the new variable v = arccosrr instead of x e [—1,1]. The function f*(v) = f(cosv) is then defined in [0,7r] and possesses a continuity modulus o;(/*; 8) < uj{f\ 8). Let us now suppose that v', v" G [0,7r] and \v" — v'\ < 8. Then we have \x" — x'\ = \ cosv" — cosv'\ < \v" — v'\ < 8, where, however, \x" — x'\ may not necessarily attain the bound 8. Thus «w(/';J)= (/•;*)=

sup

l^'-v'l^*

| / >V") )--//V V )) Il (f;6). |/(rc")-/(x')l=^(/;<J).

Since P^{v) — P n (cos v) is a polynomial in cosv of degree equal at most to n, thus being a trigonometrical polynomial in v of order equal at most to n, the coefficients of Pn(x) can be chosen so that P*(v) should be the trigonometrical polynomial, guaranteed in Theorem 2.17, satisfying the relation sup |/'(t;) | / » - P*( P n »V )|| = 0[u(T; 1/n)] = 0[w(/; 1/n)].

0
By transition from v t o x = cos v we then obtain the asserted relation. Remark. Since for the continuous function / we have u;(/; 8) —> 0 (as 8 —► 0), so the Weierstrass Theorem follows from the Jackson Theorem. The Gamma function. For a > 0, the Gamma function (r-function) is usually defined by the formula rT(a)= (a)= / Jo

e-xxxxaa~-ldx. e-

(This improper integral has meaning only for a > 0). It is not difficult to verifythat T(a) actually has the following properties: i. r ( i ) = 1; 2. T(a + 1) = aT(a) for any a;

15 15

1.2. Some Topics from Function Theory of a Real Variable Variable

3. T(a + 1) = a! for positive integral a. One can extended the function T(a) for all real a by the aid of the formula

rr((aa)) == 3*±!1. E<£±!>.

a Then T(a) (a G R) has the three properties listed above. Recall also Gauss's definition of r-function l1--22--33......((nn--l1)) r /, x r r (va ) = hm na—, T(a) —, --7 . ,— -r. ' n->oo n-^oo a(a a ( a 4+ l)(a 4+ 2 ) . . . ((a a 4 + nn -- l1)' ) Lemma 2.19. For fixed a the following formula T(n4a)

a

T _ / l \ l

(2.17) (2ir)

^ - 4 ♦"(-)]■ ~°°

holds. Proof. Denote the left-hand side of (2.17) by Tn(a). Using Stirling's formula, we derive In + (a (a -- 1/2) 1/2) In In aa -- aa + + -- ± ± -- — —-\| ,, In r(o) r(o) = = In In V2n V2n 4where 0 < 0 < 1, 2?i = 1/6 and B2 = 1/30 are the Bernoulli numbers, we obtain In T n (a) = InT(n In T(n 4 a) - InT(n) In T(n) = In y/2ir + (n + a- 1/2) ln(n 4- a) InT - (n + 4 a) - In V^r 4- 0(1/n) 0(1/n) v 7 ^ - (n - 1/2) Inn -14- nn 4= (n- 1/2) In

haln(n-+a) - a 4 0(1/n) n = (n-l/2) - + o[-L l n n - a + 0(l/n) 0(^0 Ij 4 + aalnn-a V n \n n J/ \ = lnn a + ln

I4CM-]

which proves (2.17). The Euler integral of the first kind B(p, q)= f (1- xY~lxq-1

dx

(p>0,q>

0)

1/0

can be expressed in terms of the Gamma function thus w „ ^ = r(p)r(
BM

T(£Tq)'

(2.18) (2 18)

'

Chapter 1. Preliminaries

16

Bessel Functions. The Bessel functions of the first kind of order a (a is arbitrary real) can be defined by k 2fe+ {-l) {x/2)2k+a (-i)*(*/2) ° "i^ o r(fc i)r(fc ++ aa ++ i)' i)r(* + + i)r(*

y>

J aW (J)

((2.19) 219)

This series converges very rapid for all x > 0, if a > 0, and the product xaJa(x) is an even integral function for a > —1/2. If a is not integer, Ja(x) and J_ a (x) are linearly independent. We note the special cases 1/2

JJ-i/2(x)

— JJ - V a ( * )== (( —

cosx cosx,

>

JJ

—j ) i1 // 22(x) W= = ((—

sinx. sinx.

The function (2.19) satisfies Bessel's differential equation y" x "V V + (1 - aa2 x~ x"22)y )y = 0, y = (x). y" + X" = J a(x).

(2.20) (2.20)

For nonnegative integral values of a we introduce the Bessel functions of the second kind

r„(^|^)(tof+7)-ig^^(I)-"+2" m=0 m=0 2

1 " ((-l)™(s/2)"+ - l ) ™ ( s / 2 ) " + ™/ ™/n^ 1 7T ^ n m=0 m=0

m!(n + m)! v '' v

™ lx

V ^ A ; fc=l fc=l

+

^ A J ' fc=l fc=l

where 7 = 0,57721566... is Euler's constant. This function is a second solution of (2.20). The following important asymptotic formulas are used in various applications

Ja{x) Ja{x) ==

+00 xx +00

/2 3 Ya{X)= Y a(x) = \ii)(jA' sm s i n ((zZ-?Z-l\+ - ? - j ) + 0 ( x0{x- 3 / 2'% ),

x -»• +oo. Z-++00.

+0 COS xX {^) (^k)C0S {( -T-i) -T-I)+0{^)' [xm)'

^^ ''

(2.21)

It may be useful to notice the order of magnitude of J a (x) and Ya(x) for x —> +0 J

^ = 2^TIj[1 +

^ W — 7TX ^ a YO{X)K *b(*)«

2

7T 7T

2

In-. In-. X X

0

«

0 = 11,2,3,... « '2-3'-

((2.22) 2 22)

'

17 17

1.2. Some Topics from Function Theory of a Real Variable

We have the following important formulas 2 sin a7r J'a{x) J.aa(x) - JL (x) ■ JJ'_aa(x)J (x)Jaa(x) (x) = H™£E, ^ ^ , 7TX nx

(2.23)

J' • J«(x) J'aa(x)Y (x)Yaa(x) (x) -- Y^x) Yi(x) ■ Ja(x) = = - — , aa==0,1,2,... 0,1,2,... 7TX

Differential equation. We shall make use of certain elementary Liouville formulas for the investigation of asymptotic properties of classical polynomials. Let {ui(0),u2(0)} be a fundamental system of homogeneous differential equation of the second order A{9)u" A(6)u" + + B(0)u' B(6)u' + C{0)u = = 0.

(2.24)

Then a nonhomogeneous equation A(0)u" + B(9)u' A(0)u" B(0)u' + + C(0)u C(9)u = = f{9) f(8) has a solution, which can be written in the form //»x ,/ix /m f ui(0)u UX{0)u2(t)-u U2(6)ui(t) fit) f(t) JJx 2(t) -2(0)uAt) u(8) = ciui(0) + c2u2(d) + / -77T7—7-7 T-TTT—j-r-7-r-r dt,

, uiC^W-u'aWtiiWACt)

(2.25)

where 0$ is a fixed and ci,c 2 are proper constants. Further, from (2.24) follows 'ui) + B(0)(t*iu B{0){u'1u22 - u' u'22ui) ui) = 0. A(0)(u"u2 - u2'ui) Consequently, we obtain the equation u[(e)u22(0) u'22(0)ui(0) = c3 exp - y | | | dt ,, ui(»)ti (») - u'

(2.26)

where C3 and #i are the constants. Summability of numerical series. Let; {un} (n G €Z Z+) + ) be a sequence of the real numbers and let sn (n € Z+) n n

*n = 5ZUfc fc=0 fc=0

(neZ+)

denotes the nth partial sum of a series 00

f>. fc=0

(2.27) (2.27)

Chapter 1.

18

Preliminaries

We consider a doubly infinite matrix

//a<°> 4 ° > a<°> a<0) ... a!°> a<0) . . . \ I

AA Q>Q

fli CL-t

...

&71 0>n

...

I

I I

Un Un

U-i CL-%

... ...

Oi JX I*?!

. . . . . .

I I

M M==

V v

■■• //

of numbers. With a sequence so, si, 5 2 , . . . we associate the sequence {crn} given by n)n) o-„ si«i++. .... .++a<,
Nn

n)

= £i4 i, fc=0 fc=0 fc=0

oo

An

= £4 n ) fc=0

exist (and are finite) for all n. The matrix M will be called T-regular if the following conditions are satisfied: (i) linin-^oo ai' — 0 for v € Z+; (ii) the iVn are bounded; (iii) lim n _,ooA n = 1. The finiteness of Nn implies the existence of An. T h e o r e m 2.20. If M is T-regular matrix, and if su tends to a finite limit, then an -» s. 5 + e6^, -> 0.0. Correspondingly,^ ncrn—=CTa' Let s5^ +a a'n, where v = s u, euu —> n n+ n 5 *'n = 5i4 sAnBI , <

< ', = e o 4 n ) + £e i a i n ) + . . .

Here a'n —> s by condition (iii). Let AT be the upper bound of Nn and let |£j,| < rj/2N for i/ > i/o> where 77 is any given positive number. Then K l < ( k o | l 4 n ) | + . .•.. + |e,J|a&>|) K | | a W | ) + (\a%>+1\ + |a<& 2a | + .....)»?/2JV. .)V/2N. The first sum on the right tends to 0 as n -> 00 (condition (i)), and so is less than I/277 for n > no. The remainder does not exceed M7/2JV Nrj/2N = 77/2. Hence \G„\ Kl <<77" forr n > no, cr^ — - »> 0,0, cr crnn-> -¥ 5,s, as desired. In future we usually consider the matrix A = {A< {Ainn)>,, kA; == 00, ,1l,,22, ,. . . ,. n, n, n, n++ 11;; AA00n)n) == 1,1, AA ^^ == 0,0, nn ee Z+} Z+}

(2.28)

1.2. Some Topics from Function Theory of a Real Variable

19

that is, A is a triangular matrix. In this case we denote for the series (2.27) n n) C/nn = ^tfnn(A) (A):=^Ai ix f c U := £ X^Uk

((n n €GZ Z+). +).

(2.29)

fc=0

If Un tends to a limit U we shall say that the sequence {un} is summable A (or, A-summable) to limit U. The Un are also called the linear means (A-means determined by the matrix A) of the {un} (n € Z+). One can rewrite the relation (2.29) by the aid of the classical Abel transform (summation by parts): n

nn—1 -1

X ] MfcUfc

A/XfcSfc - M m + l S m + V>nSn

fc=m+l fc=m+l

ib=m+l fc=m+l IS 7 H T 1

(2.30) (2.30)

( m = — 1 , 0 , 1 , . . . , n — 1, 5 _ i = 0 ) ,

where {/xn} (n € Z+) is an arbitrary sequence of the real numbers, and A/^fc is the first difference of {/xn} (n E Z+) (an empty sum is always defined to be zero). We have the following representation n

C/n(A) = X>Alt n ) S f e

(neZ+). (»€Z+).

(2.31)

Jfc=0 fc=0

In this case the conditions of T-regularity are the following (i) limn^oc Ajj.n) = 1 for k = 0 , 1 , . . . (ii)Efc=olMn)|
*-C: a ) * - ( " : ■ ) the nth coefficient of binomial series oo oo

11

X

5>" "=(l-*)i+° n=0

^

(°*-l,-2.-)

'

and by s" the nth coefficient of the power series 1-

OO

5 n ==

^ ^

n=0 n=0

oo °° x n' ^ZsSnXn

(1(T3^E - x)« - ' vv

''

nn == 00

(2.32)

(2-32)

Chapter 1.

20

Preliminaries

then the quotient

^ («€Z (nez ++) < == ;§ is called the n-th Cesaro mean of order a of the sequence {sn}> which we shall write in the abbreviated form (C, a)-mean. According to the product theorem of power series a£ may as well be written in the form

< = iitAn-1k°<<-

( 2 - 33 )

(2.33)

Summability (C,0) is an ordinary convergence. Summability (C, 1) _

1

1

'

» + i ; Jb=0 is called the arithmetic means. By definition of A% we have

A^^

ACt+P+1

n V ^

= ±AUtk _

ACL

A(3

fc=0

(2.34) (2-34)

for all a and p. In particular, replacing a by a - 1 and 0 by 0, we have

*s=itAk~i>

(2-35)

(2.35)

fc=0 fc=0

and, consequently, a

a

-1

^4 n _- ^J n - l—^ irl "1 . ^From (2.34) and (2.36) we obtain nn

£

1

<

n

fc=0 fc=0

^S-ik^fc.

i.e. in this case (n) (n)

^n-k

ff -- A<* ^ ^ \\

for 00 < < fc k< n for fc < n

00 otherwise. otherwise.

The formula (2.32) implies a

) . . . ( a + n) _ (/ n + a \ \ __ (a + l)(a + 22)...(q \ n ) n!

(2.36) (2-36)

1.2. Some Some Topics Topics from Function Theory of a Real Variable Variable

21 21

We infer that for a > — 1

i»g<=f;iog(i+2). iog4j = £k>g(i+!). ('+!)• We note that log(l + u) = u + 0(u2) for small u, and hence

(2(2.37) 3r)

w=th£°(*y

-

Therefore, denoting Euler's constant by 7, we have alogn| < 7 + o(l) 0(1) + I ff^o(±)\ | l o g ^ - «togn| > ( - ^ ) | < C, V VAC K X / l!

'fe=i fc=i

where C is a positive constant independent of n; hence, there exist two positive constants C\ and C2 such that Cix < ^ § < C2 C

(n > 1).

(2.38)

(We note that in the case a > —1 the numbers J4£ are positive). Consequently, A A

( n ) f -?=*■ ^ f

AA<«>

can be estimated as follows

= < K II vv

ffor o r0O<< A *: <
" "

0 0

(239)

(2.39)

otherwise otherwise

^ . „ ( f
a

°G)

n and by definition we have the following assertion Theorem 2.21. The (C, aj-summation process for a > — 1 is positive T-regular. It is useful to note that A% increasing (as a function of n) for a > 0 and decreasing for —1 < a < 0, and if a is a positive integer, then

n

( r)=

l))((n + + 2)...(n + a j «_ a _ (/ / nn-fa H+ aa \\ __ / (n n + a \ __ (n + + ll)(n a)) nn ) n } \ a! V * )/ ~ \V <* "" V <* )) ~ ~ <*«!-

AAan —

By Gauss's definition of T-function for a , a 7^ — 1, —2,..., we get

A

^f^T)

(^-i.-».-)-

(2.40)

Let a > — 1, (3 > 0. Then s" + / ? is the n t h expansion coefficient of the product

n sr-o«n* ET-o«»* 1 1 B

a

( l - x ) « {1-xY (1-x)" (l-x) (1-x)^

Chapter 1.

22 22

Preliminaries

and therefore by reason of the product theorem of power series n 6

n fc=0

fc=0 k=o

and, consequently a +/?

n n

1i

~ -JZ+0 l^t An-kAk°k An An fc=0 fc=0

•

n) By putting here AAi AA^.n) == (A^Z\A%)/(A% *) we easily infer that this A-mean is a (4:U?)/(^+(+^) positive T-regular matrix. So by Theorem 2.20 we have established the following assertion. Theorem 2.22. / / o sequence is summable (C,a), a > —1, to sum s, it is also summable (C, a + 8) to s, for every S > 0. 2. Voronoj-Norlund means. Let po,Pi,P2, • • • be a sequence of real numbers and

n

Pn = = Y,Pk^0 P„ £>*/() (n€Z+). (n€Z+)-

(2.41)

fc=0

Define the Voronoj-Norlund means by formula

or

W= A(n)

^iV-A: ±

W == AX A A(n)

Pr^k £^£

-* n

-» n

(fc == (fc

o0 , l , . . . , n„ ; n„ ee ZZ++))

(fc == 0 0, l, ,l ., . . ., ,nn; ;nn€€ZZ++)).. (fc

We denote Voronoj-Norlund method by (WN,pn). In particular, (C, a)-method is * (WN,pn) with pn = AZ-1. It is easy to see, that method {WN,pn) is T-regular, if the following conditions are satisfied: 1) limn_>oopn/Pn = 0, 2)Znk=0\Pk\=O(Pn). k=o\Pk\ = 0(Pn). In fact, by definition the method (WN,pn) is T-regular, if (i) limn^oo Pn-k/Pn = 1 (fc = 0,1,...),

(») E L o |AAln)| = 1/|P„| E L o |P»-*I = V\Pn\ E L o IPfcl = o(i)

are valid. Condition (ii) is equivalent to 2). Condition i) is equivalent to lim ^ = ^ = 0 (* = 0 , l , . . . , n ; n eG Z+). We show that this relation is equivalent to 1). In fact, denote lim - ^

n-Kx>

rn /^

ak = a*

(k = 0 , 1 , . . . , n; n € Z+).

1.2. Some Topics from Function Theory of a Real Variable Variable

23

It follows from the representation tion Pn-k

_ p Pn-k n-fc

Pn

P n -fe Pn-k

P n - 2 PPn-1 Pn-2 n-1

Pn-k Pn-k+1

Pn-1

Pn

that a,k = aa00(l - a00)fc because ,.

Pn-k

hm —— = ao, n->oo P n

Pn-k

r

,.

hm —

/(«..

Pn-fc+l Pn-k+1 \

= hm I 1 — —

n->oo P n _fc+i

n->oo ^

«..

= 1 — ao.

P n - f c + l Jy

So a 0 = 0, i.e. relation 1) holds. 3. Riesz's weighted means. Riesz's weighted means defined by A(n) = 1

_JVi

(fc = o , l , . . . , n ; n € Z + ; P - i = 0 )

•» n

or A AAi A ^n) = fg-

(* = 00, ,l l, ., ....,.n>;n ;nn 6€ ZZ+), +),

•* n

where P n is defined by (2.41). We denote this method by (P,p n )- In particular, (R, 1) is an arithmetic mean. Method {R,j>n) withppn n =={n+ (n +l)la) a- - nna,a , i.e. i.e.PPn n == (n(n+4- ll))aa, , is Zygmund's method of order a and denote (Za). In particular, (Z,1) (Z, 1) = (C,1). (C, 1). As above, from definition follows, that method («,*») (R,pn) is T-regular, if the following conditions are valid: 1) lrnin-.+oolPnl = +oo;

2)£Lob*l = o(Pn). 4. The Abel-Poisson summation process. A series 2 ^ L o u " *s s a ^ *° ^ e summable by the Poisson-Abel method (in abbreviated notation: A-summable), if X^Lo7*71^" converges for 0 < r < 1 and the finite limit OO lim

oo

>

rKUk = s

fc=0

exists, where r tends to 1 along the real axis. It is not difficult to see that oo

J2rkuk fc=0 fc=0

oo

= {1 -

r)J2rksk,

Chapter 1.

24

Preliminaries

so A-summability of a sequence {sk} (k G Z+) may be defined as the existence of oo

lim (1 -r)yV fe $fc. fc=o This method of summation plays an important role in the theory of analytic functions. Theorem 2.23. If the series (2.27) is (C,ct)-summable to sum s for an arbitrary a> — 1, it is also A-summable to s. Proof. Let {r n } (n € Z+) denote an increasing sequence of positive numbers less than 1 which tends to 1 from the left. We may obviously write f > f c r * = (1 - r n ) 1 + * f ) r * « J = (1 - r„) 1 + « f ^ g ^ r * . fc=0 fc=0 fc=0

The expression on the right is a linear mean of the sequence {erg'} (A; 6 Z+), and corresponds matrix with a^ = (1 — r n ) 1 + a r*j4£. This method is positive and satisfies conditions (i)-(iii). This proves the statement of our assertion.

1.3

Some Topics from Functional Analysis

Normed spaces. Examples. A linear (vector) set E is called a normed space if every element x € E has associated with it a real number ||x|| > 0 (the "length" of the vector x) called the norm of the element x (or \\X\\E) the following conditions {normed space axioms) being satisfied: (1) ||x|| = 0 is equivalent to x = 0 (6 is a null element); (2) ||Ax|| = |A|||a;|| (homogeneity of the norm); (3) ||x + 2/|| < ||x|| + ||y|| (triangle inequality). On writing for x, y e E p(x,y) = \\x-y\\

=

\\y-x\\

we reduce E to a metric space, that is if to any pair of elements x,y £ E there corresponds a real number p(x, y) (the distance between elements x and y) with the properties: (1) p(x, y) > 0; p(x, x) = 0 and if p(x, y) = 0, then x = y; (2) p(y,x) = p(x,y) (symmetry); (3) p(x, y) < p(x, z) + p(z, y) (the triangle inequality). In fact, p{x,y) — 0 implies ||x - y\ = 0, which is equivalent to the equation x - y = 0, this being in turn equivalent to x = y. It is clear from the definition

1.3. Some Topics from Functional Analysis

25 25

that we have symmetry for distance. Finally, the triangle inequality for distance is a simple consequence of the triangle inequality for norms: p(x,y) = = \\x ||x -- y\\ y\\ == ||(x ||(x -- z) z) ++ (z (z -- y)\\ y)\\ << \\x \\x -- z\\ z\\ ++ \\z \\z -- y\\ y\ == p{x,z) p(x,z) ++ p(z,y). p(z,y). The metric thus denned in the linear set obviously has two further properties in addition to the general properties mentioned: (1) transitivity (1) transitivity p(x + z,y + z) — p(x, y) (x, y 6 E) p(x + z, y + z) = p(x, y) (x, y € E) (distance is unchanged by displacement); (distance is unchanged by displacement); (2) homogeneity (2) homogeneityp(Xx, Ay) = Ap(x, y) ( A > 0 ;(\>0;x,yeE). x,y€E). So So

p(Ax, Ay) = Ap(x, y)

(A > 0; x, y e E).

||x|| = p(x,0).

||x|| = p(a;,0). p r0 of the space) p(xnn,,x0)0, = ||||x»n - zOil e ) if l>p 0 || -♦ 0. We write in this case x n -► x 0 or

, 7t™^Z(*7Z) -™ZiT^; ZeV:Z7^ n ZZ

lim o ror V 111 x||xn —!■ x l£o||£ lII L— n o►o u,r !oo rn5I — ,► #onnin-^oo x n -—xx x, or

00 : i{ }X 7H P •Tfn ? f thenormed rt Hspace £P is • sariHto, be h fundamental A rf , / r r r A r..!: T"f T } < ■ r r I T f I I { l l ' T n ? the normed space E is smd to be fundamental (or Cauchy) if ||x„ - z m || -> 0 (m,n -+ oo), i.e. rf ||xn - x m || < e for m,n > iST The normed space E» complete if everyfcmdamental fandamental sequence {*„} E « said to be compile is convergent i.e. if here exists a point x„ x 0 eE such that xx„ -+ xz.0_ Complete n -> spaces which are called spaces of type B or Banach spaces (after the name of the Polish mathematician S. Banach) play a particularly important role among normed spaces We shall indicate some examples.

1. It is evident that the Euclidean spaces with the usual definition of distance are Banach spaces. (-00 < oa < b < oo) of every function f(t), /(*), continuous in 2. The space C[a,b] (-co the interval [o, [a, 6] b] with withthe thenorm norm \f{t)\ | | / | | c = sup |/(t)| a
is obviously also Banach space. 3. The spaces L^a, b] with r > 1 are Banach spaces with the norm defined by (2.2) (the null element is a function f(t) is equal to zero /z-almost every where).

Chapter

26

Preliminaries 1. Preliminaries

In fact, the properties (1) and (2) are obvious, the triangle inequality 11/ + 9\\r,n < ||/||r,/x + ll#llr,/i follows from Minkowski's inequality (2.9). In order to show that L£ (r > 1) is a Banach space, we consider a Cauchy fundamental sequence {/n} (n = 1,2,...) from L£: ||/ n -/ m ||r,/i ->• 0 (n, m -+ oo). It is possible to select from {/n} a subsequence {fkn} with \\fkn—fkn+i \\r,n < 2~ n . In the case r > 1 Holder's inequality implies l/fcnW / W+1, W i ^ W << IIA n - /-f cAn n +,llr,M{ W } ^(r_1>/r ^ O= °^ Q )r ) '• Jj [ IA»(*) - -An (*)!**(*) ll/fc. 1 l k M {j^f ^rfM*)} i.e.

E // n=l'/a

i/iknW-/ikn + lW|dM*)<00.

If r = 1, this estimate follows immediately from \\fkn — fkn+i\\r,n < 2~ n . Therefore the existence /x-almost everywhere of the function

/(*) /W = E[An(*)-/*n £[/*»(*)-/w.W] +lW] n=l

is a consequence of B. Levi's Theorem 2.3 whenever. Moreover, on account of Minkowski's inequality (2.9) we obtain m

ii^= m ^ll£to»-/*« m—»oo II *—-^ "r,^t + >>l =l m-+oo II n*—-^ mn = l

1

"r,^t

oo -

m

oo

fc=l

n=l

fc=l

n=l

-

^ m i? 0O EII^-/w 1 IU<E^ = 1' / = f(t) is therefore a point of L£, and clearly ||/ — / n ||r,^ — ► 0, i.e. L£ is complete. The convergence in L£ (r > 1) is convergence in mean with the index r, i.e. 11/ — /n ||r,/x —> 0 (n —>• oo) is equivalent to 1/r

/ |/»(*) l / n ( t )-- /(*)| / ( t ) rP ddti(t)} / x ( t 1/T ) } -»• ^ 00( (n n - -> ^ o ooo). ). {{ J 4. We define the discrete space T (r > 1) the elements of which are all numer ical sequences (&,&»• • • >fn»• • •) such that the series YlkLi lfn|r a r e convergent. If we put for x = (&,&, ...,£„,...)

Ikllr \\x\\r=l =

f ( £ l & l r )11//rr f(£i^r) n=l

<

v

nn=l =l

' SUp|Cn| sup|f„| n n

11l
^ <°°

r = oo.

1.3. Some Topics from Functional Analysis

27

lr becomes a normed space (the triangle inequality in lr is in fact Minkowski's inequality (2.11)). It is easy to see that V is a Banach space. A subset D of a metric space E is said to be dense in the set Eo C E if there exists, for every x € Eo and e > 0, a point z in D such that p(x, z) < e. The metric (normed) space E is called separable if it has a denumerable dense subset. The space R n , C , L£ ( l < r < o o ) , Z r ( l < r < c o ) are separable. This is obvious for R n . We can take as our denumerable dense subset, for example, the set of points with rational coordinates. As regards C = C[a, 6], we can take as our denumerable set dense in C the set of all algebraic polynomials with rational coefficients. For, by Weierstrass's theorem 2.16, we can approximate any continuous function uniformly (i.e. with respect to the distance in C) by a polynomial, to any previously assigned accuracy. It is clear, moreover, that we can secure the rationality of the coefficients of the polynomial by varying it by an amount which can be as small as we please. Hence, given any continuous function x(£), there is a polynomial with rational coefficients as close as a denumerable dense set in the space C the aggregate of all piecewise linear functions whose graphs consist of open polygons with vertices at rational points. The space L£[a, b] is also separable. As an example of a denumerable dense subset of this space, we may take the class of functions which vanish outside a finite collection of bounded rational intervals, and take on each of those intervals a rational value. The space lr (1 < r < oo) is separable, since the denumerable set of sequences with rational coordinates that vanish as from a certain number is clearly dense in

r. Linear operators and functionals. Given two normed linear spaces E\ and E2, any mapping y = Tx = T{x) of a linear subset tt of E\ (possibly E\ itself) into E2 is called an operator (from ft C £1 to £ 2 ): T : Vx G ft -* Tx = y 6 E2. Q is called the domain of definition of operator A. The set of all elements in the form Tx — y is called the domain of values of the operator T. The operator T is said to be linear if T(axi + /3x2) = aTxi + f3Tx2

(a, /3 are scalar).

The operator T is said to be continuous at every point XQ € ft if Txn —> TXQ as x n -> xo- If T is continuous at every point of a set G C ft, we simply say that is continuous on G.

Chapter 1. Preliminaries Preliminaries

28

Lemma 3.1. A linear operator T on a normed linear space E\ is continuous if and only if there is a finite number M such that \\Tx\\ < ||Tx|| < M\\x\\ M||x||

for every x 6 EE\. x.

(3.1)

The sufficiency of the condition is obvious. To prove its necessity, suppose that the ratio ||Tx||/||x|| is unbounded. Then there is a sequence of points x n , xn T£ 0, such that ||Tx n || > n||x n ||. Multiplying xn by a suitable constant we may assume that ||x n || = 1/n. Thus xn —► 0, while the preceding inequality gives H^nll > 1) contradicting the continuity of T at x = 0. The linear operator T : E\ —> E^ is said to be bound on E\, if the relation (3.1) holds. Given a bounded linear operator mapping a normed linear space E\ into another normed linear space Ei, the number ||T|| := sup ||Tx||, \\T\\ IMI
In further we will often omit "bounded" and will write instead of "linear bounded" simply "linear". Examples. 1. We consider the following functional on the space C[a, b]:

Uu xx

n

( ) = YlCkX(tk^ fe=l

where £1, £2, • • • > *n is a system of points of the interval, ci, C2,..., c n are given numbers. U is a linear and

FII == XE >NI -. 11^11 ib=i

2. Let 1 < rr<
1.3. Some Topics from Functional Analysis

29 29

The linear functional in L^[o,6] is defined by U(x)=

f[ x(t)y(t)dfi(t) x{t)y(t)dn(t) Ja Ja when y is an arbitrary element from L£ [a, &]. In this case

\\u\\II ==llylk„Nk„. lltf The integral U(x) = J x(t)y(t) dt Ja is a linear functional in C[a, b] with \\U\\=

Ja

fb\y(t)\dt.

Dirichlet's integral 1

r

X{t)

= T, L/ >

, . .sm(2n s i n ( 2 n + l ) ( il)(t-to)/2 - t o ) / 2 ,, , _ (n Ge Z+; Z+; (n 2sin(t-t0)/2

2sin(A)/2

*

x

.

a

,

x

r

n

*o€€ [ - T , I T]] ))

*° " *"* <°

^*

is a special case of U. It represents the nth partial sum Sn(x'yto) of the Fourier expansion of x(t) at t — to. The norm of this linear functional is equal to ll«? II

1

* rf

s i n ( 2 n++ ll )) (( tt--*t o00 )) // 22 || |sin(2n dt d sin(t - 1 0 ) / 2

l*.ll*.-ft. = S^ j_ l*.ll*.-tt. j ^v I

ssmm( tt_ , 00) //22

|<

or by periodicity IK? II lie?

* r | sgi|sin(2n i nn (( 22 nn + ll)(*/2) )(t/2)| 27r J_n I sm(£/2) I sin(V2) X

\\Sn\\c2^C2ir = ^ J_^ |

sin(t/2)

I^

(3.2)

3. We consider the integral operator y = Tx: y(*) K(t,s)x(t)dt, y(s)= = /f K K(t,a)x(t)dt, Ja where K(t, s) is a continuous function. T is a bounded linear operator from C into C, whilst | | T | | = max / a<s
|A-(*,«)|*; |/f(t,5)|*;

and from L 1 into L 1 (the integral operator can also be considered as a linear operator from the space L 1 in L 1 ): | | TT| | =| = max m K jjff o
|/f(t,,)|«fa. \K(t,s)\ds.

Chapter 1.

30

Preliminaries

We shall consider the space C ^ consisting of all 27r-periodic functions defined on some interval [a, a + 2-K] and satisfying / ( a + 2-K) = f(a). Then operator y = Tx ra+2n

y(s) = = / Ja

K(t, s)x(t) dt

(s e[a,a + 2?r])

with continuous kernel K(t, s) will be a linear bounded operator CW into C. We suggest that the expression for the norm of T will be ||T||= \\T\\=

f + " \I"*"\K(t,s)\dt. K(t,s)\dt.

max a<s
Ja

Generally speaking, this operator transforms periodic functions into nonperiodic ones. A necessary and sufficient condition for T to be an operator from CW in C is evidently that the kernel K(t, s) be 27r-periodic with respect to its second variable, i.e. that K(t, s + 2-K) = if (£, s). If, finally, K(£, s) is defined throughout the plane and is also 27r-periodic in its first variable, the integration in the formula for norm of T can be carried out over any interval of length 2ir. T h e o r e m 3.2. Let E\ be a normed linear space and L a linear subspace of E\ dense in E\. Let T be a linear operator defined for x £ L, taking values from a Banach space E2 and satisfying the inequality \\Tx\\ < <M M\\x\\ ||Tx|| M

(x e(xeL). L).

(3.3)

Then T can be uniquely extended as a linear operator to the whole of E\ without increasing the M in (3.3). Proof. For let x* be a point of E\, {xn} a sequence of points from L such that Ik* — #nII -> 0, and un = Txn (n = 1,2,...). We have \\xm — xn\\ -» 0 and so, by (3.3), Htlm - U n | | - > 0 . The completeness of E2 implies the existence of a u* = lim un and we set u* = Tx*. The number u* is independent of the choice of {xn} —> x*. For if we take another sequence {x'n} —> x* and set u^ = limn_^oo Tx'n, the joint sequence {xJJ} = x\, Xi, X2, X2J . •. will also converge to x* and the number u" = limn-^oo Tx'^ will be equal to both w* and u'^. Hence u* = u*. The validity of (3.3) at the point x n , together with the relations ||x n || —► ||x*||, \\un\\ -> ||u*|| (consequences of xn —> x*, un —► u*), implies its validity at x*. Similarly, the validity of T(ax -h /3y) py) = aTx OLTX + + pTy /3Ty T(ax +

at the point x, y of L is preserved in Ei. The inequality (3.3) implies the continuity of T and there is at most one continuous extension of T from a set dense in E\. The sequence of linear operators play an important role in the convergence and divergence theory of orthogonal series. To investigate this role, we need the following concepts of the theory of point sets: a set is said to be the first category if it is the sum of a denumerable set of nowhere dense sets. In the opposite case it is said to be of the second category.

31

1.3. Some Topics from Functional Analysis Analysis

L e m m a 3.3 (Baire R . ) . In the Banach space E the complement CG = E\G of a set G of the first category is always of the second category. Proof. This is certainly the case if the complete space E itself cannot be the first category. This fact, well-known in the theory of point sets, may be proved as follows. Let F C E be an arbitrary set of the first category, i.e. F = U~ = 1 F n > F=U%LlFn, where the Fn are nowhere dense sets. Then their closed envelopes Fn are also nowhere dense. It must be shown that E \ UU^LiF %LlFnn is nonempty. First of all, in the nonempty open set E \ F\ there exists a sphere S(x\\p\) = {x G S E, E, \\x — #i|| < Pi}> constructed with a radius 0 < < p\ pi < 1 around the point x\ G E\FE\F\. V Since F2 is nowhere dense, there is in S(xi,p\) S(xi,pi) a point X2 and sphere S(x2,P2) = {x G E, \\x - X2II < P2}, S(x S(x2j2p2) ,P2) C C S(xi,pi) S(xi,pi) constructed with the radius 0 < P2 < 1/2 such that S{x2,P2) nI F2 F2 == 0. 0 . 0 such that

is uniformly sup

for all alln= n = 1,2,... 1,2,...

bounded, in other words, there is a constant

||T„a;||B ||T n x||E 22 < C for all n = 1 , 2 , . . .

l|x|| E l
Proof. We observe first of all that, if a bound for the values taken by a linear operator in some sphere is known

l\\Tx\\ | T x E2 | | £
(x (x€S(x (X€S(X 0;S)) 0;6))

an inequality can be written for its norm:

liril <
Chapter 1.

32

Preliminaries

For, we have, taking any x' with ||x'|| < 1, consequently, x = xo + Sx' € S(xo, S). So that ||Txo + <5Tx'|| B2 B2
imi < ^ .

imi < f . We turn to the proof of Theorem 3.5. Suppose the sequence {||Tn||} (n € N) be unbounded. We introduce the functional sup||T n x||. p(x) = sup||T„x||. n

It must be bounded in every sphere, since we have p(x) p(x) << Cm C in S(xo, S(XQ,6), J), we should have by above remark£ |||T | Tn||£; „ | | £2 2<< 22C/S C / ( 5for f o nr n== l1,2,... , 2 , . . . It follows from this that the set Gk Gk — = {x {x e€ Ei Ei :: p(x) p(x) >> k} is dense in E\. Further, the set Gk is open, since if xo eGk, i.e. p(xo) > k, we shall have ||T &forsome no, so that, in view l|T„x nxo||E 0 ||£,2 > k of continuity of ||Tnoxo||E2 we also have \\T ||T no £||£ > 2 x\\E > kA; for x sufficiently close to XQ. no 2 The set Gk, being open and dense in E\, is residue. By Corollary 3.4 n^ fl^Gfe =1 G fc iis residue and hence nonempty. If xXQ sup isup || = = o ooo contrary xGk, we have n ||T n^Gk, n ||T n x n0xo|| 0 6£ r\%L to the original assumption. Using the above arguments and Theorem 3.5 we have the following Resonance Theorem. Theorem 3.6 (principle of localization of a singularity). Under the hypotheses of Theorem 3.5 if sup||T su = oo, P ||r nn||||=oo, n

an element xo £ E\ can be found such that ssup||T u p | | T nnxx00| |||E E 2 2 =oo. =00. n

(3.4)

The set of elements, satisfying (3.4), is residue. Theorem 3.7 (Banach-Steinhaus). The necessary and sufficient conditions for the sequence {Tn} (n € N) of linear operators mapping the B-space E\ into Bspace E2 to be convergent on E\ to a linear operator T are that (i) the norms of the operators Tn be uniformly bounded |\\T | Tnn\\
(n€N);

(it) the sequence {Tnx} convergence on a set D dense in Ex.

Interpolation Theorems

33

Necessity. The first condition follows from Theorem 3.5, while the second is obvious. Sufficiency. We take an arbitrary x € E\ and choose xf e D such that |\\x-ABi | * - * ' | | * 1 <<£ e. For sufficiently large m, n |\\T | r mx' x ' - T n a-T ; 'n|x'\\ | B : E2 £. i <<e. Consequently ||T m x - Tnx\\E2 < \\Tmx' - Tnx'\\E2 + ||T m x - Tmxf\\E2 + ||Tnx - Tnx'\\E2 < e + (||Tm|| + ||T n ||)||x - x'\\El < < (2C + l)e. Thus in view of the completeness of the space Ei exists Tx — linin-Kx, Tnx. Obviously, T is a linear operator. Further, since l i m n - ^ ||Tn£||js2 = ||Ta;||£2 < 00, then we have sup n ||Tnx||£;2 < 00, and by Theorem 3.5, the sequence of the norms is uniformly bounded. Then ||Tx||£; 2 = lim HTnxlU^limsupllTnllllxll^, ||T n x|U 2
n->oo

from which follows ||T||oo n-»oo

Thus T is a linear bounded operator. Theorem 3.7 is completely proved. For Dirichlet's integral (3.2) the condition of boundedness is not satisfied (see Chapter V §1); hence there exist continuous functions whose Fourier trigonometrical expansions are divergent at a preassigned point (Du Bois-Reymond).

1.4

Interpolation Theorems. Estimates of t h e HardyLittlewood maximal function

In this section we shall prove Interpolation Theorems of Riesz-Thorin and Marcinkiewicz, and we shall apply Marcinkiewicz's theorem to the estimate of the Hardy-Littlewood maximal function M. Finally, we shall obtain the weighted norm inequalitites for the maximal function M.

Interpolation Theorems Before we shall prove of the Riesz-Thorin Theorem one more auxiliary is required, this time from complex variable theory. Theorem 4.1 (Phragmen-Lindelof's Principle). Suppose that f(z), z = x + iy, is continuous and bounded in the closed strip V: a < x < (3, and regular in the

34

Chapter 1. Preliminaries

interior ofV. If \f(z)\ < M on the lines x = a and x = (3, then \f(z)\ < M also in the interior ofV. Proof. Suppose first that (4.1) -+ 0 f(x + iy) -> as y —> ±oo uniformly in a < x < (3. If ZQ = XQ + iyo is in the interior of V, the inequality |/(zo)| ^ M follows from the classical maximum principle applied P, \y\ < 77, T), where v to / in the rectangle a < x < /?, 77 is so large that 77 > |yo| and \f(x ± ir})\ <Miora<x
= f(z)e(x2y-^y2/nVen2ixe2ixy = f(z)e^^n /n

(n € N),

which satisfies (4.1), and on the lines x = a and x = /3 does not exceed1 Me^ Me 7 2 //nn,, 1 2 nn where; 7 = max(|a|, |/?|). HenceB |/\U(zo)\ Me< /' ,, and ai on making n -> 00, we get n (^o)| ^< Me l/(*o)| < M. Theorem 4.2 (Three-line Theorem). Suppose that f(z) is continuous and bounded in the strip V, and regular in the interior of V, and that | / ( oa + tzyy))||< M i ,

\f(/3 + + iy)\<M2 1/09

for all y. Then for every ZQ = XQ + iyo in the interior of V we have M Hxo \f(xo + iyo) < iyo)\<M^ iyo)\<Ml:Xo) Mt MtL{xo \

where L(t) is the linear function taking the values 1 and 0 for t = a and t = (3 respectively. L(z) In fact, it is enough to observe that F(z) = f(z)/Mp /(z)/M 1z'M^~ M 2 1_L(z) satisfies the hypotheses of Theorem 4.1 with M = 1. Theorem 4.2 connects the upper bounds of | / | on the three lines x = a, x = XQ and x = (3. +00,; Lemma 4.3. Given any function f(x)j f(x)y a < <x < < b, and a numberr 1 < r < +oo we have ll/llr = sup| sup / f(x)g(x)dxl f(x)g(x)dx\, (4.2) 9 '• Ja '' Ja where where the the sup sup is is taken taken over over all all gg with with \\g\\r' \\g\\r' < < 1. 1. The The result result holds holds if if \\f\\ \\f\\rr = = 00. oo. Proof. We We may may suppose suppose that that ||/|| ||/|| rr > > 0. 0. Let Let IIgg denote denote the the integral integral on on the the right. right. Proof. By Holder's Holder's inequality inequality By |/<,l < < WfUgW* ll/llr||ff||r' < < ll/llrH/llr. \i,\ ll/llr < +00, A result true even in ||/|| r = 00. On the other hand, if ||/|| r < +00, we set ff0(x) ff0(x)

= | / ( x ) 1r 1 | ^ f o r r > l . = |/(x)r |5jMforr>l.

Interpolation

35

Theorems Theorems

We verify that (4.2) if ||^0||r' = 1, Igo = ll/llr- This proves (4.2) if ||/|| r is finite. If||/llr = +oo»we have to show that there exists a function g with \\g\\r' < 1 and such that Ig exists and is arbitrarily large. Suppose first that b — a < +oo, and let fm(x) denote the function equal to /(x), where \f(x)\ < m, and to 0 otherwise. Let gm(x) be derived from fm(x) in the same way as go was derived from / . If m is large enough, ||/ m || r is positive (and finite), so that || and I9=

f f9mdx= Ja

m m f f g dx fmgmdx=\\n\r,f.

Ja

is arbitrary large with m. If (a, 6) is infinite, (0, +oo) for instance, we define fm(x) as previously, but only in interval 0 < x < m, with / m (x) = 0 outside. This ensures the finiteness of ||/ m || r for every m, and the rest argument is unchanged. Let r = oo. The proof of \Ig\ < \\f\\r remains unchanged in this case. On the other hand, if M := ess sup |/|, and if 0 < M' < M, the set E of points, where |/| > M' is of positive measure. If we choose a subset E\ of E with 0 < \E\ \ < oo, then the function p, equal to sgn/(x)/|2?i| in E\ and to 0 elsewhere, has the property \\9\\r>= =ja\g\dx J\g\dx = =l, l, l9Ig= =j^J -±-J lldlr* E

l/ldx> >M',M\ l/ldx

so that sup \Ig\ > M. Lemma 4.3 is proved. The following generalization of Lemma 4.2 we shall use in the proof of the Riesz-Thorin Theorem. Lemma 4.4. Given any function f(x), a < x M ||/||r ,M = sup I / f{x)g(x)dn{x)\ f(x)g(x) dfJL{x)\ (1 < r < 00), oo), 9

*' Ja Ja

(4.3)

''

where the sup is taken over all simple functions g with \\g\\r',t* ^ 1/ T' is a conjugate index. In fact, the extension to Lebesgue-Stieltjes integrals does not require new ideas. That for / and g we can take simple functions is easily verified directly when r' = oo, and follows when r' < oo from the fact that S is dense in L£. Let R\ and R2 be two measure spaces with measures \i and v respectively. Let T be a mapping from R\ to R2. Then T is of type (r, 5) (strong type (r, s)) if \\Tf\\ M||/||PlM ||T/|U,„ s,v << M||/|| r , M, , where M does not depend on / . The least value of M here is the norm of the operator T (see §3). Similarly T is of weak type (r, s), if v{x \Tf(x)\ > a} < IC|l/l|r>M ^Mha

]j f,

hen sa << oo, when 00, W

Chapter 1. Preliminaries

36 36

where C does not depend on / or a, a > 0. We define weak-type (r, oo) as identical with strong type (r, oo). Hence T is of (weak, strong) type (r, oo) if ess sup \Tf\ < M||/|| r , M

(1 < r < oo).

By Chebyshev's inequality (Theorem 2.12), the notion of type (r,s) is stronger than the notion of weak-type (r, s). In future we sometimes to omit the index /x or v. Theorem 4.5 (M. Riesz-Thorin). Let R\ and R2 be two measure spaces with measures p, and v respectively. Let T be a linear operator defined for all fi-simple functions and ( l / a l f l / / ? i ) an junctions fj on R\. Suppose that T is simultaneously of type (l/ai,l//?i) (l/a 2 ,1/A2), i-e. that (l/a2,l/ft). \\Tf\\1l/f3l ||T/|| Afi||/|| 1/ftlf / / 3 l < Afi||/|| 1/aif

||T/||1/A < M M22||/|| ||/||11//aa22,,

(4.4) (4.4)

the points (c*i,/?i) and (0:2,#2) belonging to the square 0
00 < / ? < !l .

Then T is also of the type (1/a, 1/(3) for all a = (l-t)ai+ta (1 - t)ai2l+ ta2, and

/?=(!— t)px + +tfo 0 = (1 - t)0 tfo

(0
HT/IU/^ < Ml-tMtWfWv*. A# 1 1 - t A^||/|| 1/o . WTfh/0

(4.5) (4.5) (4.6) (4.6)

In particular, if a > 0 the operator T can be uniquely extended to the whole space Lja preserving (4-6). Proof. Wefixt in (4.5), and so also the numbers a,/?, and consider the functions a(z) = (1 -

Z)CLI

+ za2l

p(z) = (1 - z)0x + z02l 0(z)

which for z = 0, z = 1, z = t reduce to a\,P\\ a2,P2; <x,P, respectively. We consider z in the strip 0 < x < 1, which we call V. For any simple / , by (4.3) we obtain Tf.gdvl \\Tf\\ P V H 1/0 i A-s -sup s u p\[l / Tf-gdu\ 9

' JR JR22 JR.2

''

(ge S, (geS,

\\g\\l/(i-0) = 1). \\gh/(i-0) !)•

(4.7) (4-7)

*at |ll/H 1.:Fixing / and g, consider the integral We may suppose that | / | | 1/Q i / 0 = 1. 11== /f

JR JR22

Tfgdu.

,v We write / = |\f\e / | eiu, u,, g = \g\e \g\e™and i introduce the functions

F2z = l|/|/ al( z^>// aVe i t,t ,

(4.8) (4.8)

Interpolation

Theorems

37

Gz = | 5 |(i-0(*))/(i-/V*\

(4.9)

where we temporarily suppose that a > 0 and /3 < 1. The integral $(*) = /

TFzGzdu

(4.10)

reduces to / for z = t. If / = 0, Fz is to be taken as 0 whatever the value of the exponent; similarly for Gz. Thus, if ci, C2,... are the distinct nonzero values of / , and Xi> X2> • • • the characteristic functions of the sets where these values are taken, and if Cj = |CJ,\etUj, we have j

A similar formula, with exponents (1 — (3(z))/(l — 0) holds for Gz. If we replace Xj by TXJ in the sum defining F z , we get TFZ. Substituting this in (4.10), we see that $(z) is a finite linear combination, with constant coefficients, of exponential az with a > 0. In particular, $(z) is bounded in the strip V. Consider now any z whose real part is 0. The real part of ot(z) is then CL\. Holder's inequality applied to (4.10) by (4.4) gives |*(*)| < ||TF,|| 1 / A ||G,|| 1 / ( 1 _ A ) < Afi||F,|| 1 / 0 i ||G,|| 1 / ( 1 _ A ) .

(4.11)

But it follows from (4.8) that

\\Fzh,ai = HI/r'-Hi/a, = | | / | | ^ ° = l; the second equality being valid both for a\ > 0 and OL\ = 0. Similarly from (4.7) and (4.9)

iicji/p-*, = nipi (l - w/(1 -»iii/(i.« = y i i / 7 ^ ( 1 " w =

L

Hence \${z)\ < M\ for x — 0. Similarly, |$(z)| < M2 for x = 1. It follows from Theorem 4.2

|/| = |*(t)| <M\-*Ml

By (4.7), IIT/Hx/^ = sup^ |/| satisfy (4.6). The two exceptional cases a = 0 and /? = 1 cannot occur simultaneously. If /3 = 1, then /?i = ^ 2 = 1» and a = 0. We define Fz as before and set Gz = g; thus G2 is independent of z. The rest of the proof is simplified, since in (4.11) we may replace \\Gz\\1/{1.0l) by WgWy^^) = 1. If a = 0, and so c*i = a2 = 0, /? < 1, we set Fz = / and keep the old definition of Gz. (It may be also observed that if <*i = <*2> then (4.6) is a consequence of Holder's inequality). If T is initially defined for simple functions, and if 1 < r < 00, then there is a unique extension of it, as a linear operator and with the same value of constant in (4.6) to the whole LlJa (see Theorem 3.2).

Chapter 1. Preliminaries

38 38

It is sometimes convenient to be able to take Mi M\ = M 2 = 1. This we can siiif.a.Vilp rconstants, nn always do, if (3\ ^ fa we need only multiply T and dviy Kv by suitable while if OL\ ^ a 2 we may multiply / and d/j, by constants. Theorem 4.5 is proved. Let / be a /x-measurable function on any measure space (X, /x) and m(A) = M ( { xe€ X X ::| /\f(x)\ ( x ) | >>AA}) } ) (A > 0). »({x L e m m a 4 . 6 . If (X,/x) is a measure space, if f(x) // (X,M) 1 < r < oo, 00, the then /•OO /•oo

/

| / | pr

is fi-measurable,

and if

rOO roo

Arr dn(X)

(4.12) A rr - 11m(A)

d/i = - / | / | rf/i A d/j,(\) = =rr / A ~ m(A) dA. dX. (4.12) 7o JO Jo Jo Proof. If / is bounded, the first equation follows from the fact that the approxi mating Lebesgue sums of the first integral coincide with approximating Riemannm e secona. in m e general case, ior u > u Stieltjes sums of the second. In the general case, for u > 0

- u/Va>(A) -f \rdii{\)=l = f

r

r \f\ d» i/rd M

JO J{\f\
= =

{{

/ i ( x ) =/<*)-/<>(*)> = /(i)-/o(x), 0 0 if|/(x)|(*)>

ie.e. /o = / X X{{||//||>>77}}J. / 1i = /X{|/|<7}» / X { | / | < 7 } . *where x is a characteristic function. Then

r ri ri_r r pi \Mx)r-r rddrfx) yj \fi/o(*)i d»(X)==JJ \M*)\ i/o(x)n/o(*)ri/r ^<**», 0(x)r daw M(x) <<7 7- j Ji/r r\ — r < < 0. Similarly since n r r l\fi(x)rd^x)=j\f d^x)<^-rr\fi^ j\f(x)\rd^x), J\fi{x)rM*)1(x)\=\f1(x)p-f\fi{*)\

so / i e€ L Z P£1 and / 2 £ € £.!? LJ? with / = f/xi + + / f22. T h e o r e m 4 . 7 (Marcinkiewicz Interpolation T h e o r e m ) . Let(X,u)and(Y,v) Lei (X, /z) and (y, i/) 6e measure spaces, and let 1 < Ll + + L? LJ^1 £o < r\ r\ < < oo. oo. Suppose T is a mapping fromnLl U u-measurable functions such that

Interpolation

Theorems Theorems

39 39

(i) \T(f + g)(y)\ < \Tf{y)\ + |Tff(»)| \Tg{y)\ (subodditivity (suftoddiftvity condition); (H) v\{y \Tf(y)\ > A}| < ||/||iCb/A, (n) "l{y WfhCo/X, f/ € L\; (iii) (Hi) v\{y \Tf(y)\ > A}| < ( U / l k d / A y s /f € 1$; (when r\ = oo we assume instead of that dll/IU). IIT/Hoo < dH/Hoo). Then for 'or \1
/ 6 € !£, LI,

tuAere C r depends only on Co, Ci, r and n . Proof. Fix / € I£, 1 < r < ru andfixA > 0. Let E {y:\Tf(y)\>\}. Exx = {y : |T/(y)| > A}. We are going to get a tight trip on v(E\) and then use Lemma 4.6 to estimate ||/|| r . The clever Marcinkiewicz idea is to split / at \/2C\. Write, as above, / X { X : | |/(i)|>A/2Ci}. /(X)|>A/2CI}J / o = fX{x:

/X{x:|/(x)|
Then |T/(y)| \Tf(y)\ < \Tf |T/00(y)\ (y)| + + |T/i(y)|, IT/xfo)!, and ExcBxL>Z>A, BX = = {y (y)| > Bx {y- : |T/ \Tf00(y)\ > A/2},

where

£> A/2}. DXA = {y : |T/i(y)| > > A/2}.

Now by (ii) we have K AA ) <) < 22 %^/ |0| /| 0| |I | A/2Ci

\f\dii. \f\dn.

To estimate v(D\) we consider two cases. If r\ = oo, then ||/i||oo < < A/2Ci and by (iii) Dy = 0. (This explains the presence of C\ in the definition of /o and / i ) . If 7*1 < oo, we have by (iii)

y(Dx) < (\cx\\h\\r\ \A

<^ § ^ / y

A i

| / r «fc-

J\f\<X/2Ci

We bound v(E\) by v(B\) + v(D\) and use Lemma 4.6. The case r\ = oo is easier: We bound I/(I?A) by I/(^A) + I/(DA) and use Lemma 4.6. The case r\ = oo is easier: r 1 \f\df\d\ \\Tf\\r=r / rAr l- i/(£ A )dA < nr\r~U2^ /I l|T/||; = /o f°° r\ - v{Ex) d\ < Jof°° rA'-1 (\ 22 A^*^ J\f\>X/2Ci / J\f\>\/2Ci|/|dM |/| dfA) dX

JO

<2C0r

/•

/

/-2Ci|/|

JO

\f\ r2Ci\/\ Xr-2d\dfj, |/| /

\

A

J\f\>X/2d

r -r*£±rj\fr*H rnrr~lr

A r - 2 dAd/i = 2 r ° ° ^ 1

/ |/|rrf/x,

)

40

Chapter 1.

Preliminaries

because r — 2 > —1. Hence Tf € L£. If r\ < oo, we have the same thing plus an additional term bounding v(D\):

||T/||;< T r A ' - 1 ^ / JO

\

A

•/0

|/|d^rfA

<* y | / | > A / 2 d *

/

^|/|
The first integral we just estimated in the proof for r\ = oo. The second integral is

(2C{pr [\fp

f"

f\f\rdfi

\r-r>-U\d»=^^±

J J*Ci\f\ because r — n — 1 < — 1. Altogether this gives

ri-r

J

Iir/|| r < cP||/||P with \r - 1

r\—rj

which proves the Theorem 4.7.

Hardy-Littlewood Maximal Function If f(x) is locally integrable on R 1 , then by Theorem 2.6

lim

—!— f

f(t) dt = f(x)

(4.13)

for almost every x G R 1 . To make this theorem quantitative we replace the limit in (4.13) by the supremum, and we put the absolute value inside the integral as usual. Write |/| for the length of an interval J. The Hardy-Littlewood Maximal functions of / is

Mf(x) = sup ±- f\f(t)dt x€l \I\ Jl

for / locally integrable on R 1 . Now if / € L r , r > 1, then Mf(x) < oo almost everywhere. This follows from Theorem 2.6, but we shall soon see a different proof in Theorem 4.9 below. The proof of Theorem 4.9 will use a covering lemma of Vitali type. Lemma 4.8. Let /j, be a positive Borel measure on R 1 and let {I\t... ,In} be a finite family of open intervals in R 1 . There is a subfamily { J i , . . . , Jm} such that the Ji are pairwise disjoint and such that m

-

5>w)>2Ku?=i7>)-

Interpolation

Theorems

41

Proof. By induction { J i , . . . , J n } can be replaced by a subfamily of intervals such that no interval Ij is contained in the union of the others and such that the refined family has the same union as the original family. Write the Ij in the refined family as (oijyPj) and index them so that c*i < c*2 < ... < tonThen /3j+i > /3j since otherwise 7 J+ i C Ij and <*j+i > fij since otherwise Ij C Ij-\\Jlj+\. Therefore the even-numbered intervals and the odd-numbered intervals comprise pairwise disjoint subfamilies. Then

Y, Kb)* E^)<M(uy.i/i), jeven

j0dd

and for {J»} we take either the even-numbered intervals or the odd-numbered intervals, which gives us the larger sum. Lemma 4.8 is proved. The Hardy-Littlewood Maximal Theorem is this: Theorem 4.9. / / / 6 L r (R*), 1 < r < oo, then Mf(t) is finite almost everywhere. (a) Iff € L ^ R 1 ) , then Mf is weak Ll, |{t€R1:Af/(t)>A}|<|||/||1> (b) Iffe

Lr(Rl),

withKr

A>0.

< oo, then Mf e L r (R x ) and \\Mf\\r
where Cr depends only on r. Proof, (a) Let / £ L 1 and let A > 0. The set E\ = {t : Mf{t) > A} is open, and therefore measurable. For each t € E\ we have an open interval / containing t such that

|7f/ l/|rfs>A ' which is the same as

\I\<jJ\f\ds.

(4.14)

Let K be a compact subset of E\ and cover K by finitely many intervals J i , . . . , In that satisfy (4.14). Applying the lemma 4.8 gives us pairwise disjoint intervals Ji, J 2 , . . . , Jn that satisfy (4.14) such that m

u

J

2

| "=i ;|^ £l J >l3=1

Chapter 1.

42

Then

Preliminaries

\K\<\^ Ij\<2Y,\J \K\ <\u?l=1 IJ\<2^J j\f\ds<\J\f\ds. ^ \f\ds <j J \f\ds.

Letting \K\ increase to \E\\ gives us part (a). (b) In this case we apply Marcinkiewicz's Theorem 4.7 with (X, /x) = (Y, v) = (R^dx). The operator M clearly satisfies the subadditivity condition (i). Condition (ii) we proved as part (a). We take n = oo and condition (iii) holds with C\ = 1. By Theorem 4.7 we obtain part (b). It of course follows that Mf < oo almost everywhere. Theorem 4.9 is completely proved. In future we use the following variant of the Hardy-Littlewood Maximal Theorem. Theorem 4.10. Let \ibe a positive Borel measure on R 1 , finite on compact sets, and define M M /(x) = sup —r —- /I | / | d/j,. xei VK1) Ji Then

n({M M{MtlMf(x)>\})<jj\f\d /(x)>A})<~||/|rf/x fi and dfi
If l| //|rr d», dn,

1 < r < oo oo

(4.15) (4.15)

are valid, whenever f € L* or f G L7^, respectively. Remark. It is easy to see that the Lemma 2.14 follows from Theorem 4.10.

Weighted Norm Inequalities for Maximal Function Let 1 < r < oo and let \i be a positive Borel measure on R 1 , finite on compact sets. We consider the following problem. When is the Hardy-Littlewood maximal operator bounded on L£? That is, when is there a constant Cr such that T r T j\Mf\rdii
(4.16)

for all measurable functions /(#)? For solution this problem we use a decomposition lemma. Lemma 4.11. Let I be a bounded interval, let u £ Ll(I) and let

«>±jWdt. Then there is a finite or infinite sequence {Ij} ofpairwise disjoint open subintervals of I such that

Interpolation

Theorems Theorems

43

1) \u\ < a almost everywhere on I \ |J~ |J- Ij, Ij,

(4-17) (4.17)

Jj j\J\u\ dt < 2a,y 2) a < l/\Ij\ a
(4.18)

*;EiWI
(4.19)

Proof. We may assume J = (0,1). Partition I into two intervals UJQ = (0,1/2) and w\ = (1/2,1). For each interval u there are two cases 1/M Jt' \u\\u\dt\J 1 \u\u\u\dt>a. dt > a. Case (i) applies to the initial interval I by hypotheses. In case (i) we partition u into two disjoint (open) subintervals of length \u)\/2. For each of these two subintervals we apply case (i) or case (ii). When we get a case (i) interval we repeat the partition process. However, whenever we reach a case (ii) interval UJ we stop and put UJ in the sequence {Ij}, and we do not partition u. Since no interval in {Ij} is partitioned, the selected intervals Ij are pairwise disjoint. If x G I\Ulj, then every dyadic interval containing x is a case (i) interval. By the Theorem 2.6 this means \u(x)\ < a for almost every x € I \ Ujlj, and (4.17) holds. Each selected interval Ij is contained in a unique dyadic interval 7? such that |Ij\ = 2\Ij\. The larger interval /? was not selected and hence was a case (i) interval. Therefore a> [d,> Mdt \u\dt

w\W" -mh

i

and (4.18) holds. Because the Ij are pairwise disjoint case (ii) intervals, we have

EwsEs/j"1'"5;/1"1''' and 4.19 holds. Lemma 4.11 is proved. The solution of our problem is based on the following condition due to Muckenhoupt: weight UJ(X) satisfy (^4r) (1 < r < oo) (for brevity: u € (A-)), ^

(]7i f d')(wJr-"*-" ) (jTi /<*>~ ~ d'r *) <+c<"+ r{viJr sup

udx

1/(r 1)rf

00

If sup is taken over all intervals I C (a, 6), then u) G Ar(a, b). Let p is nonnegative function on (0, +oo) such that Cip(t) < p{2t) < C2p(t),

t > 0,

i,C2 <
x e R1.

(4.20) (4-2°)

44

Chapter 1. Preliminaries

Then the j4r-condition has a form

(/ Hence

1 r 1 r 1 w(t)
R>0. R>0.

\x\ (Arr)) *> - 1 < <&-l
Also, xaa G A r (0,1) (0, l ) <* < *- 1- l<Ja J b a

0
J\ - Ja

<+oo

J

whence our assertion follows. Similarly, whence our assertion follows. Similarly, (1 - x2)a G A r (-1,1) & - 1 < a < r - 1 ( 1 < r < oo). (1 - x2)a G A r (-1,1) & - 1 < a < r - 1 ( 1 < r < oo). In particular, In particular, (1 - x 2 ) - 1 / 2 G A r (-1,1) for all r, 1 < r < oo. (1 - x 2 )~ 1 / 2 G A r (-1,1) for all r, 1 < r < oo. Theorem 4.12. Let fi be a positive measure finite on compact sets, and let 1 < r < oo. Then the estimate (4-16) with Cr independent of f holds if and only if (i is absolutely continuous, dfx(x) = uj(x)dx, and the density u(x) satisfies Ar-condition (4-20). Before going further, let us try to understand how the (Ar) condition (4.20) comes about. If 1 0, and let V be an open neighborhood of E with fi(V \ E) < e. Then / = XV\E has / | / | r d\x < e. On the other hand, Mf(x) = 1 when x G E because \E\ = 0. Hence by hypothesis, r r r r ,jL(E)<J\Mf\ dn
so that fi(E) — 0 and /x is absolutely continuous. rf/x(x) == LJ(X) u(x) dx, dx, when UJ(X) G Ll and u>(x) Write dfi(x) u(x) >> 0.0.Fix an interval I and let f(x) = uj(x)Xai(x), xi(x), for some real a. Then f(x)=uj(x)« r

1+ra

l+ra j\f\rdfi=L(x) d^=L(x) j\f\ dx,dx,

and if x G /,

Mf fdtllfdt>±.]adt M«X)>±l„°dt.

^w\L -w\L" -

Interpolation

45

Theorems Theorems

Approximating f(x) from below by bounded functions, we obtain ( Iu;dx)( (l»° / \ j a dx d xy y (Judx)

r r < \I\rT f\Mf\ f \Mf\ dnd(* < C\I\r

1+ar1+ar fufu> dx,dx, dx, Iw1+ar

and takingj a = —l/(r - l / ( r -— 1) = 1 + cocr, we conclude that I/ / \ V(r-l) V(r-i) \ dx dx cmr dx jdx cmr dx Ur ){L\$ )^ '

Ur ){L\$

)^ '

which is the (Ar) condition. Lemma 4.13 proves. To show that (Ar) implies the boundedness of M on Z£, we need four consequences of (Precondition (4.20). The first two consequences are quite trivial. Lemma 4.14. If\ < r < oo and ifw(x) satisfies (Ar), then (a) u(x) e (As) for all s > r; 1 >), where r 1 = r/(r — 1). (b) the weight (l/a;) (l/u;) 1/^-!) /^ - 1 ) ,satisfies5 (Arr'), Proof. For (a) note that l/(s - 1) < l/(r - 1), so that by Holder's inequality

n

ui:r^mr-r~ 1/(3-1)

l/(r-l)

(r-l)/(s-l)

For (b) note that l / ( r - 1 ) = r ' - l , so that if v = ( l / o ; ) 1 ^ " 1 ) , then (l/v) 1 /^'" 1 ) = u. We proved Lemma 4.14. The other two consequences of (Ar) lie deeper, but they can be derived from the following inequality. Theorem 4.15. Let r > 1. Ifv(x) > 0, and if

{ml^TvrdxT ^iKKwJ, {w\L w\Lvdvdx-

(4 2l)

-

(4.21)

for all subintervals of some interval Jo, then

dI

iC{ a))

d

{ikf,/**) {wJ,/ )'" '' ^■"•' -"- {ml {wJ,:*)*) 1

vd

(i22) (4.22)

forr r < s < r + + 77, where w/iere 77 = 7?(r, 77(7*,K) if) >> 0. 0. The inequality reverse to (4.21), with constant 1, follows trivially from Holder's inequality. Therefore the constant K in (4.21) must obviously satisfy K > 1. Proof. We can suppose that 7IQ0 = [0,1] and that J/ /IQo vrr dx = 1. For A > 0 write Ex = {x € IIQ v(x) > > A}. A}. 0 : v(x)

46 46

Chapter 1. Preliminaries

What we are going to prove is the estimate j/

dx < rCX vvrrdx
JEx

with some constant C = C(r,K). (4.22). For s > r we have I/

v8sdx dx == f

J E\ JEi

(4.23)

A > 11,,

vdx, vdx,

JEx

But first let us observe how (4.23) easily gives

vrvs-rdx dx = = ((s-r)

J E\

vr(l/(s-r)+

f

= / vrdx + (s-r) = {s-r) JEx

J1

A55"-77*" *"1 1//

[ Jl

[

r 1 X*-Cx'-^dXJdx - dX dx) dx dx

f

J E\

r vvrdxdX. dxd\.

JEX

By (4.23) the last term in the above expression does not exceed C{s-r)

f Jl J\

Xs~2 fjI vdxdX
Hence

C(s-r)\

fj JEI JEi

Xa~22d\ dX C?-^- f vvs sdx \*iX = C?—^ dx. 5—1 S — 1 JJElEl

vdx vdx jf JO

)J^dx
vrdx.

X

Taking C > 1, we have C(s - r)/(s - 1) < 1 if s < r + (r - 1)/(C - 1) = r +17, 7/ > 0. For such values of s we then have 33 dx < II /I vv dx<

vvrrdx dx + +C Css [[

J{v
Jlo

dx. vvrrdx.

JEi JEI

r Because of (4.21) and the normalizationsJ \Io\ this provess (4.22) |io| = 1, 1> fj / / v dx = 1, this and the theorem. To prove (4.23) we set pP = = 2KX 2KX > A >> 1. 1 Since trivially

//

r

r l dx
JEX\E0 JE\\Ep

1

r 1 r 1

jf

r l r 1 vdx<{2K) vdx<(2K) - X-- X

-

JEX\E0 JEx\Ep

jf

vdx, vdx,

(4.24)

JEX JEx

proving (4.23) really amounts to making an estimate offffEEvrdx. vr dx. By the Lemma 4.11, there are pairwise disjoint subintervals {Ij} of IQ such that

< J_ / Pr<7^f F

If? Jij MilI Jij

r

r vrdx<20 VrdX<20

(4.25)

and such that:v<0i v < p almost everywhere on IQ \ Ulj. So except for a set of measure i0\uij.i zero, we haveiEp Ep C CUlj. Ulj. By (4.25) we have /I

J Ep JE P

r ry vvrrdxdx:<J2f dx<2P £\Ij\. < V\ Ivvrdx < WyVjl yJh j Jlj

j

y

(4.26)

Interpolation

Theorems Theorems

47 47

By (4.21) and the other inequality in (4.25) we also have

£

Tdx

vdx

"^{w\L: {ml/■*)"** T ^nL ml;*-This means that

i/^f/^^f^^+^i, ^jL;dx^jLE>vdx+TlIjl> lI

so that by our choice of /?, \Ii\ < \ ( \Ij\<\[

vdx. vdx.

Substituting this inequality in (4.26) gives f

JEp JEp

f

rr+11 r r r1 1 \ -vdx<2 + K X

f( vdx,

* JBx JE\ "

JEX JBx

and with (4.24) this yields the desired inequality (4.23). The proof of Lemma 4.15 is complete. Corollary 4.16. If I 0 and C > 0 such that, for any interval I, ,/

v 1/(1+5) 1/(1+4)

1+il s

f ±-JWx)) + (±J[u(x)} dx)dx)

< ^l"(x)dx <^Jw{x)dx

(4.27)

(Reverse Holder's inequality); (b) there is e > 0 such that u € (Ap-£). Proof. To prove part (a) we can assume that r > 2, because of Lemma 4.14 (a). The Holder inequality shows that

"(MGriw^+rWith this inequality (Ar) yields

s

1

jrJ.^^iwJr" m /— * (m "^)'"*)^ m • 1 -1) ^ 1 Sincei r — 1, we can now use Theorem 4.15 on the function>n V v= = u) u1l^rr~ to - 1> > 1 obtain

,< ,,i sc ,, (IJ^-^y-^l^f'-" (m^• '- *f (m/^^~

48

Chapter 1.

Preliminaries

for r - 1 < s < r - 1 + 77,77 > 0. Taking 1 + 8 = s/(r -1), we have (4.27). To prove part (b), we apply (4.27) to (l/uj)1^r~1\ which is a weight function satisfying condition (i4r/), r' = r/(r — 1). We get

( U/(=) / 1

*)

) / (rr"-1l)) r / x1 v\ ((1l++< *5 )/(

\

((rr - l1)) // (( l1+ +« 5 ))

"

*(a/er*r(

C

r / 1 \ V(r-l)

\

, r r-1 "1

(;)""""

Setting e = ((5/(1 + 6))(r - 1) > 0, so that (r - e) - 1 = (r - 1)/(1 + 8), and multiplying both sides of the above inequality by ( 1 / | / | fjUidx) now yields

(A/~)(A/(=r-"*r "(A/-)(A/(=r**p and oj has (A r _ e ). This gives (b) and we finished Proof of Theorem 4.12. We suppose u(x) has (Ar) and, writing d/j, = u(x) dx, we prove (4.16). The converse of this was proved in Lemma 4.13. Applying Holder's inequality to fu;1^ and uj~l/r and noting that r*/r = l/(r — 1), we have

m/l/"fas(m/l/H'"(m/(i)"<""^)1""rThe second factor can be estimated using (Ar) to yield

m/l/^£C(mi^)""r(m/l/r*)=c(i^)/l/r*)'"' since d/x = u(x) dx. Taking into supremum in the last inequality over all intervals I containing x, we get 1/r Mf{x) < C(M4\f\r))1/r. (4.28) Since v(x) G (A—e) the estimate (4.28) can be replaced by Mf{x)
Nl/(r-e)

Using this inequality and (4.15) with the exponent r/(r — e), we obtain

f |M/r dp < cr y"(MM(i/r-e)r/(r-£) ^ < c r J i/r ^, which proves Theorem 4.12.

Chapter 2

Orthogonal polynomials and their properties 2.1

Orthogonal systems in a Hilbert space

A vector space 7i is called an inner product (or scalar product) space if to each x, y 6 % there is a real number (x, y) € R 1 satisfying the following axioms: (i) (y,x) = (x,y) (Vx,y€H); (ii) (ax + £y,2) = a(x,z) + / % , z ) V x , y , z e U \ Q , ^ G R 1 ); (iii) (x, x) > 0 with equality if and only if x = 0, 0 is a zero element from W. Remark. One can introduce a complex inner product: x, y € H »-» (x, y) e C. 7n £/ws case property (i) (y,x) = (x,y), tu/iere the bar means as a complex-conjugate number. Throughout this book we consider real space only, and a case of complex will be consider separately. We prove the inequality: |(x,y)| 2 <(x,x)(y,y)

(Vx,y€?{). (Vx,y€^).

Proof. Fix x,y eH. Then, for all A G R 1 , by axiom (iii) 0 < (x + Ayix + Ay) = (y, y)A2 4- 2(x, y)A + (x, x). But, the quadratic in A will be nonnegative in A iff its discriminant is > 0, which condition is the asserted inequality. Remark. One can show that ||X|| = ||x||

y/^x) ^/(^x)

defines a norm on H.

49

Orthogonal Polynomials

50

Examples. 1. The space I2: oo

*=(6,6,-..,&,...) and

oo

(x,y) = ^ ^ n 7 / n ,

with

Y^£<0°

Kn€R1,n6N),

n=l

y = fai>*fc>---i^7n,...) fan e R 1 , n e N ) .

n=l

2. The space L 2 : x{t)eLl\a,b} x(*) € L^[a,6]

with

/ x2(t)dfi(t) < oo

and (x,y) = / x(t)y(t)dfi(t). Ja Remark. In the case of complex spaces I2 and L2^ one can define Remark. In the case of oo complex spaces I2 and L 2 one can define ! (x, y) = ^2 $nfin and (x, y) n=l Ja (^y) = $^fn^» and (x,y) =

x(t)y(t)dii{t), x(t)y(t)dfj,(t),

n=l ^ respectively. respectively. We shall also notice some simple propositions that refer especially to scalar We space: shall also notice some simple propositions that refer especially to scalar product product space: 1) continuity of a scalar product: xn -> x, yn -> y (n -> oo) in W implies 1) continuity of a scalar product: xn -> x, yn -> y (n -> oo) in W implies 1 (x„,y„) (xn,yn) -> (x,y) (x,y) in R ;

2) Hz ||x + y|| y\\2 + \\x ||x - y|| 2/||22 = 2(||rc|| 2(||x|| 2 + ||y|| ||2/||22) (Vx,y (Vx)2/ 6€ «H) ) (the parallelogram tity). identity). Let M be a linear set in 7i. Then n n

£(M) = {x; x = 2Z^kxk\ xi,X2,...,x n are arbitrary elements from M, £(M) = {x; x = ^AfcXfc; x i , x 2 , . . . , x Afc 6 R 1 , k = 1,2,...,n} Afc 6 R 1 , k = 1,2,...,n} is called the linear span of set M. The closure of the linear span of J£ is defined by n

£(M)

=

{xeH;

x=

lira V Ain)xfc, zfc € H,

n—»oo *—* *"

A£°

e€

1

R , * = l,2,...,n; n e N } .

2.1. Orthogonal Systems in a Hilbert Space Space

51

The system of elements {xa} G W is said to be complete in H, if £({x = «, H, Q }) = £({*«}) i.e. set of all linear combinations of the elements {xa} (a G A) is dense in H. The linear set %i C H is called linear subspace in 'H, if it is complete. The inner product space H is called a Hilbert space (after the German mathematician Hilbert D.) if it is infinite-dimensional complete and separable with respect to the metric p(x, y) = \\x ||x -- y\\ 2/|| == ^{x y/(x -y,x-y) -y,x-y)

(Vx.yeW). (Vx, y G H).

The spaces Z2 and L 2 [a, 6] are Hilbert spaces. Hilbert space is a direct generalization of Euclidean space, its "geometry" comes closer to Euclidean geometry, in particular, one can introduce the fundamental conception of orthogonal elements. Given x, y G H, we say x is orthogonal to y, denoted x_Ly, if (x, y) = 0. We say x G H is orthogonal to the set E C H, denoted x±E if x_Ly, Vy G E. We say the sets i?i, #2 CH are orthogonal if x±y, Vx G # i , Vy G i?2-

Basic Properties of Orthogonality 1. xJ_yi,2/2 implies x±(ayi + /?y2) (Va,/? G R 1 ); 2. x±y n and y n —> y (n —> oo) implies x±y; 3. x ± £ implies x_LX(£); 4. x ± £ , where E is complete set in H, i.e. if £(2£) = H, then x = 0; 5. Let EL = {x G %, x_L2?}, for fixed subset E oiH (E1- is called the orthogonal complement of the set E in H). Then, E1- is a linear subspace of H. The following theorem is fundamental in Hilbert space theory. Theorem 1.1. Let Hi be a linear subspace of a Hilbert space H and lefH.2 = %i> Then, every x G H can be uniquely expressed in the form /; x = x' x' + xx", ,

x' G € Hi, x' « i , x" x" eG H ft2.-

The vector x' is the one in Hi closest to x, i.e. ||x-y|| |\\x-x'\\=p{x,H{) | x - x ' | | = p ( x , H 1 ) == inf ||x-y|| y€Hi

and is called the projection of x on Hi, denoted x' — pr^x. Similarly: x = pr-H pr^i22x. x. {x n } (n G N) Proof. By the definition of p(x-,Hi) there exists a sequence {x„} N) in Hi such that 2 2 (nG | | x* - xxnn|||| 2<
Orthogonal Polynomials

52

We will prove that {x n } (n € N) is a Cauchy sequence in H, and that its limit x', guaranteed to exist in Hi by the completness of H and Hi in H, satisfies x' JLX" = x — xf. Now, by the parallelogram identity, ||x„-xm||2+||(x-x„) + (i-a:m)||2

=

2(||x-xn||2 + ||x-xm||2)

<

4p2(x.,Wl) + - 2 + - 2 -

Since ||(x - x m ) + (x - xn)f

= 4 jlx - ^ ± ^ | f > V ( * , « i ) («* ^ £ = 1

6

n^j ,

we have 2 2 lp22{x,H + -^2+4-4p ^ + 4 - 24p (x,'H1 1)) ||x-x < 4p (*,'Hx1) + (*,'H \\X-Xm\\m || vnr n* 2

= =

so

2 —r 2 + m2 —r +

vnr n* 2 — 22=: -> 0 (as n, m -» oo), n —=: -> 0 (as n, m -> co),

x' x' = = lim lim xx nn n—>oo n—>oo

exists and

\\x-x'\\=p(x;'H1). To see x" = x - x' € W2 = 'Hi', let y ^ 6, y € fti and observe that for any A € R 1 Ax,Hi)

< = < =

| | x - ( x ' + Ay)||2 = ||x"-Ay|| 2 (y,y)\2-2\(x",y) + (x",x") (y,j/)A 2 -2(x",j/)A + p2(x,'H1) p2(x,Hi)),

(as (x",x") = (x - x',x - x')

SO

0<-2(x",y)A + (y,y)A2. Taking A = (x",y)/(y,y)

yields -(x",y)2>0,

so(x",y) = 0.

Finally, the decomposition x = x' + x", x' e Hi, x" e H2 as unique. For, if x = x' + x", then x ' - x ' = x"-x", and ||x'-x'|| 2 = ||x"-x"|| 2 = (x'-x',x"-x") = 0, so x' = x1 and x" = x". This completes the proof of the theorem. A set of elements {x a } (a € A) C H is said to betotaZin %, if there is no nonzero element x eH such that (x, x a ) = 0 for all a € A.

2.1. Orthogonal Systems in a Hilbert Space

53

In particular, the set of functions {xn(t)} is total in L£, if from the relation

(n e Z+; t e [a, 6], - o o < a < b < oo}

f/ xxnn(t)x(t)d»(t) (t)x(t) d»(t) == 00 (ne (n e Z+) Z+)

./a Ja

follows x(t) = 0

fi /x — almost everywhere in [a, 6]. b}.

An example of a total system in L^[a, 6] (—oo (—oo << aa << 66<< oo) oo) isis the thesystem system { « » } (n ( » €€ ZZ+). +). {«»}

(1.1) (1.1)

In fact, let the function x(t) e L£[a,6] satisfy the equation n I/ x(t)t x(t)tn dfi(t) dfj,(t) = = 0, 0, Ja

(n (n eG ZZ++)). .

./a

Integrating by part, we have from from the the last equalities /

/./a

F(t)t F(t)tnndtdt = 0,

(ne6 Z+), Z+), (n

(1.2) (1.2)

Ja

where

F(t) = / Jo

x(r)dn(r).

By Weierstrass Theorem we find the polynomial P(t) such that max

te[a,b]

\F(t)-P(t)\<£.

Thus, using (1.2) and Schwartz-Bunyakowskii's inequality, we obtain Thus, using (1.2) and Schwartz-Bunyakowskii's inequality, we obtain 2 F22{t)dt /" F (t) dt :=

f F (t)dt Ja Ja

=

f F(t)[F(t) - P(t)] dt<e f \F(t)\ dt

f F(t)[F(t)-P(t)]dt<e Ja Ja 2

< eVb^i{ f F (t)dt}1/2

f Ja Ja

\F(t)\dt

Ja whence

rb

/ F2{t)dt F2(t)dt<e2(b-a). Ja Since e > 0 was arbitrary, then F(t) = 0, consequently, x(t) = 0 /x-almost everywhere. Corollary 1.2. A necessary and sufficient condition for a system {xa} (a € A) of elements to be complete in the space ri is that {xa} (a e A) is total.

Orthogonal Polynomials

54

Proof. The necessity of the condition follows from the basic property (4) of orthogonality. To see totality implies completness suppose, if possible {x a } (a G A) is noncomplete. Then £({x a}) -= Hi C H, so by Theorem 1.1 there exists an £({*«}) x G H \Hi, x ^ 0 with x = x' + x", x' G Hi, and x" ^ 0, i.e. x"'±xa for all a G A and x" ^ 0 - a contradiction. The elements ^xi,X2,... 1 ) %2 j • • • ,x j ^ nn G H are said to be linearly independent if a relationship of the form n

71

y^AfeXfe XA: = 0 ]]TA P fcAfcXfc

(AfcGR1; k = = 1,2,.. l,2,...,n)

k=l fc=l

is only possible when Ai = A2 = . . . = An = 0. Otherwise, the elements xi, X2, • •., xn are linearly dependent. An infinite system of elements is said to be linearly independent if every finite set of different elements of the system is linearly independent. An example of a linearly independent system in the space L^[a, b] (—00 < a < b < 00) is the system (1.1). The Gram matrix of a system of elements;s xi,X2i... x\, X2,...,x,x space I n n of a Hilbert H is the matrix {ui^ij^i, G = {uij}ij=i, {w»j}t,j=i,

where Uij Uij == (xi,Xj) (xi,Xj) (i,j Uy fax,) ((i,j t , j == 1,2 l , 2 .... . . , ,n). n).

The Gram determinant of a system X2,..., stem xi, £i,#2, • • •x>n xn(xk fakG€H,11,k k==1,2,...,n) 1,2,... , n) is the determinant of the Gram matrix G A n ==

A n

I (x\,xi) ((zi,£i) zi,£i) (x2,xi)

({xi,x x i , x2)2 ) (^1,^2) (x2,x2)

.... .. ••• ...

({xi,x x i , xnn))n ) I {xi,x (x2,xn)

\(X2,X1)

(X2,X2)

...

(X2,xn)\

I (xn,Xl)

{Xn,X (xn,X2)2)

...

(x n ,X n ) I

( n = l , 2 ,. ..... ), AAo0 == 1l). ).

{ n = 1 ) 2 ;

I (xn,Xl) {xn,X2) ... (x n ,X n ) I Now we get a criterion of the linearly independent of a system of elements of a Hilbert space. tern xi, xi,xx22, ,. . . ., ,cH, xxnn CC7i, %, 1 11
]y]P aiXi otiXi ==90 and i=l

n

^ J ai

>

0-

ii=l =l

Multiplying of both sides by Xj, we get that the numbers ctj is a solution of the following homogeneous system of equations n

^2aj(xi,xjj(x ) i,xj= 0 0(j = "%2a )=

j=i 3=1

ll,,22 , . . . ,, nn))..

(1.3)

2.1. Orthogonal Systems in a Hilbert Space

55

This system has a nontrivial solution, so A n = 0. Conversely, if A n = 0, then the system (1.3) has nontrivial solution a i , civ, • • • , ctn. Put n n

/ = 53aia?»One can rewrite the system (1.3) in the form ( / , * i ) = 0 (j (/,*i)=0 0 == 1,2,...,n). l,2,...,n). Multiplying these relations by a i , a 2 , . . . , a n and summing we obtain ( / , / ) = 0, (/,/) i.e. / = 0, and therefore, the system x i , X 2 , . . . , x n is a linearly dependent. The Lemma is proved. A system {xn} (n 6 N ) of nonzero elements of a Hilbert space ri is said to be orthogonal if any two distinct elements of the system are orthogonal: (xnnj, xmx)m) = 0

(n ^ m; n, m € N). (n^m; n,m€

If, in addition, the norm of every element xn is unit, the system is called orthonormal, that is, for orthonormal system Snmnm ((xxnn,Xrn) , xm) = <$

(n,m (n, m €e N ) ,

where 8nm is Kronecker's symbol. If {xn} (n e N ) is an orthogonal system, then one can obtain an orthonormal system by dividing each element by its norm

f_x» \

{IIWIJ■} \\\Xn\\)

(n € N ) .

• M Note, that any orthogonal1 system is linearly independent. In fact, if J.

n

y^AfcXfe 53 *kXk= =^,0, fc=l fc=i 2

2 then forming the scalar product of both sides with Xk would give us;AAfc||#fc|| 0 fc ||x fc || = o accordance with our assertion. A total orthonormal system in a Hilbert space ri is called an orthonormal basis in ri. T h e o r e m 1.4 (Orthogonalization T h e o r e m ) . Let { x n } (n € N ) be sequence of linearly independent elements in a Hilbert space H. Then there exists an orthonormal sequence {
56

Orthogonal

Polynomials

(i) every element ipk ipk is is a a linear linear combination combination of system (i) every element of elements elements of of an an initial initial system %li ^2j • • • > %ni

(ii) every element Xk (k element element Xk Xk (k (k Proof. Put Proof. Put I (Zl>#l) (xi,X2) I (Zl>#l) (X\,X2)

(ii) every (ii) every l,
,J = A fc=

^

= 1 , 2 , . . . , n ) is expressible linearly in terms of = 1,2,...,n) 1 , 2 , . . . , n ) is is expressible expressible linearly linearly in in terms terms of of ... ...

(xi,Xk-l)xi (xi,Xk-l)xi

I I

(x2,Xi) (^2,^1)

(X2,X2) (^2,^2)

... ...

(X2,Xjk_i)X2 {x2,Xk-i)x2

I (Zfc,Zl) I (Zfc,Zl)

(Xfc,X 2 ) (Xfc,X 2 )

... ...

(Xfc,Xfc_i)Xfc I (Xfc,Xfc_i)Xfc I

/ fc = 2 , 3 , . . . \

( k = 2,3,... \

I ri = «i )■

(1.4).

(

(1A>

We are taking the latter determinant as the sum of the products of elements of the last column with the corresponding algebraic complements. So,

*4 = 2 > ( f c ) * ,

(1.5) (1-5)

i=l

with AJk' = Afc_i ^ 0 by Lemma 1.3, i.e. assertion (i) is valid. Hence the assertion (ii) follows from it. The system {y^} (A; = 1,2,...) is orthogonal, as, by definition, ( v 4 , S i )== 0o (Zi) Consequently

(i (z = ll ,,22,,. .. . ., ,f *c - l! ))..

(1.6)

(v*.vi) = o (i(« == l1, ,22, ,...... ,, ** -- l! )) .. (*>fcVi)=0

Set y/fc =

iiKir* ^i

(* = 1,2,...).

Then {*>**)Afe

(fc = l , 2 , . . . , n ) (*

and, therefore, on taking into account (1.5) and (1.6), we arrive at the relationship fe) 0<(v' Afc = A fc^_A 0 < ( f*e ,v' 4 Xf c)) = Aifc) 1 A*fe

But A 1i = ( x 1i , x i ) > 0 ,

(fc (* = l , 2 , . . . ,, nn )) ..

(1.7) (1.7)

2.1. Orthogonal Systems in a Hilbert Space

57

so A* > 0 (k = 1 , 2 , . . . , n ) , in accordance with our assertion. Corollary 1.6. The orthonormal system $ = {<£*} (k e N ) has the following representation 1*
I (#i>zi) 0&i,xi) ((z2,a;i) x2,xi) •••

((^1,^2) xi,x2) (x2,x2)

... ...

(xi,Xfc_i)xi I ( x 2 ,x , x ffcc_i)x -i)x2 (x

I (Z ^fc,Xi)

(x*,X2)

...

(Xfc,Xfc_i)x (Xfc,Xfc_i)Xfc f c I|

Corollary 1.7. Linear span generated by elements { x n } (n € N ) coincides with a linear span C({
6e a representation of a normalized system. If we require AJ^ > 0 ; then {
system

k

= £.

vl =

tt=i = iL t=i

satisfying the stated conditions. Putting t. = Vk

x(fc) k

A

^r

*

we obtain, that f can be express in terms x i , x 2 , . . . ,Xfc_i, and, therefore, is orthogonal one's self, i.e. f = 6 and <£* = &/?£. Since ||y?fc|| = || 0 and /zj^ > 0. Corollary 1.8 is proved. T h e o r e m 1.9. Every Hilbert space H has a countable orthonormal basis. Proof. Let ^ l1 ,, ^^ i2- ,. .- ., .^, n^ i, . . -.

be a countable dense subset in ri. Then a complete set of linearly independent elements / i , / 2 , . . . , / n , . . . can be selected from {tpn} (n 6 N ) . In fact, we need only eliminate from the sequence {i/jn} (n € N ) all elements tpk which can be written as linear combinations of elements ipi with smaller indices (i < k). Applying the orthogonalization process to / i , / 2 , . . . , / n , . . . we get an orthonormal basis. Examples. 1. The system {e n } (n € N ) , en = ( 0 , 0 , . . . , 0 , 1 , 0 , . . . ) is an orthogonal system in I2. The system is clearly complete.

Orthogonal

58

Polynomials

2. A finite or denumerably infinite system $ = {ipn(x)} (n € Z+; x G [a, 6]) of L^[a, 6] - integrable functions is said to be orthogonal (briefly: O.R.S.) with respect to the measure /x(x), or with respect to the distribution d/x(x), if n G eZ+) (
/ ip2n(x)dfi(x) = l Ja

( n€e ZZ+) +) (n

are also fulfilled. So, the system $ = {ipn(x)} (n. G Z+; x G [a, 6]) is orthonormal O.N.S.), if (n,m (
(briefly:

where £ n m is Kronecker's symbol. Trigonometrical system {1, cos no:, nx, sin na:} nx}

(n (nG € NN))

is O.R.S. in L\^\ and

{iv^F' 7^'WTirj "TIT'' 1

cos nx isin cosnx s i nnx n x l1

, _TX (n€N) (nGN)

is a total O.N.S. in L\^. Orthogonalization process. Let the real functions / o ( xz ) , / i (W x ), ., " . .>. , /fm(x) m(x)

(TO (m is finite or infinite)

be a system of L^-integrable functions and linearly independent. Then an orthonormal set
= EAin)/*(*),

A ^ > 0.

fc=0

The set $ = {ipn} (n G Z+) is uniquely determined. Moreover, for the orthonormal functions
Elementary Properties of Orthogonal

Polynomials

59

where, for n > 1

Gn(x)

=

I (/o,/o) (/0,/0) /o) I (if/ Ii ,Jo)

((/o,/l) /0,/l) (/l,/l) iflJl)

(/„,/l) | (/„,/0) (/»,/o) (fnjl)

...

( / 0o , / n - l ) / 0o ( x ) I

/l,/»-l)/l(x) ••• ((/l,/»-l)/l(x)

••• (A,/»-l)/»(x) I|

(fiJj) = J fiWfiMM*)

/

n = l,2,...

\

v G0(x)=/0(x);'

(»,i = 0,l,...,n)

(1.9) (1.10)

and, for n > 0, ^ n = [(/t,/j)kj=0,l,...,n>0

(1.11)

are Gram's determinant. We write JD_I = 0. The procedure of deriving system {
= 5Z a i n Vfc(^)

(0 < m nn < m nn + i ) ,

fc=0

such that { S m „ } converges to f(x) in the sense of L^-metric. It follows from Corollary 1.7, if the system { / n } (n € Z+) is complete in L^[a, 6], the O.N.S. {»)= f(t)
2.2

Elementary properties of the orthogonal polynomials

Definition of the system of the orthogonal polynomials We suppose that all measures /z (or all distributions d/i) occuring in this book are positive, bounded, monotone increasing functions with infinitely many points of increase in the finite interval [a, 6], where derivative /z'(x) > 0 vanishes at most in a set of measure zero (in the sense of Lebesgue). In this case we may denote

Orthogonal Polynomials

60

by Ho{x) the absolutely continuous part of /z(x). If E is an arbitrary set with a /x-measure n(E) = 0, then we have 0 = n(E) = I dfi> I dtio(x) = / /io(x)dx. ii'o(x)dx. JE

JE

JE

Since on E /i'(x) = Mo(x) > 0 holds almost everywhere, the last integral cannot vanish unless E is a set of measure zero also in the sense of Lebesgue. Therefore the sets of measure zero are at the same time sets of measure zero in the sense of Lebesgue also. For this reason in what follows we may by the term "almost everywhere" mean simply "in the interval (a, b) everywhere with the exception at most a set of measure zero in the sense of Lebesgue". Lemma 2.1. Let /z(x) be a measure and II(x) be a not identically vanishing polynomial, taking nonnegative values for x £ [a, b]. Then I II(x) U(x) dn(x) > 0. Ja Proof. Since II(x) can have only a finite number of zeros and suppd/z (i.e. the set of points of increase of /x(x)) is an infinite set, there exists a point xo £ [a, b] with n(xo) > 0; then there also exists an interval [xi,X2], containing xo such that U(x) > l/2II(xo) holds for x £ [xi,x 2 ]. Then

n(x)<*/x(x) >> f/ 22 n(x)d/z(x) n(x)d/z(x) >> in(x in(x00)[Mx )[Mx22)) -- M*i)l M*i)l >> °> °> f/ u{x)dtA(x) in accordance with our statement. Remark. It follows also that if II(x) takes nonpositive values for x £ [a, b] and if it does not vanish identically, then I n(x) dfi(x) < o0

Ja

holds. Lemma 2.2. If in the system of (n + 1) polynomials F 0(x),Fi{x),...,F (aO,Fi(z),...,F n(x) B (as) every polynomial Fk(x) (k = 0 , 1 , . . . ,n) has a degree k, then an arbitrary polynomial Qn(x) (deg Qn(x) = n) can be uniquely represented in the form n X Qn(x) = ^^A kF (x). Qn(a;) f ckF f c (x).

fc=0

(2.1)

Elementary Properties of Orthogonal

Polynomials

61

Proof. Denote Fk(x)

=

aJ fc) +o ( 1 fc) » + . . . + o|!fc,xfcl

Q„(x)

=

a 0 + a i xx 4+ . . . + a n x n ,

4fc)^0,

a n ^ 0.

One can obtain the following system of equations for the determination of unknown coefficients A*: ( an — — \Anan(n> , I Q"n — A n a n j ^ . ! ', I a n _ i = Ana^^! \ («) 4. A \ n - i a (n-1) \ (*0 4-. A \ n - i a (n-1) (n-2) a n _ i = Ana^^! ^ . ! ', , \
I a0 = A n 4 + Xn-i4 ~

+ • • • + Ao4 -

The determinant of this system is equal to ) I) W) «i* *o. a ^ a«$ri ^ . - ....«. a ^ ^ O .

So, numbers {Afc} (k = 0 , 1 , 2 , . . . ) in the relation (2.1) are uniquely defined. Lemma 2.2 is stated. Using results of the preceeding section and the above Lemma we can prove T h e o r e m 2.3. Given a measure /x (or distribution d\i) as described above, there exists a unique sequence of polynomials Po(dfi; x),pi(d/i; x),pi(dxx; x),p 2 (d/z; x ) , . . . ,pnn(dp\ x ) , .•. •. • Po(rfM; (d/i; z)> with the following 1.

(2.2)

properties: n n pPn{dfi] . . .• x) == fc knnxxn 44- fc fcnnxxn_1* 4+ .. n(dfi] x)

(2.3)

is a polynomial of precise degree n with the positive leading coefficient; 2. the system P = {p n (d/z;x)} (n € Z + ; x € [a,b]) is orthonormal, that is / pnn (rf/x; xx)p ) pmm (dp; x) dp(x) = Smn

./a Ja

(m, n € Z+),

where 8mn is Kronecher's symbol1; 3. an arbitrary polynomial II n (x) can be represented in the form nn

nn (( xz ))

n n

= £^aaffcc Ppffcc((xx)),, = fc=0 fc=0

1

In future, we sometimes will omit the dependence of /A: pnn(x) (x) == pPn(dw x) == p np(d/i)n(dw x) n(dn).

(2.4)

Orthogonal

62

where

r b

a* = ak(dn) =

(x)pkfck(x)dfjL(x) (x)dfjL(x) / U n„(x)p (x)d/x(x) nn(x)p

a^(dfj)

/ Ja

Polynomials

(k (k == 00,,1l , . . . ,,n); (fc n);

4- the system P = {pn(dp>)} (d^>)} (n (n€€ Z+;x Z+;x GG [a,b]) [a,6]) is complete in L*[a,b]. Using the general formulas (1.8)—(1.11) with /fnn(x) (x) = i n ( n G Z+) we obtain, for n > 1, I Co

pn(rf/z;2:) (d^x)^{D=n.lD( nD)-n1f_2\i D n ) - 1 / 2

where where

ci

ci c\ Ci c2

... ...

cCnn_i _i cn

1 a;

cCnn

cCnn+i +i

...

cC22nn -1 -i

xXnn I

I cn

Cn+i

...

cn== / xnndfj,(x)

(ne

Z+)

1 /1/2 2 Po(dn,x) Po (<*/*, x) == DQ £>01/21 / 2 == c 0CQ , 1/2 c 1/2 Po(dn,x) = £>Q" = o" 5

cn = / x dfj,(x)

c 2 n -i

(2.5) (2-5)

z I

(ne Z+)

are the moments, and I Co D A in ==-

Ci

...

C2 C 2

••• •••

Cn+i Cn + i

...

C ll C

I Cn

C n _i _i I CC n

"

>>> 0o

C2n-1 C2n-1 |

are Gram's determinants. We call pn(x) the orthonormal orthonormal polynomials polynomials associated with the measure measure \i or the distribution d\i (for the sake of simplicity in the notation we will write: O.N.P.S.). If / pnn(dfi] (dfi] x)pmm(dfJi' (dfji'1 X) x) (f/z(x) cf/z(x) = 00 Ja

(n (n ^ 771; m; 771, m , n71 €€ Z+) Z+)

we call pn(dfi;x) (dfi;x) the orthogonal orthogonal polynomial polynomial system system (simply: (simply: O.R.P.S). O.R.P.S). A A similar similar definition holds in the special case dfjt(x) == iy(x) c?)u(x) w(x)c?x, dx, where w(x) is nonnegative and measurable in Lebesgue's sense and

rb

f

/ w(x) dx > >0 Ja Ja (briefly: (briefly: w(x) w(x) is is aa weight weight function function or or w(x) w(x) is is aa weight). weight). We We also also call call of of the the ororthonormal (orthogonal) polynomials associated with the weight function w(x). thonormal (orthogonal) polynomials associated with the weight function w(x).

Elementary Properties of Orthogonal Polynomials

63

Theorem 2.4. A polynomial Pn of degree n is orthogonal with respect to the measure n(x) p,(x) on the interval [a, b] if and only if the following condition is valid /I Pn(x)Q Pn(x)Q m(x)dp,(x)=0 m(x)dp,(x)=0

(2.6)

Ja

for each polynomial Qm(x) of degree m, m <',x) (dfJi]x) belongs to the system (2.2). By (2.4) m m

Qm{x) =

^2akpk(dfji]x) *=o

and then equality (2.6) holds. Inverse assertion is obvious. Remarks. 1. Above we normalized Pn(dfi; x) by llftill llailkp = 1 ||Pn|| = Ibnll^

(n€Z+).

Sometimes other kinds of normalization are appropriate, such as fixing the value of at x = o or x = b (below we prove that pn{p) ^ 0, pn(b) ^ 0), or fixing the highest coefficients of p n (x) and so on. 2. Let [a, 6] be an interval symmetric with respect to the origin, that is a = —b, and let us consider a distribution d\i of the type w(x) dx with an even weight function, that is, w(—x) = w(x). Then pn(dfi\x) is an even or an odd polynomial according as n is even or odd: pn(dfjL\-x) (dfjL\-x) = =

n (~l) (~l)npn(d^x).

It can contain only those powers of x which are congruent to n (mod 2). Indeed, we have for v = 0 , 1 , . . . , n — 1

f

a

ro ra

Pn(dfi; —x)xuw(x) dx = (—l)u pn(dfjL] x)xuw(x) dx = 0. // '/> u (—l)vu I/ pn(d^] x)xuw{x) dx = 0. J—a Pn(dfi; —x)x w(x) dx = (—\) -a pn(—x) possesses the same orthogonality «/—a Consequently, property as pn(x) (see Consequently, the the same orthogonality n(—x) possesses Theorem 2.4). pTherefore, comparing coefficient of x nn , property we obtainas pn(x) (see Theorem 2.4). Therefore, comparing the coefficient of x n, we obtain Pn(dfi; -x) == const constppnn(dfi\ (dn;x) pn(dn;x). pn(dw -x) x) == (-l) n(-l) p n (d/i; x). 3. The linear transformation x = pxf + g, p ^ 0, carries over the interval [a, b] into an interval [A, B] (or [2?, A]), and the weight function w(x) into w(pxf + q). The polynomials n 1/2 (sgnp) |p| 1/2ppn(px'+ (sgn p)n\p\ q) n (px' + g) [^4, B] (or [£, A]) with the weight w(px'+q), w(pxf+q), and the properties are orthonormal on [A, of orthonormal polynomials don't change at appropriate points. So, in future, we often will consider orthonormal polynomial systems on the interval [—1,1].

Orthogonal

64

Polynomials

Extremal properties of orthogonal polynomials Recall 7rn = {II(x) is a polynomial, deg II < n}, that is, II(x) runs through all polynomials of degree at most equal to n with real coefficients (see Notations). Theorem 2.5. The integral

/f nn2n2(x)d»(x), n(x)d^x),

Ja

(2.7) (2.7)

where IIn(x) ranges over the set of all 7rn with the leading coefficients 1, becomes a minimum if and only if x n nn(x) = T-Pn( -r-Pn(x) ) (n (n € Z+; x € [o,6]), [a, 6]),

Kn

where

pPn{dK xnn + . . . , knfc„>0(ne >0 (ne Z+) n(rf/z; x) = pn(x) = knnx

belongs to the system (2.2). Proof. Let us write II n (x) in the form (2.4) n n akpkkp(dfi;x) n(z) ^2a n ( z ) == ^ajbPfc(d/x;x). Un(x) k(dfi;x). fc=o fc=o fc=0 n

Comparing the leading coefficients of both sides we see that an = l/kn. Thus, by orthonormal property of of {P„> {pn} (n (n €€ Z+) Z+)

f

/.&

'a •/a

Ja

n.2n(x)dn

i _ i_ K

*n"

nn-l -1

i=o i = 0 i=0

and the expression on the right-hand side becomes minimal iff ao = oi = ... = a n _i or in other words, Un(x) n„(x) == n„(x) == T-Pn(z), T-Pn(z), T-Pn(x), «n

as we have stated. Remarks. 1. By this Theorem, among all polynomials of degree n with the leading coeffient kn the polynomial pn(x) has the smallest distance from the zero point in the metric of the space L2 [a, b]. 2. Minimum of the integral (2.7) is equal to l/k„ and from (2.5) we find for this minimum the value Dn/Dn-i, since

m"*-

-fcn( =% f ) " !

Elementary Properties of Orthogonal

Polynomials

65

Theorem 2.6. The exact upper limit of the values taken at the point xo e [a, b] by the square Il£ (x) of any polynomial IIn(x) of degree not higher than n and satisfying the condition

f

2 / U2U n(x)dfi(x)
Ja Ja

is

(2.8) (2.8)

n

n

2 ^2p k(dn;x). 0). Y,Pk(d^x 0

(2.9) (2.9)

**k=o =0

This bound is given by the polynomials extr

(dn; EE L oLPof ePk(An; ( ^ ^ o ) px0f)p ; a ; )x) ^ ^ > Q) c (k^ (po(xo) > 0). r -.1/2

n r (g)= n^ix) nf* (*) =

r

ELoPfc(xxo)

I A/.

{2M) (2.10)

Proof. By the Cauchy-Bunyakowskii inequality and the expansion (2.4) we have quai n

n

;E ;E

;(i< ) < ^ a 2 ^ p 2 ( dMW;z). i). n^(x) n2(z)2f>2(d fc=0 fc=0 fc=0

Jb=0 Ar=0

.1 ~~«.

On the other hand, by the integral condition tion (2.8) one obtains n

J24<1 E< k=0

(n € Z+). (nez +).

fc=0 fc=0

Therefore, it is clear that Ylk=oPk(dw %o) is an upper bound of the values Il£(xo). „2 On account of the orthonormality it is easy to see that the polynomial II„ r (see 2.10) satisfies also the prescribed integral condition. Thus this statement is established. The following assertion plays an important role in future. Theorem 2.7. The minimum of the integral

/f nl(x)d»(x), n2n(x)dn(x),

Ja Ja

where IIn(x) runs through all polynomials from 7rn, for which n n (xo) == *1 (( nn€6 Z+; Z+; x x00 € € [o,6]) [o,6]) is equal to

rrr nn n L

-- .. -- 1 1

^^ pp || (( dd // xx ;; xx 00 ))

fc=0

,,

Orthogonal

66

that is,

r n

-.-1

2 min / nH*(x)dii(x) £rf(<*Ai;*o) 5^Pfc(rf/x;a?o) n(x)dn(x) = L J nn(xo)=i,n (x0)=i,nnn€7rnyJaa fc=0 [j^-j j Moreover, this bound is given by the polynomials eXtT o0extr =

nn =

Polynomials

.•

(2.11) (2.H)

x Z2=0Pk(d»;x ELoPfe^M^ o)Pk(dfJi\ x) 0)pk(dn;x) E LOPK^^O) ZUP&dKXo) '

Proof. We have

n n

nn n0*0

^afcfcj>ik(rf/z;x), n0*0 = = ]Ta j>fc(d/z;a;), fc=o fc=o fc=o

where

n n ^akkPk(d/jL;xo) Tinn(x (xo)^2a Pk(d/jL;xo) = = 1. 1. n 0) Tin(xo)^2a = 1. kPk(d/jL;xo) o fc=o fc=0 Jk=0

Since, as above

n

J24= Ja/ 6 n^(x)^. n* (*)<*//. Ja

fc=0 fc=0

So we obtain the following problem to find n

Efe=o Efe=o ET.

min

aafc fc

ffc c xx

/Jafc-

=1 =1

P P ( °) ° ) ife=0 i f efc=0 =0

Cauchy-Bunyakowskii's inequality yields

a l<(J2 l)(Epl(d^x )(X^^( rf ^0x)). °))(E4) X

So

7

fc=0

lrr nn 2 ±4> n

fc=0 fc=0 fc=0

^p2(d//;x [l> * ) L

Lfc=0

fc=0

0

-.-I -,-1 -IJ

>

and the signum of. inequality if i n e q u a l i t y attains a t t a i n s if 11 and a i i u only c _

Pk(d^x Pk(dfi;x pk(dfi;x 00)) 0)

Ek=oPUd&xo)

(k = 0 , l , . . . , n ) .

0 extr ^From this follows (2.11) and the extremal function is n n Theorem 2.7 is proved. Corollary 2.8. Suppose that ii\(x) and fi2(x) are measures and

Mi0*0 "- Mi(y) < ^0*0 - fifi22{y) {y)

(2.12)

Elementary Properties of Orthogonal

Polynomials

67

holds for any pair of numbers x > y ; if' {p {pn{dp.\\x)} (n 6 Z + ) and {pn{dp>2] x)} n{dp,\\x)} (n e Z+) are O.N.P.S. with respect to the measures HI p \ and p>2, respectively, then rr nn

i-|- ~.--111

L L

JJ

5^p2(d/n;x) 5^pJ(d|ii;x)

fc=o fc=o Lfc=o

-I

rr

n

-,- j. -- 1

< ^pl(rf/x2;x) Lfc=on

.

(2.13)

-IJ

LI

Proof. By Theorem 2.7 and the condition (2.12) one can obtain r nn

-| — 1 l""

fb

-.-1

T

^ p | d(V>2',x)\ d/i2;x) \Y,Pk( LfcTo J

=

222 min mini n / UU (x)du2(x) m TL(x)du2(x {x)dp,2(x) n€7Tnn,, n (n(xo)=ij n€7T a r 0 ) = l 7 aa

b > min min /f > *o)=l JJ/Aa ~ n€7r n€7r ~ n€7rn ,,min , rn(a:o)=l n(so)=i ~ Il€7rnnn, n(a:o)=l Jna 111J

rr n r nn r n

U Ii22(x)dui(x) U2(x)dui(x)

-I 1- , -- 1 1

d/ii;x)

L

»-fc=0 fc=0

J

in accordance with our statement.

ChristoffePs formula T h e o r e m 2.9. Let {pn(x)} = {p {pnn(dp;x)} (d/x;x)} (n {n e€ Z+) be 6e the £/te orthonormal mials associated with the measure p. at the interval fa, [a, 61. 6]. Also let p(x) = x ^xX^O O^ O p(x) = c(x — — x\)(x x\)(x — — X2) X2)•.. ... (x (x — —xi), £/), Xl),

(2.14) (2.14)

be a 717 which is nonnegative in this interval Then the orthogonal als {pn(x)} = {pn(dp,\x)} (n € Z + ) , associated with the distribution p(x)dp(x), can be represented in terms of the polynomials as follows: (x) I p n(x) IPn{Xl) Pn(^l)

PPnn++l l((X^ll))

Pn(xi) I Pn(^f)

pnn+$x{) P +l(z/)

p(x)p = \ P(x)pn(x) n (x) =

p Pn+i(x) n+l(x)

.••• .. Pn+l{x) Pn+i{x) I ••• Pn+l{Xl) Pn+l{X$

•'•

... •••

pPn+l(Xi) n+l(xi)

polyno-

.

polynomidp(x) =

(2.15) (2.15)

\

Proof. T h e right-hand member of (2.15) is a 7rn+/ which is evidently divisible by p(x). Hence it has the form p(x)pn(x), where pn(x) is a 7rn. Moreover, it is a linear p{x). combination of the polynomials p n ( x ) , p n + i ( x ) , . . . ,p n +i(x), so that if q(x) is an arbitrary 7r n _i, then / p(x)p p(x)pnn(x)q{x) a>(x) = = 0. (x)q(x) dp(x)

Ja Ja

(2.16)

Finally, the right-hand side of (2.15) is not identically zero. To show this, it suffices to prove that the coefficient of p n +/(x), that is, the determinant [p n + l / (x / i + i)],

Orthogonal Polynomials

68

i/, /x = 0 , 1 , . . . , / — 1, does not vanish. Suppose it vanishes; then certain real constants Ao, A i , . . . , A/_i exist, not all zero, such that j _ i p nn++j_i(x) j_i(x) AoPn(z) + A iip nn ++ i ( x ) + . . .. + AAj_ip

(2.16)

vanishes for x = x i , X 2 , . . . , xi. Hence (2.16) is of the form p(x)G(x), where G(x) is a 7r n _i. We have the relation

l /

p(x)G(x)G(x)

dp{x) = 0

whence G(x) = 0, a contradiction. R e m a r k . By using some special properties of dp(x) or of p(x), formula (2.15) can be simplified. For example, let dp,(x) = w(x) dx, w(x) and p(x) be even functions, ve ha\ and a = — b. Then, instead of (2.15), we have the representation (Zisiseve even)

p(x)p (x) p(x)pnn(x)

n(x) I Pn(x) I Pn Pn(xi) *0*l) == \ (Xl^ I Pn{Xi/ 2) fa/2) I Pn{Xi/2)

Pn+2{x) Xi) Pn+2(xi) Pn+2(xi)

Pn+4(*) 4 (x) Pn+4(^l) Pn+4{xi)

.••• .. • •• ...

Pn+2{xi/2) Pn+2{xi/2)

Pn+4{xt/ fo/2) 2) Pn+4{xt/2)

... •••

Pn+l{x) I pn+l(xi) (*l) Pn+l(xi) II 'J Pn+l(xi/ ^2/)2 ) | Pn+l(xi/2) |

;ros of , tal se1 where { ± x i , ± X 2 , . . . , i ^ z / 2 } is the total set of zeros of p(x).

Recursion relations The following relations play a fundamental role at the treatment of the convergence of expansions in orthogonal polynomials. T h e o r e m 2.10. The following recursion relation holds for any three consecutive orthonormal polynomials xpnn(dp; (d/j,\ x) = anpn+ i(dfj,; i(rf/x; x) + unnppnn{dp\ {d^\ x) + a„_ip an-ipn(dp\ n+ n +i(d/x; nn(d/x; x)

(n €G Z+; a_i = 0, p_i(d/z; x) = 0)

(2.17)

with fen

ann = ann(dfi) (d/x) = -T—^-; ^-; fen+1 fcn+l

un = un(dfi) (rf/x) = 7^ ^ - -^ ^ &n «n

&n+i »n+l

(n G Z+),

(2.18)

where kn = kn(dp), kn = kn(dp) (n G Z+) are the coefficients of the corresponding p n (d/z;x) (see (2.3)). For the proof, we first expand the polynomial xp n (x), which is of degree n + 1 , in the function of {pn} (n € Z+): xpn(dp,;x)

n+l = ^afcp ^ajkPfc((f/x;x)(n f c (
69

Elementary Properties of Orthogonal Polynomials

Multiplying both sides of this relation by p3(x) and integrating, we notice that for s < n — 2 the polynomial xp3(x) is of degree not higher than n — 1 and for this reason the integral on the left-hand side vanishes by orthogonality, while on the right-hand side we obtain CLS. Itt follows that a3s = 0 for s < n — 2 and a therefore xpnn(dfi; xp (dfi; x) = = aannn-ip -ip _ip -i(dw _i(d/x; annPn(^M; p nn(d/x; x) +i(c^; x). x). nn-i(dw n n +ip nn+i(c^; (d/x; x) x) + +a x) ++ aau+iPn+i(c Since both sides of this relation are polynomials of degree n + 1 , by comparing the coefficients of xn+l we get that kn fen a«n+l (n € €Z+) <*n+i ("> Z+) *+i = = TTI fen+1 fcn+l fcn+1 and consequently fe: A Tr^-Pn+iidwx) =■■ (x annn)p (dfi)x) (d/jL; x)-x) - - aaa -ip _i(rf/x; x). T-^-Pn+i^;^) (a: -(x a-n)p )p (d/x;x) -ip n(d/jL; nn n_ip n.nn-i{d^x). «n+l

Q i=

Multiplying here again both sides by pn-\{x)

and integrating, we obtain that

-r

-i(dfjL] dfi(x) a«„-!=/ xpnx)p (dfi;x)p x)dfjL(x)' l(dfi\x)p nn-i(dfjL;x)dfi(: n - i = / xpnr(dfi\ n-i(dfi'1x) but

fen-l (d^;x) _ip _i(
aC*n-1 = -T n_i =

So we have A: fe fcn-i fe fen-1 xpn(dfi]x) = —^-p fcn+1 n+1(dwx)

+ anpn(dKx)

A; _ i , fcn-1 + -£—p _i(d/i;x). nn_i(d/i;:r). A; n

B y comparing the coefficients of xn we obtain that n ~kfen — a &nkn fen ^ nfc ^ nn === fen **nn++ll kn ~~ T T

«n+l

and therefore fen ^n _ fen+1 ^n+ 1 Qn <*n = =

an =

IK z— Ifen+1 z— n

^n

«*n+l in accordance with (2.17), (2.18). T h e recursion formula (2.18) is valid for n = 0, if we write p _ i ( d / r , x ) = 0 (x <=[<*,&]). Theorem 2.10 proves.

Orthogonal

70

Polynomials

It seems sometimes suitable to norm the orthogonal polynomials by multin plying them by a constant in such a way that the coefficient of x should be equal to 1: 1 pn(dii\ x) = kk^pnid/i; x) == xxnn ++ .. . . n pn(d/JL;x)

and p-i(dfj,]x) = 0 . p_i(d/x;x) With this notation the last formula gives Corollary 2.11. The polynomials {pn} (n e Z+) satisfy to the recursion formula

Corollary 2.11. The polynomials {pn} (ra G Z+) satisfy to the recursion formula pn+i(rf/x; x) = (x - A n )p n (d/r, x) - £ n _ip n _i(d/x; x) (n = 0 , 1 , 2 , . . . ) pn+i(dfi; x) = (x- An)pn(d/ji; x) - £ n _ip n _i(d/x; x) (n = 0 , 1 , 2 , . . . ) with with

An = Un An = un

and

*lc22 ^n-l = % 1 > 0 . Kn

Corollary 2.12. Orthonormal Orthonormal polynomials (d/x; x) = =ppnn(x) € [a,6]) [a,b]) polynomials pnn(dfjb) (x) (n € Z+; x G satisfy the following satisfy following five-term five-term five-term recurrence recurrence relation relation x2pnn(x) (x) + (a^_ x + a\ un)pnn(x) (x) +ip a* + u£)p (x) = annaannn+ip +ip (x) + an(unn + un+i)p +i)Pn+i(z) n+ 2+ n+i(x) nn+ 22(x) + a n _i(u _ i ( u n _i _ i + unn)p )pnn-i(x) _i(x) + a n _ 2 a n _ip _ i pnn__22 (x) (x) n-i(x) p_2(x) P-i(x) = p_i(x) (n G Z+; p= 0, a_ 2 = 0, a_i = 0). 2 (x) = 0, p-i(x) (2.19) This follows by applying the recurrence relation once more. L e m m a 2.13. The recursion recursion coefficients coefficients form a Jacobi Jacobi (tridiagonal) (tridiagonal) matrix matrix Lemma

(

u0

a00

0

0

0

... ... \

o°0° ZZ Su2 aa°22 0o ::: :::

S

(^ = "»(*)) CM) (2.20) (^S M4*);*» M^);*»="»(*))

with the following with following properties: properties: 1) ann >0,u >0,unnee R 1 (n eG Z+); 2) b

r

3)

3)

a>n= a>n= xp n(dwx)p n+i(dii;x)dfi(< <x) nn(dwx)p nn+i(dii;x)dfi( a>n=Ja xpxp (dwx)p +i(dii;x)dfi( <x) b Un= \ Xp Xpl(dfJL\x)dfl{x) n(dfJL\x)dll{x) Un= Ja Xpl(dfJL\x)dfl{x) Ja an < const,

an < const,

\un\ < const

1 \\

\

(n€Z+).

(n€Z+).

(2.21)

(2.21)

)

)

(n G Z+)

\un\ < const (n G Z+)

(2.22)

Elementary Properties of Orthogonal Polynomials

71

4) if [a, b\ b] is an interval symmetric with respect to the origin (a = —b) and dp,(x) = = w(x) w(x) dx dx rvith rvith w(—x) w(—x) = = w(x), w(x), then then dp,(x) un = = 0 (ne Z+).

(2.23)

Proof. The properties 1) and 2) are evident. Applying the Cauchy-Bunyakowskii inequality to (2.21), we have

U>{/• b

a n <max(|a|,|6|)j

>nr >r< N s

1/2 ,

pl(dfjL;x)dfjL(x) \

and

-6 ~b

pl(dn;x)dn(x)\

x^ 1/2

Pl+i(dl*;x)dii(x)[

< const

I / ppln+l+ 1(dfji\x)dfi(x)\ (d//;x)d//(xU

< const

|unn\| <<max(|a|,|6|) max(|a|, \b\) / pPn(dn, x) dfx(x) < < const. \u n(d/z, x)d/z(x) For the proof of 4) we notice that it follows from formulas (2.21) and pn(dfjb] —x) = (~\)npn(d^x), that

A

oa

pa

xp^(rf/x; x)w(x) dx = un = t)[(-irPn(d^t)]2w(-t)dt / /J —a xpl(dfi;x)w(x)dx= - l » n ( d Pn / i (dn;t)} ; i ) ] 2 «2;w(-t)dt (-t)dt J/—a ( - t ) [ ((-t)[(-iy •a J —a -a and by symmetry of the weight w(x) we derive and by symmetry of the weight w(x) we derive un = = -- / tpn(dfi; tpl(dfji\t)w(t)dt t)w(t) dt = —u - u nn, J. /—a -a J-a as stated in (2.23). Thus, for an arbitrary O.N.P.S. {pn} (n e€ Z+) we correspond the Jacobi matrix T (2.20) with bounded coefficients an,un (n e€ Z+ (see (2.22)). Consider inverse problem: given matrix T, and define polynomials by three-term recurrence relation (2.17). Does there exist a measure dv such that {pn} (n € Z+) is the system of polynomials orthonormal with respect to the dv {dv is a monotone decreasing function whose derivative may, however, vanish eventually not only in a set of measure zero!)? We shall consider T as the matrix of a certain linear operator T in a (complex) Hilbert space H. ri. For this purpose we take in K ri an orthonormal basis {e^} (k € Z+) and in the first place define the operator T for the unit vectors e^ ek by the equations Tekfc = ak-iek-i

+ ufckeekfc + afcefc+i

(fc = 0 , 1 , . . . , e_i = 0). (k =

Due to its linearity the operator T is then also determined for all finite vectors oo

9 =■E ^2xkek. fc=o fc=o fc=0

Orthogonal

72

Polynomials

Since by virtue of the definition (Tcni, en) e n ) = (e m , Ten) (Tem,

(m, n = 0,1,2,...)

it follows that for any two finite vectors / and g we have the equality (f,Tg). (Tf,g) = {f,Tg). But the set of all finite vectors is dense in H. Therefore the operator T is symmetric and by (2.22) is self-adjoint bounded operator. Eigenfunction g of T one canfindfrom Tg = xg. Let a solution of this equation be the vector oo

9 = fc=0YlPkekfc=0

Then

oo oo

E

>,Pk(a>k-iek+i + ujfcefc + afcejk+i) = fc=o fc=o fc=0 fc=0 fc=ofc=0 whence it follows that whence it follows that

oo oo oo

xypuek,

fc=0

xpk afc_ipfc_i + +U ukppkk + a,kPk+i (k (k = = 1,2,...) Xpk = = CLk-lPk-l + CLkPk+l 1,2,...) xpo = xpo = UQPO UQPO ++ dopi. aopi.

We have been let to the three-term recurrence relation (2.17) with the condition (2.22). To order our problem we apply Fundamental Spectral Theorem to the operator T associated with ^"-matrix (2.20). We remind this statement. By a resolution of unity we mean any one parametric family of operators Ex, x € R 1 , satisfying the following conditions: 1) Ex are projectors, that is, Ex = E*, E% = Ex; 2) for #2 > xi the difference EX2 — EXl is a bounded positive operator, that is, EXlEX2 = £min(xi,x2) (orthogonal); 3) in the sense of convergence in norm on every element we have Ex+o = ExX (-oo < x < oo),

iiLoo E-oo = 0, i£+oo ^ + 0 0 = I/ (unit operator).

By Fundamental Spectral Theorem, to each self-adjoint operator T in a Hilbert space H corresponds a unique resolution of unity Ex, x e R 1 , such that the domain DT of T consists of all vectors / for which

/

JRi

2 x2xd(E d(Exf,f)<+oo xf,f)<+oo

Elementary Properties of Orthogonal

73

Polynomials

and T=

f

■JL

xdEx.

Moreover, if T is a bounded self-adjoint operator, then the last representation means as uniform limit of the integral integr; sums, in addition _ Jo *~\j

forxsupSpT,

where Sp T is a spectrum of T. Put i/(x) = (Exeo.eo), x € SpT. It follows from the properties of set Ex that dv is a monotone nondecreasing function. The moments of this measure are

■x*di/(x) = (Tfceo,e0), S' n = / ySpr

k e Z+,

i.e., we get the momentum problem associated with ^"-matrix. In a (complex) Hilbert space I?v we define the operator U:

LI-+H,

that every function / G L\ corresponds vector >nds vec / ( T ) e 0 =. // f(x)dExe0, JSpr iSpr where the integral means as a limit of the integral sums by norm of 7i. We prove that U is isometry,, i.e., i.e., (Uf,Ug)=

(

(2.25)

f(x)g(x)dv(x)

7spr JSpT soy 1

(2.24)

forall/,p€L2. for a l l / , 0 € 1,2. Because it is Because the the set set of of continuous continuous bounded bounded functions functions is is dense dense in in L£, L£, then then it is sufficient to verify (2.25) for these functions. Computing the integral (2.24), as a limit, o f Ri*>mnnri-St.ip>lti*>e int.pcrral eiims xut± Horiw* limit of Riemann-Stieltjes integral sums, we derive n n

(Uf,Ug) = ]im(£,f(Xj)(EXj

nn

- £ x ._Je 0 , $>(**)(£** - EXh_x)e*)

j=l

iik=l k=l

n n

=

im Z£) f(/xj)9(xk)((EXk

lim

- ^ . J f f t , - E.j.Jeo.eo)

».fc=i

= lim ]T/(xjMxjKvixj)

-

/•Af

V(XJ-I))

= /

Jm

/(x)^(x) du(x)

Orthogonal Polynomials

74

(here m < XQ < x\ < ... < xn < M is subdivision of SpT = [ra,M] because projectors EXk - EXk_x and EXj - EXj_x are orthogonal for k ^ j). T neo, e 0 , n ge Z+}, Z+}, Since the operator U translates polynomials to the set {{T which is dense in ri, i.e., image U coincides with W, consequently U is unitary operator. In particular, polynomials are dense in L2. Let {pn} (n e Z + ) be the polynomials, associated with ^"-matrix. We will show that (2.26) ek=p *eZ+. =Pk(T)e *€.Z+. k(T)e00, , We use induction by k. For k = 0 equality (2.26) is evident. Let (2.26) is proved for A: == 00 ,, 11,,.. .. .. , n. n. Then, Then, by by definition definition T, T, one one has Tpnn(T)eo Tp (T)e (T)e00 = Te Tenn = a n e n +i + unneenn 4+ aann__ie _iieenn__i. _ii..

(2.27)

On the other hand, from recurrence relation for pnn it follows that Tpn(T) (T) = = anPn+1 (T) nPn+1(T)

4+ unPn(T)

+ 4 a n _ip _ i p n _i(T), _i(T), Tp (T)e0 + w unnnp/>n(T)e 4 a n _ip _ i p n _i(T)e _ i ( T ) e 0. (T)e pnn++i(T)e Tpnnn(T)e (T)e 00 0 = a n + ii(T)e n(T)e0 0 4np

(2.28)

Comparing (2.27) and (2.28), we obtain (2.26) for fc A; == nn++ 1.1. Relation Relation (2.26) (2.26) isis proved. Notice that (2.26) is equivalent to the following relationship U:p U:pk(x)^e k, k, k(x)^e

k€Z+. k€Z+.

It is not difficult to see that the polynomial system {pnn}} (n € Z+) is orthonormal with respect to the measure v. In fact, by (2.26) and the Fundamental Spectral Theorem we have ^>M -M 4 , n = (e 4,n ,enn))n) = (pk(T)e (pk(T)e00,p,pnn(T)e (T)e = = (x)dExxee00\\ / ppnn(x)dE (x)dE j fa,n (e/k,e (pjb(T)eo,p = (f / pk(x)dE (x) dj& kk,e 0)0) xxee00 xeoj n (r)eo) • m

= /

rM

222

Pk(x)p P Pk(x)p (x) d(\\E d(\\Exxxee000\\\\)) = = / k(x)pnn(x) Jm Jm Thus, we obtain the following assertion. Jm Jm

Jm

ppkpk(x)p (x)pnn(x)du(x). (x)du(x). k(x)p n(x)du(x).

Theorem 2.14. Polynomials {pn} (n e Z+) associated with T-matrix form an orthonormal system with respect to the measure dv, and this measure is (a unique) solution of momentum problem associated with P-matrix. We discuss the following problem: ^"-matrix with bounded entries corresponds the self-adjoint bounded operator. Now, it is natural to pose the question: how wide is the class of symmetric operators which can be generated by ^"-matrix? A self-adjoint operator T in a Hilbert space ri has a simple (Lebesgue) spectrum if there exists an element g e ri (generating or cyclic), that is Tkg e D(T) (k € Z+) and the closure of the linear hull of g, Tg, T2g,... is coincide with H. Theorem 2.15. Set of all self-adjoint bounded operators with a simple spectrum is coincided with a set of all operators generated by a bounded Jacobi matrices. In addition, unitary equivalent matrices arenft differ.

Elementary Properties of Orthogonal Polynomials

75

Proof. If T is a bounded selfadjoint operator with a simple spectrum and g is a cyclic element, then the system g,Tg,T2g,... is linear independent. By Gram-Schmidt orthogonalization procedure (see §1) applied to the sequence {Tng} (n G Z+), we obtain the orthonormal basis in H: e€nn == 7n, 7n,o0 oP + ln,\Tg + .... . 4- 7n,nTnp,

n G Z+. Z+.

(2.29) (2.29)

Nevertheless, one can assume that 7 n , n > 0. We get the matrix of operator T in this basis. According to (2.29) Tenn = Te — aann,,00eo eo++ .. . . ++ aan>n +ienn+i +i n>n+ie and by symmetric of T an,j = = (Te n ,ej) , ej) = (e n ,Tej) , Tej) = 0

(j < < n - 2)

because vector Tej for such j belongs to the linear hull eo, e i , . . . , e n +i, and, consequently, orthogonal to e n . Introducing ann = ( T e n , e n + i ) = 7 n , n / 7 n + l , n + l > 0,

wn = ( T e n , e nn))G €R1,

nG € Z+,

and taking into account of a n>n +i = a n , a n n = uni an)Tl_i = a n _i, we obtain, that the matrix of operator T is coincide with (2.20).

Representations of the Dirichlet kernel We introduce the Dirichlet kernels (kernel polynomials) P n (d/x,t,x), that plays a fundamental role at the treatment of Fourier expansions: n

Vn(d»;t,x)

71

(d/x;t)pfcfc(d/x;x) (d/r,x) = Vn(t,x) := ^p fc (d/x;t)p

t,x G [a,6]), (2.30) (n G Z+; *,x

fc=0

where j>fc(d/r, x) (A; € Z+; x G [a, 6]) is orthonormal polynomial system with respect to the measure \i. Notice the symmetry relation D n (d/x;t,x) = Vn(d/j,',x,t). The kernel P n (d/j;£,x) possesses the following reproducing property: for an arbitrary II(x) G 7rn is valid j U(t)V 6]), n(t)Dnn(dfi\t,x)d^(t) (du:t,x)d|x(*) == n(x) n(x) (n ( nG e ZZ+, + , x G [a, 61),

./a

(2.31) (2.31)

in particular, / pn(rf/x; (dw t)Vn(dw t, x) dfj,(t) dii(t) = = pn(x) (x)

Ja ./a

(n G Z+, x G [a, 6]).

This relation completely determines the kernel polynomials. Lemma 2.16. Let XQ,XI,X2 be finite. Then the following assertions are valid:

76

Orthogonal Polynomials

1) if xo < Q>> a, xi >xi>b, then the the polynomials {Dn{D (dp;xi,x)} orthogonal withwith b, then polynomials orthogonal n(dp;xi,x)} are are respect to the distribution (x — — Xo) Xo) dp(x), (x\ — x) x) dp(x), respectively; 2) if [a, b] C [—x2,x2], then the polynomials are orthogonal to arbitrary U(x) £ 7rn_2 _2 with the distribution (x 2 — x2)dp(x). In fact, the first statement follows from (2.31) by writing x = = xo, II(x) II (x) = (x — — xo)Po(x) xo)Po(#) or or xx == xi, xi, II(x) II(x) == (xi (xi — — x)Pi(x), x)Pi(x), where where Po, Po, ^i Pi are are arbitrary arbitrary 7r 7rnn_i. _i. If x = x 2 and II(x) = (x\ - x 2 )P(x), P(x) £ 7r ?rn_2, then we obtain by (2.31) 2 2 / T>n(dp; Vn(dp; x 2x,2x)(xl , x)(x\ - -x2x)p(x) )p(x)dp(x) dp(x)==(x\ (x\- -x2z)p(x)\ )p(x)| x=x2x = x 2==0.0. Ja Ja

Theorem 2.17. For the Dirichlet kernel of an arbitrary orthonormal polynomial system P = {pn(dp;x)} (n £ Z+) the Christoffel-Darboux (summation) formula is valid Pnn+\(dp; d x) d ~ , , x)=a x „ pPn+\(dp; t)pnn(dp; (dp; pnn(dp; (dp; t)pnn+i(dp; +i(dp; x) x) V /Ida-1 ^d/J,; t)p *)P"( ^ xx) ) -"- pPn( P; t)p *)?n+i{dp; x) M t, x) = a PnnK [dp; n ' ' ' t-x t—x t—x (n £ Z+, Lx t,x £ [a, \a.b]). b]).

(2.32)

In particular, n n

rf X x Vn (dp;x,x) (dp; x, x) == ] T^Pk{dp;x) p\ (dp; x) == ana nb[pUi A*; )Pn )Pn(d(dp; n + 1 (rf( W P; x)x)

ib=o fc=0 k=0

d

+i(dp;x)\ - Pn( M x)pn+i(dp;x)]

(2.33) (2.33)

(n £ Z+, x £€ [a, 6]),

tu/iere an = an(dp) (n £ Z+) are where are the £/ie recursion coefficients. Proof. This important identity can be easy derived from the recurrence relation. For, if we multiply (2.17) by pk(dp; t), interchange t with x and substact obtained equations, then one obtains (t - x)pk(dp; t)pk(dp; x) = ak\pk+x(dp; t)pk(dp; x) - pk+i(dp; t)pk(dp; x)] - a>k-i\pk(dp; t)pk-i(dp; x) - pk(dp; x)pk-X(dp; t)\. Summing all these relations over k = 0 , 1 , . . . , n and using p~\(dp;x) derive (2.32). It follows from (2.32) that

= 0, we

n

Y^,Pk(dp;t)pk(dp;x) Y^,Pk(dp;t)p k(dp;x)

fc=o

\pn+i(dp;t) -pn+i(dp;x)]pn(dp;x) - \pn(dp; (dp;t)t) = anant —x t —x and by the THopital rule we get (2.33). and by the THopital rule we get (2.33). =

-pn(dp;x)]pn+i(dp;x)

Elementary Properties of Orthogonal

Polynomials

77

Corollary 2.18. For the Dirichlet kernel of the O.N.P.S. P = {pn(dp,\x)} Z+) the following representation is valid {dp,;t,x) Vn(dfjL;t = yx)

2; //iiii((nn;;tt,,xx)) an-i+an fln-i + a n

(n G

n

" - *\ n" [h [h22(n;t,x) (n;t,x) ++ /i/i33(n;t,x)], (n;t,x)], an_i + on an_i + an

w/iere w/iere /ii(n;t,x) /ii(n; t, x) == pn(dfJL; pn(dfJL;t)p t)pn(dfi; x) } n(dfi;x) /ii(n; t, x) = p n (d/i; t)pn(dfi; x) "j I lI Pn(^/x;*)[p +i(d/z;x) p i { d f i ; x ) Pn(d/i; *)[p +l(d/X; X) Pn-ljdfi] /i (n;t,x) = Pnfe^^ n ^^x) nn i f e ^ - p n /i (n;£,x) = 2 (n;t,x) /i2 = Pnfe^^ife^-pn-^^x) - > I 2 tt — — —X X X I I = ft (n;x, t) /i 3 (n; (n;t, t, x) = ft22(n; x, t) J J

((nn e +. € zz Z+; (n e +.

^t ^, x €€ [ [a, ) ^ ^ € [ aa 66 ]]6]). )

Proof. It is easy to see that , Ir2> 2>nn(fl ((f/x;t,x) Dwnn._i_(iid(d/i;t,x 1 __ Q a nn_- i |2> /i;t,x) D /ai>; t; t, ,gx) )j11 1 n(d/i;t 9 x) | P P t,>x) — i '- ++ + -zPn(dn; t)pn(dn; x) n (rf/i; * Dn(d/*; *) = =———7T— 2 [ an a n _ i J +2~Pn(dn; t)pn(dn; x) 2 [ an an-i J 2 + +

fl

\

tt

" 1 P n (n-< *n-t MPn(
Q Qn

On the other hand, by Christoffel-Darboux's formula one can get

On the other hand, by Christoffel-Darboux's formula one can get /i (n; t, x) + /i (n; t, x) h22{n\ t,x) h33(n;t,x)

and consequently and consequently

= =

[p +i(d/x;x) -i{dfj,;x)x) 7[ Pnn(d/x; ( ^ ; t)p * ) - Pn(d>H', t)pnn-i{dfJL; \Pn{dwt)p p np(d/x;t)p n (d/x;t)p n-i(d/x;x) n +l(d/X; n+i(d/x;x) t —X x -pn-i{dfi; t)pn(dfjL; x ) + p n - i ( d / i ; t)pn{dfi; x)] -pn-i{dfi;

t)pn(dfjL; x) + p n - i ( d / i ; t)pn(dfi; x)]

1

—Vn(dn;t,x) H V 2>n-\(dn',t,x) i 33((n;£,x)]. n;£,x)]. n _i(d/i;£,x) = - f / i 2 ( n ; t , x ) + / /i an a nn _ -i So 2> y ^ t M2 (n] n ; t, P n (rf/i; —Y*~[h - / i i ( n ; t,t, x) x)) = — *, x) x) 44- /i/i33(n; (n; t,t, x)] x)] ++ -jhi{n\ x) n (d/i; tt,, x 11 aa nn __ ii --f lann ^ / , . x + o Pn(aV;t,x). Z>n on 222 O n O n

This establishes the statement. Lemma For the the Dirichlet Dirichlet kernel L e m m a 2.19. For kernel of an arbitrary arbitrary O.N.P.S. O.N.P.S. {pn{dii\x)} (d/x; x)} (n € Z+) £/ie the following following formulas formulas are are valid valid (x 2 - t2)Vn(dn\

t, i [np{dw t)pn(dfjL\ x)] t , xx)) = a naon+n +i\p ( /^i ; *)Pn(d/x; *)Pn+2(dM; ^)] n + 2 (d/r, x) - p nnn+2(d/*; ++2 2( d n ( d / i ; t)p o n _ i a nn[[pn-i(a>;t)p p n _ i ( d / x ; t ) pnn+i(dwx) ( d ^ ; x ) - ppnn++i(a7x;t)p -i(dfjL\x)] On-iOntpn-iCd/x;t)p -i(dfjL]x)] + On-io i(dM;t)pnnn-i(dfjL]x)] n+i(dp,\t)p n++ii(d^;x) + ao>n(u flnK t)pn+i(d|x; i)[pnnn(dn; (d/z; t)p + uunnn+i)\p (dn; t)pn+i(d|x; x) n{un n + ++i)\p n +i(a7x; x) ~Pn+i(d/i;*)Pn(d/x;x)] (n eG Z 61). - Pn+i{dwt)pn(dwx)] Z+; + ; t , x € [a, 6]).

Orthogonal

78

Polynomials

In particular, n n

2x^Tpl(dMx) 2x^TpUdwx)

= = aannaann+i\p' +i\p'nn(diJ,;t)p (diJ,;t)pnn+ +22(dfi;x) (dfi;x)

fc=o fc=0

-p -pnn(dfi;t)p' (dfi;t)p'nn+ +22(dp,x)] (dp,x)]

+ n - l f l n b n - l ( rrffM; M ; t)p i(d^\ x) --p'p'+i(dw n+i(dw + aan-iflnbn-i( t)pn++i(dn\x) n

n

t)pt)p x)} n-i(dfi; -i(dfi;x)]

i)[p^(d/x;t)pn+i(dn; x) - p'n+l{d^\ + an(un + unn++i)[p^(d/x;t)p

n

t)pn(dn\x)],

where {a n }, {un} (n € Z+) are the recurrence coefficients for pn(dfi; x) (see (2.17)). As in the proof of Theorem 2.17, assertions of our Lemma follow from the relation (2.19), Abel's transform and by the PHopital rule.

Elementary properties of the zeros of orthogonal polynomials The following remarkable properties of the zeros of orthogonal polynomials is used in the theory of orthogonal polynomials. Theorem 2.20. The zeros of the orthogonal polynomials p n (x) = pn(d/z;x) associated with the measure \i on the interval [a, b], are real and distinct and are located in the interior of the interval (a, 6). Proof. From

fb

I

/ Pn(x)dii(x) = 0 (n > 1) Ja /a we are assured of the existence of at least one point in the interior of [a, b] at we arepassured of thesgn existence of at least in thenumber interiorofofpoints [a, b] of at which {x) changes (the function //(#) one has point an infinite n n x which p {x) changes sgn (the function //(#) has an infinite number of points of n If x^ , x increase). denote the abscissas of all such points, the product 2 »• • • > \ IfXY')(X Xj , x2— ,.. •,)Xj denote of all such product pincrease). (x)(x — £3 • • • (x — Xjthe)abscissas has a constant sgn points, (that is,the is nonnegative n porn{x){x — XY')(X — x2) ... (x — sgn 2.1 (that is, is nonnegative nonpositive throughout [a, 6]), weXi) havehas I
I

,6 rb

/I Pn(x)(x Pn(x)(x — — x\)(x x\)(x — — xx22)) ...(x ...(x — — xi) xi) dfi(x) dfi(x) > > 00 or or < < 0. 0. Ja Ja On the other hand, if /I < n this integral is equal to zero by orthogonality (see Theorem 2.4) and this is impossible. Therefore I = n. Corollary 2.21. For the zeros of the orthogonal polynomials pnn{x)}} associated with the distribution d\i on the interval [a, b] we have

and

n) B) n) x
(2.34) (2.34)

pPn(b) > 0, 0, pp2n (a) > > 0, 0, pP2n+i(&) < 00 (n (n >> 1). 1). n(b) > 2n+i(b) < 2n(a)

(2.35)

In fact, (2.34) follows from Theorem 2.20. Since the leading coefficients of pn(x) is positive and Xn is a last zero of pn{x) we obtain the first inequality of (2.35). The behaviour of p n (x) at the point a one can get in a similar way.

Elementary Properties of Orthogonal Polynomials

79

By recursion formula (2.17) polynomials pn(x) and p n +i(x) have not common zeros. Moreover, Theorem 2.22. Let x£n) be the zeros of pn(x) (see (2.34)), 4 n ) = a> 4 + 1 = b. Then each interval [ 4 >xfc+il {k = 0,1,...,n) contains exactly one zero of pn+\{x) =p n +i(d/i;x):

4U) < 4U+1) < * &

(A: = 0,l,...,n).

Proof. It follows from (2.33), that x Pn+i( )Pn(x) - p' tfn+i(*)Pn(x) Pn(*)Pn+i(z) n (x)p n+ i(x) > 0 (x € [a,6]).

(2.36)

If f and 7), £ 0, -Pn(0Pn+l(0

--jtf n/ (v)Pn+l(v) ^ n f a ) P n + l 0 ? ) > 0,

so that Pn+l(0Pn+l(v) <<°0Pn+liOPn+liv)

This indicates an odd number, that is, at least one zero of p n +i(x) in f < x < rj. Now let £ = Xn the greatest zero of pn{x)\ then p„(£) > 0, and (2.36) yields Pn+i(€) < 0. Since p n +i(6) is positive (see (2.35)), we obtain at least one zero of Pn+i(x) on the right of f = Xn , and similarly at least one of the left of the least zero Xi of p n (x). Consequently, we can have only one zero of pn+i(x) between 4 n ) and x j ^ (Jb = 0 , 1 , . . . , n). By interchanging the role of p n (x) and p n +i(x), we can prove as before the existence of at least one zero of p n (x) between two consecutive zeros of pn+i(x). Consequently, zeros of p n (x) and p n +i(x) are alternated. The other simple consequences of (2.36) are Theorem 2.23. Let c be an arbitrary real constant. Then the polynomial pnn+i(<*M; +i(d/x; x) x) x) -- cp cpnn(dfjL, (dfi, x) has (n + 1) distinct real zeros. Ifc>0 (c < 0), these zeros lie in the interior of [a, b] with the exception of the greatest (least) zero, which lies in [a, b] if only c <<

, ppPn+i(dfi;b) i(rffi;6) n+i(ri/x;fr) w+ Pn(dfi;b) Pn(dfi;b) Pn(<W&)

+i(dfjL\bU | c >> ppnPn+iidftbU n +i(d/r,6)| [ - Pn{dfi;b) pnn(dfi;b) JJ

Indeed, the function pn+i(d/z;x)/pn(cfyz;x) increases from -oo to +oo in the intervals x^ < x < x^x (k = 0 , 1 , . . . , n), where xy = -oo, x ^ = +oo.

80

Orthogonal

Polynomials

Theorem 2.24. The following decomposition into partial fractions holds pPnjdwx) n(dn;x) ^ ^ T pn+i(dti;x)

n

=yy^

lhh

J = E -a {r' W -+ir 0x-x^ fc=0

h > 0 (fc = 0,1,2,...,n), I*>0(*-0.1,2 »),

where {x^ } denote the zeros o/pn+i(d/z;x). By (2.36) we have /

P»(^;x) *»(»;*)

p..

tfe (™+i) rr-j : rr = Ik = = KeS KeS** (n+i) pn+i(dn;x) =

** **

p n +(dn;x) i (dM; x) Pn+1

+1) pPnn( ( ^ ; 4 n + 1 ) ))

+1) . .. p'n+1(dn;x■i/ (dn; x[ n+1 (d^;x^ )) +1) n+1 n+1)

w+1) +1) +1) _pUi(^4 )Pn(^;xl" )-pU^;xi )p)p» ) ).^ 0ft n+1 (d/x;4" _pUi(^;4 w+1) )Pn(^;4" +1) )-Pn(^;4 n+1) + i(^;4" n+1) 2 [ P ' n (
Theorem 2.25. If c < d and /x(c) = /z(d), then pn(d^;x) (n € Z+) has at most one zero in the interval [c, d\. Proof. In contradiction to our assumption, suppose that c < xj+\ < x\n' < d. We form II n _2(x) = Hk^iti+i(x - x^). Then we have x)IIn _2(x) forxx<>dd pPn(dfj,; n _2(x)>>00for n(d/z; x)II and by the orthogonality ° = / Pn(dM;x)n Pn{d&x)Iln-n2-2{x)dn(x) (aOd/z(x)

/Ja Ja

-f

= / Pn{dwx)nnn-22(x)dn{x)+ = Ja Ja in contradiction to Lemma 2.1.

2.3

/ p n (d/i;x)n (d/i;x)IIn-2(x)d/z(x) n -2(x)d/x(x)

►jf Jd

Jacobi polynomials

Definitions and algebraic properties In this section we consider the classical Jacobi polynomials orthogonal with respect to the Jacobi weight w«tp(x) = (1 - a;) x) aa(l u>a,/?(x) (l + xf

< 1, a > - 1 , ft p > > -1). (-1 < x <

We introduce the following function +n (-1)" P»(a,/J;x) = ^LI£(1 +»)-/»[(! )«+»(l + xf Pn(a,0; x) = (n 1 -_ *»))--*((!! + s)-0[(l -- xx)«+"(l * ) ' +)^ »]«

n!2

(3.1) (3.1)

2.3. Jacobi

Polynomials

81

(in this formula for fractional a and (5 we shall consider the principal values of expressions), and we shall show that the right-hand side of the last expression is a polynomial of degree n. Lemma 3.1. The function Pn(a,/3] x) for all x e [—1,1] can be represented in the form ,n g*. . TT> x P (n rn\a,p,x)-

n

r ( a++ nnn-H)r(j9 +nn++ H-1) r(a l)r(ff ++ 1) A A __ T(a ++ l)rQ8 1) Z. n,2n

fc=0

r (

fc j k4k(x £ ( x--- !)»-*(! l1)»-*(1 ) n - f c ( l ++ +x) (x x)fck AA a + 7 l _ f c + 1 ) r ( / 3 + Jfc + 1)>

(3.2)

where T(a) is T-functiont A* are binomial coefficients. Proof. By Leibnitz's formula we uave have n

a+n (fc)

+n (n_fc)

a+n 0+n a+n {k) a+n [(1 x)a+n + xx)0+"]("> x) }^ == £f >i^l£4t (l(l(l + l --- x) x) ]]]W [(1 [(1 [(1 - x) y»+»]<»> *[ ([(1 ]]( n]_{nf-ck)) ... [(1 x)a+n [(1 +++ x)^xf+xfn +n

:0 fc=0 Jfe=0

It is easy to see that

( n )a+1 fc -=(-< t - ^:: --;:v-* U V -"-^

c a +a+n n f kck a++nn ((fc) ]] (f cf c) ) ==( (-l) n)(a -~[(1 -- x) x)aa+n (a n)(a ++nn- -l )1)... 1)(1 x)a+n -(-l) l ) ffefc ((a a ++n)(Q . . . ( a(a ++nnn--fHck + l+)1)(1 ( l --- i )x)

' T(a + n-k

+ iy

and exactly in a similar way

, (n-

[[(1 (1 + x) ,+ + 11 (n- ff cc )) =

r|inii2 r^nil| rQfl + n + 1)(1 ++ i)/3+fc :c),+fe T(l3 + k + iy

Consequently, the following representation one can obtain from (3.1) P (a g.x) irn^{a,p,x)a.p.x)-

T(a + +n n+l)r(8 + nn+l) (h -i )yy:ff- ii pM* nMfc r(a T(a + n+l)T(l3 +l)r(^ + + + ll) ) 2_A ^ ^ A »r(a »nrr(a ( a ++ n-fc nn--f ck++ l)r(/? n , 2 n 2^1 l)r(/3++fc fc++ l) 1) nn ffc c x (1 - x) x)fck i)"--fc (l + x)

and formula (3.2) follows from the last representation. Remark. If x > 1, then for Pn(a,f3;x) instead of (3.2) we have the following formula P a P»(a,/?;«) = ^ ( xx - l ) - a( *x + 1 ) " ' [ ( * - l1) a + " (+xn + xlf++n]in\ 0 0

»( '^*) = i J r ( - V~ ( + ^""K* " > °

<

^l

.

(3.3) (3.3)

where we also mean the principal values of multifunctions. Corollary 3.2. The following formulae are valid n Pnn(a, 0; -x) = (-l) P„(/3, a; x) (a,/3;-*) (-l)"Pn09,o;*)

and

n a P p n. (a,/?;l)=( ( a , A i ) - ( » +; « )J

)

(3.4) (3.4) (3.5) (3.5)

Orthogonal

82

Polynomials

Pn.(a,0;-l) P ( « , f t - l=) -(-ir( < -nl ) " (+B 0^ )y.

(3.6) (3.6)

Proof. Replacing x by — x in the equality (3.2), we get P (a,/3;-x) P„(a,/3;-x) n(a,0;-x) =

_

T(a r(a+ n n ++l )l)T(/3 l)r(/? r(/3 + n n ++ l1) ) V n!2" ^ n!2"

n

c ) ffcc ^ A v (-1)*4[(1 ((--ll))f M ^*( l( l + »xx)"-*(l ) " - *f c((!l - xx) »)* |^ r ( a + n-k n - i k + l)T{/3 l)r(/3 + fc ^ k + l) oT(a

••Si fc=0 ( ~- ll ) ^ ( ll - -x»)x)"-*(l x )x) rT(a (( aa + n ++l )l)rpg r ( / ? + nn ++ l1)) y , > ) "- -**( (l !+ + i) l)T{/3 ^ (~1)M*(1 1 " £^ rj r( (/ 0? + n-Jfc n - * + l ) r ( a + ifefc+l) + l) ;;

fe

n!2

= (-l) n P n (/?,a;x).

fc

n

f cfc fc

Identity (3.4) is proved. If x = 1 in (3.2), one has 2^ l)r(/J + n + l) >n Tr ((aa + n + l)r(/? l)r(/3 2" p» /,n -. . ,fl. P ; n "^ ' ' " n!2" T(a n!2 r ( a + l)T{0 l)r(/? + + n n+lY + l)"

Hence

1)) r> / /D i\ _ rr(a r ( a +n n++ +1) Pn(a,/3;l) P ( a / 3 ; 1 )= = ^r|a + 1| nT(a + +1) " ' nlT(a l)

,

„ . '

H666Z +)) , (((n ZZ+), + "

(3.7) (3.7)

whence the formula (3.5) follows. The last formula (3.6) can be deduced exactly in a similar way. Corollary 3.2 is completely proved. By the aid of formula (3.3) one can calculate the coefficients of polynomial Pn(a,P;x). Lemma 3.3. / / nn _l 1 Pn(a, ft x) = anxnn + b6„x (3.8) P„(a,/J; (3.8) nx ~ + . . . , then

+ /? ^ /? + + 22nn++ll)) _ 1 rrT(q (( aa + n *" - ^n!2 / j/? y9+++n nn+ l) ^ r r ( (ar a(+a + ++i) l)

an =

and

- _ n(a-p) p n ( a - / ? ) rr(a ( a + /? /3 + 2n) n!2« " n!2 n r ( a + /? /3 + n + l )

bn= 6 n

,

,_ ,

( (3.9) 3 9)

( nn €€ € ZZZ+) ((n ++ ))

Z ++ )) v{ n € Z

'

( (3.10) 10)

"

*-

Proof. For x > 1 we have the representation /

\ a+n /

(x - l)a+n(x + lf+n = xa+0+2n (1 - i j =

xa+^+2n +

(£ _

\ /3+n

{1 + 1 )

Q)x«+/J+2n-l

+

2.3. Jacobi

Polynomials

83

Differentiating n times we obtain n) [(x - l ) a + n ( x + !)"+»] = Anxa^+n l ) " + n ] ((n)

a+(i nn 11 + Bnxa+(3 ++ --

+ ......,,

(3.11)

where A 1 Ann = (a + /? + 2n)(a + /? + 2n 2 n-- l1) ). . ...((a a + p/? + n + 1), l), Bnw = (p0 8 - aa)(a ) ( a + /? /3 + 2n 2 n -- l l)(a ) ( a + /J /? + 2n 2 n -- 22))......((a a + pi8 + n). n).JJ Next, one finds

(3.12) (3 12) '

-ternr H)"' " ( 1i + -; )

<*-D-<«+i)-'=*—'(i-|)

\

/

x

./

V\

'

Q a-0' fLi++«2 -_ Q (>.(a ++1) 1) ^ L

x

x X X X

X x^ X^ X X2^2

x

J

r!_£ ../ ££+!) ... il +++... 1_£+ M+2) L

/? , /?(/?■ +1 2 X'x5z2 X X X X X

. A

(3.13)

h ..

OL-3 X

Substitution of (3.11) and (3.13) into (3.3) gives

p Pn(a,frx) = ^ 1+ h + +^

*to*')-rtF\ -ir ---\ + = ^{Mn

..][Anx»

+

Bnx»-i + ...]

+1^»(« - 0) + B »]* n_1 + •••}•

By (3.12) one can derive _

ttn

Ann _ 1 r(a + /? + 2n + l) ~ n!2» ~ n!2 n!2» n!2"n r(a + /? + n + 1) l)

and 6n = ±[An{«-0) + Bn\ = ^ ^ n( al + /? + 2 n - l ) n!2 (a + + 00 + + 2n +n n+ + 1) 1) xx (a 2n -- 22 )) .. .. .. (a (a + +P /? + _ n ( a / 9 ) r(a + /? + 2re) n ( a - 0? ) r(a + /? + 2n) ~ n!2n r(a + /? + n + 1) in accordance with our statement. Lemma 3.4. Polynomials Pn(a,(3\x) (n € Z+; x G [-1,1],a,/? > - 1 ) form an orthogonal system with respect to the weight wa^(x). We shall call it Jacobi orthogonal polynomial system.

Orthogonal Polynomials

84

Proof. To obtain the orthogonality relation we need the value of the following integral JJmn-= (l-x)a(l mn:=[ f {l-x) / :

3 + x)< x)f3P Prnm(a,fcx)P (a,0;x)Pnn(a,0;x)dx (a,l3;x)dx

(m,n€Z+).

By formula (3.1) one can deduce Jmn ^ f Jmn = ^

£< f/

a+n

' P Pm n (a,0; ( a , ft x) x ) [(1 - Xx)° ) +n(l(l

+n (n) x ) " +] " ] ( n ) dx. dx. ++ xf

Integrating by parts, one gets Amn := yJ

a+ n) P m (a,/3;x)[(l - x)a+n "(l (l + x)" x)"++"] "]((n) dx

= P m (x; (x;aa, 0) [(1 - *)«+»(l + *)*+»] (B " 1) | l x - J^1 ^ P4(a,/J;x)[(l ( a , / 3 ; x ) [ ( l - x)° + "(l + *)"+"] »y»+»](' ^ ,-1) dx dx ^(o,/J;*)[(l Pj,(a, # * ) [ ( ! -*)«+"(l - * ) « + - ( ! + *y»+"] *y» + "] ((nn - ||)) dx.

= - £

Integrating by parts (n — 1) times, we get n

J:

Ann = (( --1l )) " j| A nn =

Q+n P£Ha, dx. P i , n ) ( a , (3; ft **)(1 ) ( 1 "- xx) ) Q + n ( (l l ++xx)"+" ) " + " dx.

1 (a,/?;x) = 0. Consequently, in this case Jmn If m < n, then P5J P^T\a,P]x) mn = 0, that is

I>

/3 |j (l-x) ( l - xa(l) a ( l ++ x) x)0P P(a,/3;x)P 0 (m (m
Z + ) is orthogonal, as stated above. By Theorem 2.4 the system {Pnn(a, (a, /?; x)} (n € Z+) Lemma 3.5. The norm of polynomials PP„(a,/?;x) is equal equal to to n(a,/3]x) is a ++ ++11 a+0+1 ^^r(<* r(a nn ++ l)r(^ nn ++ ll)) )V' 1.1/2/ «\ M r(a ++n+l)T{0 l)r(^ ++n+l) \ 1 2/ 2 1.1/a/ iIIi D // AA M II /f/ 222° 2 fl\ ft n/ (a,/3) {a ( a 0) S= l||Pn(a,/3;x)|| | P P ( a / ? ; X ) l 1 2 2 ,^ == ** >' ^ " l l " » ^' * ) " > — ' = \(n!(a /?/3++ +22n ) )) j |nnl!((aa+++/g 2nn+ + l l)r(a )lr) (r a( a++/?/3+ +nnn+ ++l l)

Z.J (n €e Z+) (3.14)

moreover ft (a,/3) ~ — (as ( o snn -->• f ooo). o). hnn(a,0) n

(3.15)

2.3. Jacobi Polynomials

85

Proof. In order to prove the formula (3.14) we put put m = n in the preceding calculations, then a (l + xfPl(a, xfP^a,0;x)dx Jnn == J j/ ; (1 --x)x)°(l 0; x) dx

= = =

xr+n(1 +x)0+n dx dx ih ^ i: Il \xPp"""n)(Q)(Q''##**)(1)(1"x)a+n(1+x)0+n P^(a, 0; x){l - x)a+n(l

+ x)0+n dx

T(a+0 fl1 _ T(a r(a +ft /? + 2n+l) 22nn + l1) ) 0+n (l-x)a+n{l+x)(1+X) dx. 2n n!2 r(a +ft nBMXa ZJ++ n + ll )) . / . / 1 X)

■jj

0+n

dX

'

We introduce a new variable t by the transformation t = (x +1)/2. In this integral x = 2t-l

f

+n n a+/3+2n+1 a 0+nn 0+n aa+na+n +n (1 ( l --xa+n x) ) a(la+n + n(l( l ++ xf xf x)^+ dx = = 22a+l3+2n+1 +^+ 2n+1 f/ t^+ (\(l-*) t) + n
/o

By (I. 2.18) the right-hand side of the last relation is equal to ((aa + + l)17(ft l)r(/? ++ 7in ++ 1) l) a+/*+2n+i rrr(a + n Tl + In + 2) r(a +ft P + 2n

2 a + 0 + 2 nl++ l1

Hence r(a++Pft 2+ n++1)l1)) tt+/g+2w+1 ^r(a ( a++nn++l)r(ft lJIX/J++nn++1)l) j7 __ «h , (n , „ *m* _ r(a ++2n2n +0+2n+l T(a l)r(ft 1) n (a, ^ r(a + ft + JJnn n n - nn(a P) p) - n , 2 22n n n r ( a + p + n + 1 } n!2 r(a +ft+ n + 1) P +2n2n++2)2) " > " n!22nr(a r(a + /J «+/g+ir(a + wn + + l)r(ft l)r(ft + 1) 1) l)T(p + + nn + 22«+/»+ir(q p+ " n!r(a + 4-ft+ n + l)(2n + a + ft + 1)' 1)' that coincides with (3.14). Next, we estimate the normalizing multiplier /in(a,/J). n (a,ft). By (I. 2.17) one can get 2.17) one can get nn^ _ n!(2n + a +ft+ l)r(a l)T(a +ft+ n + 1) 7 a^a - =G Cn, :GC nCn, 1 2«+/5+ir(a + + !)r(^ + i) 2«+^+ r(a nn + l)r(ft 2*+£+ l)I\ft ++nn + 1) " n n ~ ' ' i.e.

hnn(a,p)>dn-\ (a^)>Cm-\ (a,p)>dn-\

where the constant C\ > 0 is independent of n € Z+. On the other hand, applying ormulaa (I. ( 22.17) Applying the tut; formula lurmuia yi. &.n) ito (3.14) once more, we obtain hnnn(a,(3)=o(±\. (a (a,0)=o(±\. h (a,0)=ol±\. iP)=0\±\

• &

(3.16)

We infer the limit equation (3.15) from the the last two • estimates. Lemma 3.5 is is completely proved. Theorem 3.6. Let Let {p {pnn(a,ft;x} (a,/3;x} be be the theorthonormal orthonormal Jacobi Jacobipolynomial polynomial system. system. Then the following statements are valid:

Orthogonal Polynomials

86 1.

,/ „a s\ /n!(2n /? + n + l1))) „pp ,_ /nl(2n M 2 n + a + /0? ++ ll))rr(( a + ft ^ a_^s ^ p ? r1 ( a nV( a^A > ); ( a , / ? ; * H y / 2 ^ W n H ) r O ? n l ) ^ Pn *Y a^->r(a 2«+/ + r(a +++ « ++ i)ra?+n-n) l)r(/3 ++ n ++ l ) " '' " ».(«,&«)-^

. ((3.17) 3 J17) 7) (3

-

((neZ n € Z ++ )) xxe€ [[-1,1]); -1,1]); 2.

the leading coefficients of p„(a, pn(a, (3; 0\ x) is 2 / a ++ ft i. - i. / a\ /Q /^ ++ 2n 2 n++ Tl fcn = kn{a,0) = —yj tt+/g+1 nn|!22a+/M-l

r ( a + /? + 2 n + l ) rfr > x - _ *■» + +* /* + + »» » ++ ■ i> l)T(B + n ++ l l)T(a l)r(/? lD yv /rr(( a + n + l)r(/3 ))rrH(( aa + ft /30 + n + 11) l) 3.

(3.18)

(( n- 6. Z Z +. ) ;;

(3 < 3 18) 18>

-'

orthonormal Jacobi polynomials satisfy to three-term recurrence relation

xpnnn(a,P;x) (a,0\x) (a,fcx) (a,/?;x) =a =annppn+ (a,/3',x) pnu(a,/3',x) + a n _ip anaa-ip _i(a,/3;x) xp i(ct,0;x)+u (a,(3;x) -ip -i(ct,0;x) nn+i(a,0;x)+u nnn-ip nn-i{a,f3]x) npnn+ n+i(a,0;x)+u + i(a,/?;x) n p n (a,/?;x) + n-i{a,f3]x) ((p_i(a,/?;x) p . 1 ( a ) f t i ) = 0,0 ,o_i 0, , n e€e ZZ+), (p-i(a,P;x) oo_. ii==00, (p-i(a,0;x) Z+) + ) )

(3.19)

w/tere where (n + ll)(n ) ( n + a 4-f ft ) ( n + a + /?l3j 3++ 1) + l)(n l)(n 0 4+ ll)(n „n _ „„ / ,„„ / m « __ o / l)(n + fl ll)) P ; ann anMWa,p, ^y ( 22 (2n 2 ) 2)2(2n 22) n + a+ +a /? Y(2n Y y (((2n 22nn ++ Qa ++ ^/3+l)(2n ft / 3++1l )l)(2n )((22nn+ +a a+ +/? ^ 0+ + + +/ft 3 ++3) 3) 3)

(n €G Z+) and

(3.20)

"»""»(^" ( a B+a+ f+"2)faB+a+ ^) (" eZ + ) (3-21) «n = «n(a,/3) =

2 22 ft tf2-a -a

(2n + a + /3 + 2)(2n + a + /?)

(n G Z+)

(3.21)

(if fi/ a = — 0 one has UQ = 0). Proof. The representation (3.17) follows from formula (3.14). By (3.9), (3.10), and (3.17) one can calculate the leading coefficient ofpn(a, 0\ x) in the form n! ln\(a /?P + 2n 2n + l ) r ( n + a + 0/? + 1) rT(a 2n + 1) kfc = = k k( (a,0) 3) = //n!(a 3)= ( a + ff ( qa + 0/^ + 2n n n a 3 1r ( n + a + l ) r ( n + /J + l ) _ (a,p) y 2 « + * " n\2"T{n + a n n ~ Y +/ + r(n /3 n!2^r(n y 2 a + / 3 + l r ( n + Q + 1 ) r ( n + /3 + 1) n ! 2 n r ( n + a + /?++ 0/?1)+ 1) n_ nlo,^

nTT _ ^ l_ laa + 0P + 22n"+T 2n+l _1_ "2»V ~

n!2«+/*+i n!2«+^+i

which coincides with (3.18).

r ( a + /3 / 3 +- j -22nn++ll ) ^ / r ( a + n + l)r(/? + n + l ) r ( a + ^ + n + l ) '

2.3. Jacobi Polynomials

87

It remains to prove formulae (3.20) and (3.21) for recurrence coefficients. By (2.18) and (3.18) one finds

( a , ^ ) = —■£aOn ann(a,p) - —2-2n = o ^n+l ^n+l fcn+l =

/1 T(2n + /? 0 +4-1) 1) 4- a + 4- 0P + 4- 2)r(2n + l) 4- a + j8 3 l 1l 2 Y n\2<*+P+ Y{n + a 4+ l ) r ( n 4+ p0 4+ ll)T(n + q 1) n!2«+/ r(n 4) r ( n 4a + 0p +4-1) n\2*+P++r(n X

/(n 4) I > + p0 44- 2) + 22)r(n 2)r(n 4+ l)!2 l ) 1 2a»+/^?++11rr(n ( n + aq 4+ 2)T(n + a 4+ P 0 +~2)~ + q a + 0p + + a 4+ P y T(2n 44- 4)r(2n 40 4- 3)

By a straightforward calculation formula (3.20) follows from the last relation. On the other hand, for determination of u n ( q , 0 ) we compare coefficients of x in both sides of (3.19). Taking into account (3.8), (3.17), (3.19), one can obtain

Rn + l)!(q 4- pP 4- 2n 4- 3)r(n /(n l)!(a + 0 + 3)T{n + q a + 0 + 2) n+1 nn Y 0 + 11 rr(a (a + 4- 2)T(n +- 22)) y «+/'+ 4- n + «+^ 2)r(n +4 -p00 4422«+0+

afl

=

- yY y "

0 4+ l)T(n l ) r ( n + a 4+ P 0 4-1) + 1) r /n!(2n + aq + 4- /? 11 r(n 2«+0+ (rn( n+4 -q aa 4+- l ) r ( nn + 0/? i8 + l ) n 22<*+0+ *° + /W * +1rr(n /?4-l) uu

+ a 4+ /? + a 4+ 0pff + 1 ) ,. /(n)!(2n 40P 4- l ) r ( n 4111i an an an a 2*+0+ r(7i q + i ) r ( n 0 ** «^+ r(n + Q + i)r(n + 2 2«^+ + P + r(n r ( n + a 4 l ) r ( 7 i + //? ?+4+-il)i) ) 2

-» yy

Using (3.9), (3.10), and (3.19), we get

. / + 0P +4-1) 1) (n 4+ l ) ( n 4+ aq 4+ l ) ( n 4+ - 0 4+- l)(n 4+ a 4(TI Y (2n + q + 4 -00 4+- ll)(2n l)(27i ) ( 2 n 4+ - aqq ++ 0 4+- 22)) 2 (2n (2n+ q + 4-0 0 + 3) X

j(n //(n ( n + l)(2n 4+ aq + 0/3 + 3)r(n + 4- a + 0 + 4- 2) (n + 4- l ) ( a - 0)r(2n + 4- a + 4- 0 + 4- 2) T(n Y T(n r ( n + aq + 2)r(n + 4- p0 4+ 2) ((nn + 1)12^+^(71 + aaq + P 0 + 2)

_ j(2n /(2n + aq + p0 4+ l ) r ( n 4+ aq 4+ 0 4-1) + 1) (a - 0 ) n T(2n + a 4+ 0) n + aq 4+ l ) r ( n 4+ p0 4-1) + 1) + aq + 0 +4-1) l) " Yy r ( n 4n!2 rT(n ( n 4Wn

l(2n /(2n + q + 0 + l ) r ( n + q + 0 ++ l 1) ) 1 r(2n + q + 0 + l ) y r(7i l) n!2" ( n + q + 0 + l1)) *' T(n n!2 r(n + q a + l ) r ( n + 0/? + 1) n!2»n rT(n

Orthogonal

88

Polynomials

A simplification gives + qa +4- l)(n 40 4- l)(n -f- a +4-ft /?-f-f1)1) / (n 4- l)(n 4+ft Y (2n 4- a 4+ ft /? 4- l)(2n + a + + ft ^ +4- 2)22(2n 4- a 4- ft 0 44- 3) 3) /(2n 4- a + P /? 44- 3)(n 3)(n 44- a a ++ft P 4- l)(n -hi) (a - ft)r(2n 0)T{2n 4+ aa + ^ + 2)(n 4-1) 4- 1) Y (2n 4- a 4l)(n 44-/3 4-1) r(n 4- a 4-ft p +4- 2) 2) Y + 0 +4- l)(n +4- aa+4- l)(n ft41) T(n , ^ r(2n + a + /?) ft) r(2n 4+ aa +4- /? + !) P 4-1) "nla W r ( n + a +ft /? + + l)l) M n rr(n ( n ++ a a ++ j/ 8?++ll)) ' "T X

It is easy to see that from the last relationship follows (n 4- a 4-ft /? 44- l)(a l)(a --ft)(n /9)(n4-4-l)r(2n l)r(2n 4-4-aa4-+ P/?4-4-2)2) 0 +4- 2)(2n 2)(2n 4+ a +4-ft ft4-4-l)r(n l)T(n 4-4-aa 4-4-ft ft4-4-2) 2) (2n 4+ a 4-ft ^mx T(2n T(2n 4,/ _ m f4-aaai 4/4-?P) )ft) T{2n r(2n 44+ aa +4-ft j08-hi) + l) T(2n 4-1) (a Wn Un r ( n 4+- aa ++ /? a + ft + l ~ vl"a "'"r(n )9 + l) !) ' nr(ri rr(n (n + + /? + 4- a 4-ft4-1)) * A bit of simplification gives - 0) (a-0)(n l) n(a-p) (q-/?)(n + l) n(q-j9) (a-/?)(n l) = n(a + a +4- ft " n^ ' + a +4-ft p +4-22 " 2n 40 Wn 2n 4ft which coincides with (3.21). Theorem 3.6 is completely proved. Corollary 3.7. The following relations are valid

<^)-W-^:l\P^O(n-^ ">.«-j + i ^ T r f i + 0 < " ^

(3.22) (3.22)

A(al)=a A ( 24n) = A(a =2naa-al 22n-+1 -al n - 33)) a2 +l+ 1 = 0 (0(n-

(3.23)

= ^^ ^^ +- +0 0( (n n"-33)) ,,

(3.24)

A(u„) unn - u n +i 0(n~3 3)).. A(ttn) = w 0(n" + i = 0(n

(3.25)

and u n(a,/3) (a,ft) Un

Remark. Using (3.17) and properties of the T-function, after elementary transformations, one can establish the three-term recurrence relation for (nonnormalizing) P n (a,ft;x): 4+ft 4-- aa+ 4-ft)P ft)P +i(a,+ft; ft; x) ^ ^ jj 2(n 4- l)(n 4- a + /? 4- l)(2n H /?)P„ i(a, /?;x) x) nn+i(a, 22 22 > + 0/3 + 2)(2n 2)(2n4+a a4+ft)x 4- aa -- /? ft 0+ 0 +42)(2n 4a 40)x + + 0 ](2n + aa ++ /J (a,/J;x) > = [(2n + aa +4/?)x ](2n + 4- l)P l)Pnn(a,ft;x) (n + /3)(2n +4-a a ++ /?-f2)P (a,^;x). (a,ft;x). J J - 22(n -h a)(n a)(n ++ ft)(2n /J)(2n + -hft 0+ 4- 2)Pnnn___i(a, ^; x). 11 (3.26)

2.3. Jacobi

Polynomials

89

Differential equation Put + n a+n n v(x) = ( l - x ) Q(l-x) ( l +(l+xf ^ + +n .

It is easy to see that \p - a - (2n + 4- a + /?)x]v(x). 0)x]v(x). (1 - x2)v\x) )v'(x) = [/? Differentiating this identity (n + 1) times, one gets n+1 n+l n+i

E' 4U ^n+i[^ ^)x]W[ (x)]( -- >. = ^ n +i [i £^ " - a« -" (2n + a + ^)x]W[t;(x)](" ). =EE^ 2 +2 fc ^ ^ ++11 (l-X ( l - x 2)W ) WVV(x)(" ( x ) ^ 2 --f c ) E^

fc=0 fc=0 n+l n+1

V

n+1 +1 fefc

fc=0 fc=n

A calculation shows that n+2 n+1 (1 - x2X )[v(x)] [a-0+(a + 0x ) ] " + a + [a-/?+(<* / ? - 2)x][t;(x)] 2)x][i;(x)]«+1

+ (n + l)(n + a + /3)[v(x)] /?)[v(x)] = 00..

(3.27)

iFrom ^From (3.1) and by the definition of v(x), v(x), we infer P»(a,/J;x) = tLj ig fl ( (1 Pn(a,0;x) l - x*))-- (( l + x)-*[»(x)]<»). x)-*[t/(x)]<»>. So [i»(x)]W x) a (l + x)"Pnn(a,/3; (a,/3;x). [«(*)]<"> = n!2»(-l)"(l n!2"(-l)»(l - x)°(l x). Substituting the last relation to (3.27), one obtains 2

(1 - x 2)[(l - x)"(l x)°(l + xfP (a,P; x)]" xX)^P„(Q^;X)1" f nP„(a^;x)]" aa

a

]| 0

[a-(3+(a f3-2)x][(l-x) x) xfPPnn(a,fcx)}' (a,0;x)}' + [a-0+(a a - j 9 + ( a + /0-2)x)[(l-x) 8 - 2)xl[(l -(l(lx) (l + xVP»(a,0; x)Y \\ a Q 5 l# JJ + (n + l)(n + a + 0)[(l x)°(l xfPnP(a, x)] = 0. 0., /3)[(1 - x) (l + x)' x)"P (a, 0; ft x)] n(a,/3;x)] We calculate the first and the second derivative of the expression o

((i-*) l - x ) ° (i+xyp,(o,fti) ( l + xyP„(o,/9;x)

(3.28)

90

Orthogonal Polynomials

and substitute to the left-side of (3.28). As a result, we receive 222 J 0 ))[a(a [ a ( a --- 1)(1 ) aaa~" l + ) < x) PP ,0; f t xx) (1 - xxx~)[a(a )[a(a ~222(l(l((.1 (a,0;x) n (na n(a, 1)(L -- xx) x)" ++ xxfP xfr p; x)) n(a, a 1 1 l 0 l l xf){Ji+xf(a, n --a0(l-x) aa0(l ^ l ----xfi(lrx H l-+^^++P - xy'{fi,pM) PPnnxf-iPnfafix) f aP.nft(a,0;x) A i x) ) -ap-il-xy-^l a 1 a 1 Q 00l l3ll 1l 1 a 1 1 3 0 a al l a l 1 -lQ0 + a 11n(a,0; -a(l-x) -~ (l+xf(l xfP' ax)x) x) (l+x)< P'nP' (a,0;x)-a0(l-x) a(l x)*-\l x) -n{c,ftx)-a0(l-x) x) ~ 0; x) n(a,0;x) nP n(a, 0; x) - -a(l-x) a ( l -- a(l-z)*+VxfP^(a,0;x)-a0(l-x) xfKicft *) «(a,0;x)-a0(l-x) f-xfP' t-(ll(l" « ) a~_ 1(l( l -+ P-PnP (l+x) P^(a,0;x)-a0(l-x) - ~(lna0(l +(l+xfxfP{a, n(a, 222 0 111 aaat t 3 2 aa a 2 0 a 2 l +- 0(0 0(0 --- 1)(1 x) (l + xfPP (a, + 0(1 x) !*„(«,ft x) + x)(l(l + xf~P x)< (a, - P0; x) x) x)°(l xf~ x)"(l xf' K(<*, P\ x) n(a, n(a,0;x)+0(l-3 + ""xf-!*„(«, ft *) *) 1)(1 -- x) x) (l(l + + xfPPnnnn(a, (a, 0; x) + 0(1 --- x) x)"(l + x) K aaall 1 a ,l a a a l l1 ll ax) l P' -a(l-x) -a(l-x) --(l+xfP (l +h xfP^(a,0; xfP' (a,0;x)+0(l-x) (l+xfP'nnn(a,0;x) (a,0;x) xf 0P' ^„(a, x) ++(a,0;x)+0(l-x) xf~ xfP' (a,0;x)+0(l +(l+x)< xf~ xf~3xfP^(a, ( aa(l(l, /0;JxfP' ;x) x )+ /0(1 J (al(l- xx) ) ° ((l] a+ n -a(i-x) (l+xfP' (a,0;x)+0(l-x) n* n(a,0;x)+0(l-x) ■a(l-x) -x(l-\l nn(l(a,0;x)+0(l-x) + -P^( P^(a,0;x n

0 a +-{l-xFil+xfPZfaftx)] (l-x)aa(l: )(l-x) (l+xfP' ;(a,0;x)) - ( l ++-x)^'(a,/?;x)] x ) ^x) ' (0aP^(a,0;x)} ,/3;x)] r(a,/?;x)] Qa _11x +-\a-0 [a-0+(a +++0022)*][-a(l ) x ]][[--aa((l l ---a xar) xr)x) )Q - (((l ll++ +x+)xfP x)" P P0xfP (a, a 0; 3 ;x) xx) ) (a: + 0I^ - 22)*][-a(l 2)x][-a(l ~\l x)0x) P0(a, 2)xl[-att -Hl P x) {a-0 >+ ++(a (a ) a-a_1 (a,0;x) I /0; n((a,0; nn(a,0: x) nnn Qa 0 a ll x xfx\xl0P -nnnn(a,0;x) P^(a. (1(1- -x)°(l x)x\ (l(l + x)x^PL(a. P^(a, 0; x)} 0;Bx P (a,0; +- 0(1 xf(a, *)] 0(1 --- x) x)a(l (l L+ ++xf~ xf~ P (a,0; 0;Q:x) x)x)+ ++(l(lx)°(l ++x)*J*(a, x)*J*(a, 0; x)\ a 00 n)[(l -- x) (l + x) xfP + (n + ll)(a )(a + + 0 + n)[(l (a, 0; x)] = 0. x) PP (a,0; x)} nn(a, n

This identity gives

(1 x 22))

( a , / ?f3; xx)) p p ( a a ft; x ) - * [\t=xY a ^ P nPn{a »-<^' *>-~ rh^^x) ^x^fr*)+^$Pn(a,0-,x) ' ' "~rS rrp + m-^Pn(a,0.,x)

+ ^K(<*,P;x)

x) + K(<*,fr*)\

'[]

+ (a + 0-2)x}\-^-P 0- 2)x] —?-P 0; x) + -^—Pn(a,-^—P 0;nx) (a,0;x)+P' + P^(a, 0; x) n(a,0;x) n(a, n(a,0;x)\ =) 111 — 1l + — xX + xx + (n + l)(n + 0 + a)Pn(a,0;x)=O. 4 ( n 4 l ) ( n + /? + a)P n (a,/?;x) = 0.

+ [a-0 {a-0

Collecting similar terms, we deduce from the last relation Collecting similar terms, we deduce from the last relation 0; x) + [ - 2 a ( l + x) -f + 2/J(l +a a -- p 0+a axx 4+ # r - 2x]/^(a, 2x] J*(a, /?; 0\ x) (1 - x222)P'Jl(a, )P;'(a, /?; 2)9(1 - x) 4(1 - x )P'Jl(a, 0; x) + [ - 2 a ( l + x) + 2/J(l - x) + a - 0 + ax + # r - 2x] J*(a, 0\ x) a

, ) - 2 a / ? + (1(^I +4bxx)^ - - « - ^ (1«1 - _ X+ />-»)« v + I[^LZ^ (1 < - x) v(1 + ' ) vv( 1 g' ) " aa xT a - ff 4- (a 4- p - 2)x

I (1 - x)

'

(1 + x)

'

1- X

qa- -) g£++ ( tjta++i£ 9 --222) g) i + ( n + + +| ^ ^-P+fr + P)* + {n+ i ) ( n + Q + J Pn(a,ft,x) = 0. 1 4- x

T

An elementary argument shows that the coefficient of P n (a,/?;x) can be represented in the form n(n 4 a 44//??44 l1). ). Finally, we get )P;'(a, /?; x) \p -- a a - (a (a + 4 0 /? + + 22)x]P^(a,0; 2)x]P' (1 - xx22)K(a, ) ^ ' ( a ,ft0; x) + + [0 ) x ] ^n{a, ( a , 0; x) x) 4-n(n (a,/?;x) = 0 . + n(n + + a-h/? a + 0+l)P 0+ + l)P l)P n(a,0;x)=O. nn(a,0;x)=O.

({(3.29)

'

'

2.3. Jacobi

91

Polynomials

We obtain the following statement. Lemma 3.8. Jacobi orthogonal polynomials P n (a,/?;x) satisfy the equation Lemma 3.8. Jacobi orthogonal polynomials P n (a,/?;x) satisfy the equation (1 - x22)y" + [ / ? - a - ( a + /? + 2)x]y' + n(n + a + p + l)y = 0 (3.30) (1 - x )y" + [ / ? - a - ( a + /? + 2)x]y' + n(n + a + 0 + l)y = 0 (3.30) or, equivalently, or, equivalently, 4- ( l - x* ) a ++ 1H( l + ^* ))/ '3++ 1 3! r^ + n(n + a + /? + l ) ( l - x ) a ( l + x)/3y = 0. (3.31) ax ax Thus, Jacobi polynomials are eigenfunctions of the Sturm-Liouville singular differential operator.

Uniform estimates Lemma 3.9. Let y(x) y{x) satisfy the differential equation

|| f c( x( x) S}) gJ+ ^+ ^( x)) yy ==00)' ^r dy] dx J

(•

(332) (3.32)

where k(x) > 0, tp(x) > 0, and both functions k(x), (p(x) have a continuous derivative. Then the relative maxima of \y\ form an increasing or decreasing sequence according as k(x)
2

(x) + ^ [ y ' ( x ) ] 2 .

If y'(xo) = 0, then v(xo) = y2(xo). It is not difficult to see that f 2 WM rr W M + k'{x)[y'{x)) fe/W[y/k(^)] + 2k{x)y{x)y>{x) 2fc(x)y(a:)^(x) .v/ (x) M -= W VW + (xW(x)]^2k(x)y(xW(x) 2y(x)y 2y(x)y (x) + -j-r v (x) = 2y(x)y (x) + -7-7 ip{x) fc(xV(x)[y'(x)]2 22 k(x)
In view of (3.32) we have 22 -i2 rr "l "i «'(x) vt'(x) / ( * )==-[fc(xMx)r =--[[ffcc(( x« M Mx«))]r'^^ •. r

Hence sgnv'(x) = -sgn[fc(x)y>(x)]', whence the statement follows.

Orthogonal

92

Polynomials

Lemma 3.10. Let the following condition q := max{a, 0} > > -- -q:=max{a,f3}

(3.33)

holds. Then the maximum of the absolute value o/p n (a,ftx) on [—1,1] attains at the point ± 1 . Proof. We apply the preceding Lemma to Jacobi polynomials Pn{a^P\ x). In this case (see (3.31)): a+1 0+l k(x) Ax) = (1 - xx))a+l (l( l + x) k(x) = kQ^(x) *)0+l

and

xf.

Consequently, kaAx)
=n{n + a + p+ 1)(1 - x) 2 a + 1 (l + *) 2 / ? + 1

and [kaAx)
_

X0 =

0-a

^TpTi-

3 35 <(3.35) - >

Thus 2a 20/ 3 [k*A*)<PaA*)Y = = --2n(n+a+p+l){a+p+l)(l-x) [kaA*)<PaA*)Y 2 n ( n - h a + ^ + l ) ( a - h / ? + l ) ((ll - x ) 2 o t ( l + x) x) 2 (x-x (x-x o). 0 ). (3.36)

The auxiliary function v(x) is 11- x 2 v(x) = vv*j,(x) +~f P n 2 (a,ft *) + n{n a,ft»)]aa,0(x) = PZ(a,0;x) 1)(lK(«,l3;x)}\ w ( w + pt t ^ + 1 ) K n(n + a + p + 1) It follows immediately from the definition that vVaA*) > pn(«,ft *) a,0(x)>P^(a,l3;x)

(for all xa; € h[-1,1]) i , 1])

(3.37)

and First, we consider the case First, we consider the case

i/«*(±l) = P (a f /J;±l). va>0(±l) = Pnn(a,p;±l). _ 11 2' <x>-\,P>

o

11 ~2 -\-

It is easy to see that - 1 < x 0 < 1.

(3.38)

(3.38) (3.39) (3.39)

2.3. Jacobi

Polynomials

93

By Lemma 3.9 Ua,/j(x) < V 0 ,/j(l) VaAx) < t>a,/?(l)

((xx € € (X 1)) (x 00 ,) 1))

and x >V v<*,/3(x) tVa,/?(-l) V>aa, ,/ 0j ((~- l ) > ,(3( g(x) ) a0)j

(x )). (X e 6€ (( -- 1 1 ,, XxX000)). )).

In view of these inequalities and (3.37), (3.38) we obtain \Pn{ct,0;x)\ (a,/J;l)| |P»(ot, ft X)| < ft 1)| (X 1)) \Pn(<*,ftx)\ < |P„(o, |P w (o,/J;l)| (x € (Xo, (x 0 ,l))

(3.40)

and \P
(x ( x e ( (- -l l, ,xx00))))..

(3.41)

Thus, Lemma 3.10 is proved in the case (3.39). Now, let a>-\,-l<0<-\. a > ~ , - K / J < ~ .

(3.42)

Let us put Waf/j(:r) = (a + 0/? + l)x 09 - a). wo.^(x) a). <*>a,/?(x) l)x - {0 (0 Obviously, for all x € (—1,1) x x > 0. W a ,fl(x) Va,0( UaA )) > °-

It follows directly from (3.34), that in this case [k all xx €€6 [-i, [-1,1]) [A:a^(x)^a,/?(x)]' a,0(x)<pa<0(x)]' < [k*A*)v*A*)X < 0° (for ( for a11 1]) and Lemma 3.9 shows |P„(o, ft x)| < |P n (a, ft 1)| (for all x € [-1,1]), [-1,1]), \Pn(a,ftx)\ (a,/J;l)|

(3.43)

i.e. in the case (3.42) our assertion is valid. Exactly in the same way, if 0 > - \ , -- l K< aa << -- \ , 0>-\,-l-K then the estimate

** -

2'

| P n„((aa,,/fJt;xx))| << | |PP„n„(((aaa>,,/ffJtt;---ll))|| |P„(a,ftx)|

--1! < *x << 1) ((-1

(3-44) (3.44)

holds. Notice, that in the case a = 0 = - 1 / 2 (see the end of this paragraph) our assertion is trivial. Lemma 3.10 is completely proved. Theorem 3.11. Assume that the condition (3.33) is valid, then for orthonormal Jacobi polynomials pn(a,0;x) (a>0;x) the following uniform estimate ||p \ppnn(a,0;x)\
(-1 x<<111) (( --11 << *X ))

(3.45)

Orthogonal

94

Polynomials

holds, holds, where where the the constant constant cc > >0 0 is is independent independent of of n n€ € Z+ Z+ and and x x ee [—1,1]. [—1,1]. MoreMoreover, over, a+12 2 max \pn(a,0;x)\ = |p |p„(a,/3; c(a,P)n (3)na+x '' n (a,/3; 1)| ~ c(a,

(if a > /?) f3)

(3.46)

(if 0), ( t /aa < /?),

(3.47)

-1<X<1

or p+1 1 22 max \p |pn(a,0;x)\ (a,/?;x)| = \|pPnn(a,/3; (a,/3; - 1 ) | ~ - c(a,0)n c{a,P)n^ ''

— 1<X<1 —1<X<1

w/iere the positive constants c(a, p) depend only on a and p. where Proof. First of all, we notice that from (3.17), (3.40), (3.41), (3.43), and (3.44) one can infer for orthonormal polynomials pn(a,P; x): |p (x000<x
.}

/ aa > > ---4I | \ \ P > -- J\ y (v / ? > ))

and \Pn(<*,ftx)\<\pn(a,ftl)\ W«,ft«)l
(a > - 1 / 2 , - 11 < > --11//22, , - 1 < a << --11//2 ) .

( ~- 11 << X X << 11 )

| p n ( a , # x ) | < | p n (a,/?;-l)| |Pn(a,/?;x)|<|p (a,#-l)|

On the other hand, by (I. 2.38), (3.5) (or (3.7)), (3.6), (3.15), (3.17) one gets pnn(a, (a, 0; 0; 1) - c(a, p)na^2

-> oo) (n -+

and

+1 2 ~ - c ( a , / ? ) ( - i r n " + 1 / 2 (n -► oo), where where constants constants c(a, c^a, p) p) are are independent independent of ot nn 66 Z+. Z+. By Lemma 3.10 and by the combination By Lemma 3.10 and by the combination last last inequalities inequalities and and limit limit relations relations yields our statement. yields our statement. Theorem Theorem 3.12. 3.12. Let Let (3.48) - l < a < - i , --!<-i K/?<-i -!<«<-!,

(c*,P;-l) Pnn P (a,#-1)

then maximum of |P n (a,/?;x)| on [—1,1] attains at the point of a relative maximum, nearest to XQ, defined by (3.35). Proof. First of all, from (3.48) immediately follows - 1 < XQ x 0 <<11 and

a+p P ++ K K 0. 0.

By Lemma 3.9 and (3.36) one obtains vv

x

afi( afi(x) ) >> 00

(~1<X< (-Kx<

X x 0)

2.3. Jacobi

Polynomials

95

and v v

x x a/j( (X 0 <X<1). afi( ) )<° < ° (xo<x< 1).

Hence, relative maxima of vap{x) form increasing sequence if point x tends to xo (from the left and from the right). A similar assertion is valid for |P n (a,/?,x)|. Consequently, the absolute maximum of |P n («, 0, x)\ on [—1,1] attains at the point of a relative maximum, nearest to XQ (from the left or from the right), as stated above.

Asymptotic properties and weighted estimates For the investigation of asymptotic properties of Jacobi polynomials we use the Liouville-Steklov method. By the aid of asymptotic formulae we get a very important weighted estimate of Jacobi polynomials. Jacobi polynomials satisfy the following differential equation (we put x = cos0 in (3.29)): sin220PZ(a, fccos0) 0}P^(a, /?; cos 0) sin 9P^(a,0; cos9) ++ [/3-a-(a [0 - a - (a ++ l30 ++ 2) 2)coscos9}P' cos9) n{a,0; + n(n + a + p0 + l)P cos 0) + l)Pnn (a, (a, /?; 0; cos 0) == 0. 0.

(3.49)

For application of the Liouville-Steklov method one can consider the following auxiliary function a + 1 22 +1/2 2 u = u(0) (sin0) a+1 // (cos0) /PP„(a,/?;cos0) u{0) = (sin0) (cos^/5+1 n (a,/?;cos0)

(3.50)

and we shall prove, that this function satisfy the differential equation 2 „ [ l / 4 - a 22 11/4-/3 /4-/32 / a ++ P^++ ll\V2 ll U ++ nn ++ r u++ [[4ii 2 7^ + 4isW2) + (< • ^ ~ j uJ 4

[ih?m) ^m) {

In fact, In fact,

a+1/2 / e\a+1/2 a

)

/

A

^ r - )\ =°■

)

>

(■

0

<3-51)

+1/2 $V0+1/2

u'(0) = I/ sin d-\ j / I cos - e\ j Pn(a, 0; cos 9) :) ( ( u'(0) = I sin - j f cos - j Pn(a, 0; cos 9) Ua cos(fl/2) [ft + 1/2) sin(fl/2)1 [< ++ 1/2) [(a l/2)co S(fl/2) _ (/?+l/2)sin(fl/2)1 X X I2sin(0/2) 2cos(0/2) I 2sin(0/2) 2cos(0/2) JJ - II sins i n - JJ

/ocnN (3.51)

3

cos' (II■ cos-

jJ

J*(a,/?;cos0)sin0. P^(a,0,cos6)sine.

(3.52)

K

' '

Orthogonal

96

Polynomials

Using (3.50), we obtain r(aa ++ 1l/2)cos(fl/2) / 2 ) c o s ^/ 2 ) W-- *W W'I[ [ 2sin(0/2) ^^72) * W 2iinW2)

O9++ V2)sinW2)1 l/2)sin(0/2)l (^

/^[( 7 /7^-t T U t v

cos(0/2) 22cos(0/2)

JJ

aa+ 2 + 11/2 1/2 /2 + ^£++V 1 222 4cos (0/2)J (0/2) [4sm (0/2) 4cos (0/2)J +1 22 0+1 2 -sin^(sin(^/2)) a+1 / (cos(tf/2))^ (cos(^/2))^+1 P i ;(a,/?;cos^) -sinff(8in(ff/2)) // i^(a > j8;c»sff)

. fl™ f W

22 W [4sin ' 4sin2(fl/2) (0/2)

cos(fl/2) [(a + 1/2) cos(0/2) Ua II 2sin(0/2) a+l/2

X

/

o\ e\

(

(/? + 1/2) sin(0/2)1 sin(fl/2)1 (0 sin(0/2)l 2cos(0/2) JI +l/2 l/2

oV oV*

[cos 9P' (a, 0; 0; cos cos0) 0) -- sin sin220J*'(a, 0P%(a,/J;/J;cos cos9)]. 0)]. 0P'n(a, (3.53) Now, one can substitute the second derivative from (3.53) and function (3.50) on the left-side of (3.51). Taking into account of (3.52), we derive the following representation of the left-side of (3.51) - I sin - J

I cos - J1

[(a + l/2)cos(fl/2) 1/2) cos(fl/2) (0 + 1/2) r(a l/2)cos(g/2) (/? l/2)sin(fl/2)1 l/2) Ssin(fl/2)1 in(g/2)l 2 „2sin(0/2) . /y> ,„. ,„ ,„. I[ 2cos(0/2) n\ ,H,costf), 2sin(tf/2) 2cos(»/2) J\ r'nia.p ; I 2sin(0/2) 2cos(0/2) I n v 'M' f(o + l / 2 ) c ^ / 2 ) _ Q?+l/2)rin(g/2)1 (/?+l/2)sin(0/2)1 o ■ J f(a (a + +1/2)cos(0/2) l/2)cos(0/2) (/?+l/2)sin(fl/2)1 ^ n 0 2sm<, [ 2sin(0/2) 2cos(0/2) JJ ^ '* ' " ' 0 " ' ) " [ 2sin(0/2) 2coB(ff/2) + 1/2 [fl aoa +

(3.54) (3-54)

^ ++l 1/2 / 2 1] „ fl_fl+1/2 ]nt,

[W^) ~ " -" [i^m

4^W2)JPn(aPnMcose) ^;C°S^ 4^m)\

[cos0P^(a, P\ cos 0) - sin2 OP^ipc, 0F%(a,/?; /?; cos 0)} - [cos9P' [cos«P^(a, /8; cos^) f3; cos0)] 9) 0P,i'(a, cos9)\ n(a, 0;

1/4-/9 1/4-/? [l/4-a 1/4-/? / + ■ + [ 4sin l i(«/2) ^ + 4cos2(^/2) + l( n+

t

+

2

22

[ l * ? ^ 4^72) [ 2

+

+

2 + /?^++lA \ 2 l] a +

0-

j JPn(a,/3;cos.).

?+ s dhid\;p(i^a a, /,?f c; c o s »90 ) /f3/ + MM s isni n^ ^++c co os d r2(2f af a++ i j icjocso s^ -^2-f22^/? 22

2 2

= sin sin 0i*'(a,/?;cos0) ^'(a,/?;cos0) T* \ — ' -

Q

-^—) J^.ft«-»)2

By aid of By the the aid of (3.49) (3.49) we we deduce deduce from from this this 22 sin cos 0) 9) sin 0P%(a,P; 9P%{a,0; cos -

0"

a + /3 + l V l . ,

2 2

i)-

0»

( a + - Jj ( l + c o s 0 ) -■ (([■ / ? + - J ( l - c o s 0 )) + cos0MiP^(a,/?;cos0) P^(a,^;cos0)

-[(■ 2

,

9P%(a, 0) 0+ + 2) 2) cos cos 0]/^(a, 0]/*(a, = sin = sin2 0P%(a, 9P£(OL, 0\ 0\ cos cos 9) 9) - [a [a -0 0 + + (a (a + +0 9)P n(a, /?; /?;cos cos0)9) = --n(n = -n(n n(n + +a a+ + /? / ?++l)P l)Pnnn(a, (a,0\ cos0). 0). /?+ l)P (a, /?;0\cos cos 0).

97

2.3. Jacobi Polynomials

Therefore, sum (3.54) (without multiplier Pn(ayfi\ cos 0) may be represented in the form of 2 (a + l/2) 1/2)22cos cos2 (fl/2) (0/2) + 1/2)Q9 + 1/2) (q ((qa + 1/2)Q8+1/2) 2 z 4 sin (0/2) (0/2) 4sin 2 (ft a + 1/2 ft /? + (/? + 1/2)2 sin2(fl/2) (0/2) + 1/2 1/2 + 2 2 2 4cos (0/2) 4 sin (0/2) 4cos (0/2) 4CUB-IP/£/ isin \v/£) tws-yy/t)

+ /? + i \ 2 * 22+ l ) J - n (Kn + a + / ? + 1 )} ')' 2 2 (a + 4-1/2) 1/2)2 ((aa +++ l1/2)Q9 1/2)09+1/2) / 2 ) ( f t+ + 11/2) /2) = (a + 1/2) - a - 1/2 + 1/4 - a 22 4 sin(6>/2) 4sin (0/2) 4 22 (q + /ft+1) (ft / 2 )22 - f/ t3t ---111///222+++11/1/4/44--f-ft/t3222 (ft+1/2) Q8 ++ 11/2) Q8+1/2)22 (a (a-f ?ft+1) + l )222 n 225 4 cos (0/2) (0/2) 4cos 4 4 4cos(0/2) 4 4 Thus, function (3.50) satisfies the differential equation (3.51). For brevity, set .. a +ft+ 1 jr_n+£±!±I. (3.55) JV = n + — 2^ — . One can present the equation (3.51) in the form +

1 / 4-- aa22 11/4 / 4 --/ft 3 22 1/4 2 + 22 44sin sin (0/2) (0/2) 4cos 4 cos (0/2) (0/2)

/ \^n (•

a + a

*u"H- (( ^ + i - ) ' ).■ _ [[ 44fftt22--11 2 [l6cos'(«/2) [l6cos (0/2)

+

We shall prove, that the function

a

A ^4

(») U Ull(B)

a

2\(l

( »2 J )\9 \^» )■

1I \1 \] 22 U4sin («/2)JJ°' 4sin (0/2) ; |J 4sin"^/zwj

(3.56)

o)]-

= VOJaiNO), y/9Ja(N9),

(3.57)

where Ja(x) is Bessel's function of the first kind of order a (see (I. 2.19)), is a solution of homogeneous equation u"+(^j£ U + 1 ++ 7 N V 22\u U =0 0.

(3.58) (3.58) ) =•)• ad substitute s In fact, one can calculate u" and into the left-side part of (3.58):

" ( 1^

3 2 V9N2J'^{N9) ( aW ) -~ \9' a(N9)+ Ja(N9) + (f ^ ^ " ^ + N/ 22\NV2jV9J V9JQa(N9). J'^N9) + ^ ^=J' (N0) \o-Wj l ) ( iFVom here, putting x = N9, one has

902N2J'l{N9) - a22 + 92N2)Ja(N9) JZ(N0) + 9NJ'a(N9) + ((-a = x2J2(x) J2(x) + xJ^(x) xJ'a(x) + (x 2 - a2)J )Jaa{x) (x) = 0

Orthogonal Polynomials

98

because Bessel's function Ja(x) satisfies Bessel's equation (I. 2.20). Thus, function (3.57) is a solution of homogeneous equation (3.58). If a is not integer, then the second linearly independent solution of (3.58) is u2(0) = y/0Jy/0Ja{N0). a{N0).

(3.59)

If a is integer, then (instead of (3.59)) the second solution has a form u2(0) = y/0Y y/9Y a(N0), a{N0),

(3.60)

where Ya{x) is Bessel's function of the second kind (see Chapter I, §2). Further, one represents the solution of the equation (3.56) by formula (I. (I. 2.25). In our case, formula (I. 2.26) can be represented in the form u'i(0)u ui(0)w2(0) (0)ui{0) = c. 2(0) - u'2(0)ui(0)

(3.61)

For determination of the constant c, one can use well known relations (see (I. 2.23)) x x J* J-a(x) JLax(x)UX) J'a( )J-ci( ) ~~-J'-a( )J<*(x) = = a{x)

1 1TX

^ (a

2 J' a(x)Yaa(x)-Y^(x)J a(x) J„(i)y (x) - Y^(x) Ja(x) = =-— TTX

c= c= I*

Hence

ls n o t

(3.62) (3.62)

integer),

(a (a is is integer). integer).

(3.63)

1TX

r ^ ,..

( I = < | I

V

2 . sin air, if a is not integer — sina7r, * 2

IT 7T

,

if a is integer.

Thus, is a is not integer formula (I. 2.25) can be represented in the form ciVOJa{N0) y/0J-a(N0) u(0) = CiV0J (N0) + Cc2V0J-

f

fe r r + Vd Vt[J (N0)J- {Nt) + y/0 JOo I yft[Jaa(N0)J-aa(Nt) JOo

- J- (N0)J (Nt)]F(t)u(t) - J.aa(N9)Jaa(Nt)}F(t)u(t)

dt, dt,

3 64 ((3.64) * )

where the function .,.x f 4/32 — 1+ (l a 9 \ (1 1 \ 1 7T F{t)= ( 2 + a 2 2 [^ (\' ~ I[l6cos ) / yt 4sin (t/2)yj 16cost2** \^4 \ 4 j(hi^^)]^ \t2 4shr(t/2) I \ 2sina7r 2sina7r

<3-65) (3.65)

is uniformly bounded for 0 < t < n — e. We determine arbitrary constants ci,C2 and #o so that instead of u(0) one can substitute the concrete solution from (3.50). Since a > — 1, then one can put 0o = 0 in formula (3.64), and the integral exists as improper. Consequently,

2.3. Jacobi Polynomials

99

equation (3.64) can be represented in the form 1 2

e-"- '

■(• f sin -

r J

!T

-1/2

Pnn(a, (a,/?;cos0) 0; cos8)

( cos - j

'(•

c20-aJ-a(m))

= ci9-"Ja{N9) + ,0

+ 0~a / I• Vt[Ja(N6)J.a(Nt) Jo , Jov a + l / 2 i v 0+1/2

i)'

x [(■s i n - J

0

I cos- j (■

JaJ(Nt)Ja{Nt)J. a(N0)]F(t) a(N6)]F{t)

(3.66)

P«(a.0: Pn(a,/3;cost)dt. cos rt
Since Bessel's function Ja(x) for small x is equivalent to xa (see (I. 2.22)), then integral (3.66) has the following order of quantity as 0 —> 0 (n is fixed): *0

£

/ Vt(Oat-a + o-ata)ta+lt2 Jo

dt

f'

I"

a 2

+2 = 0a f tdt + 0~a f t2a+l dt = 2-0a+2 4+ — - — fl 0 a+ . + - i 2* Jo Jo 22a a + 2 2a + 2 If a > 0, then the absolute valueJO of the second term in formula (3.66) increases to infinity (as 0 —> 0), and the other terms on the left and the right-hand side of (3.66) are bounded. Consequently, c
0f)

i / a. 2 — / 2 p n ( ^ ; 1 ) = CXlr(a f ^(£) + l)l Further, using (3.7), one finds

r(a-hn + l) v (a + l)(a + 2 ) . . . ( a + n) - i / 2 N - r ( a + l ) ( a + 1)(Q + 2 , ) - ( a + n ) « E f c + H ± l ) . (3.67) n! Analogous result one gets, if — 1 < a < 0, because one can divide equation (3.66) on a quantity 0~ 2a , and then we have C2 = 0, as 0 —> 0. In this case (i.e. ci = 0) from (3.66) follows (3.67), if a is negative and - 1 < a < 0. Thus, it follows from (3.66), that the following formula is valid Cl = 2

{s

0" ( w 0+

O-^l O - ^ f ssiinn- jJ

-j ((■CcOoSs2)

PPn(a,0;a»O) cos 0) Pnn(a,0; (a,/?;cos0)

= ^- » r^

=

r(tt r(tt

1)jr j g 9)J-a(Nt) (N9)]F(t) a(Nt)J-aa(NO)]F(t) /^rJ" / ? r ^ a1 ) j«( « ( W> ^ ) + / V~t[J AMNO)J-a(Nt) {N0)]F(t) a(NO)Ja(Nt)- - JJ aa(Nt)JV2n\N<* V2n\N Jo a Jo y/2n\N Jo

( 0'

< J si

v o+l/2 / v a+l/2 /

sin - J sin-1

v 0+1/2 v 0+1/2

I)' cos - J I cos-1

(■

Pn(a,f3;cost)dt. P P„(a,/?;cosi)
(3.68) (3.68)

Orthogonal

100

Polynomials

Since asymptotic properties of functions J-a(x) and Ya(x) axe the same, the proof remains valid, when a is integer. In this case it is necessary to substitute into (3.68) Ya instead of J_ a into (3.68). We consider integral (3.68) as n -> oo. Applying Schwartz-Bunyakowskii's inequality and taking into account the boundedness of function (3.65) for 0 < 0 < ir — e we find the following estimate for this integral (we denote it by An(0))

r9

2 2 2tdt A {0) /?)> (3.69) aa(N6)Ja(Nt) aa{Nt)Jaa(N9)\Hdt a(Nt) - - JJ A2nnn(0) tdt •■ (a,/3)f (3.69) a(N0)J. a(Nt) a(N0)\ Jo Jo Jo /o where hn(a,/3) is defined by (3.14), and the relation (3.16) holds. If 0 < NO < 1, then for integral in (3.69) (which we denote by Bn{0)), using the order of magnitude of J a (x) for x -> +0 (see Ch.I, §2), we derive the following estimate t9 a aa a 2 B Bnifi) (e)
r9

2a

2a 2a

2a

2a 2a = c // (d {o + 22 ++ ttt2a2a$
'f

(* ( Jo

#2-2a/i2 fl2-2a a2

a2

zi2+2aa2+2a \

(3.70) \

\ 2a 2 2 + 2a / - 2 2 2 a2 - 2 a 2a T ^ + %• + *-**?—;0 7T-Z+ %• + 0^-— << cO<*. . 22 -- 22 aa 2 22 + ~ 2 +2 2a a i1 ~ Now, let 1 < NO. We apply the tne asymptotic iformula for Bessel's function (I. 2.21) JJ a{x) ( x = ) =( ^ ) U*)=l^) a (^) \irx I

■)■■

(-f-i)-

3 2 3 23 2 cos > aa >> 00.. cos(x-^--^)+0(x/ x),0(x" c o s( x^ -- ^^ - ^- +j 0) (+ - / /) ), , x* x>a>0.

(3.71)

We will break the interval [0, n]r]

=M

r 11

T i i

: 0 ^[O'Jr]' [ '^J' '

folk first integral Thefirst integral in (3.69) (3.69) can can be estimated estimated byr the following following way

c 9w c c

,2

m ( ia+ma tdt v r a + (Art)a

/j7k~ m\ 7k| fdt ^" ^^•L L\^ c -~-L[-itir N0 '-m~\ ~NT\ S\L [~^0~No -Nr\ c

= =

+

++

+ tdt

J£± I

*'"

JA C C /(

„ * 7iy-2+2a V"~2+2a

" "

■""

1 1

idttdt

(3 72)

-

J

ivr-2-2a\ KT-1-1(x\

„r

2a_L j v 2a -2a+ + N2a < _L +— — < — —22' m y N- - 2 - 2 a N2222^ + N - 2 + 2a 2OL)~N JWl 2 2 a N 2 + 2 a J NOy 2-2a N 2 +2a ) ~7 V N2''

2.3. Jacobi

Polynomials

101

The second integral in (3.69) has an estimate

IJalVNeVNi i f I 1

/ Nx ^

i_ 1

., — — = == H-I y/NoVWt

CnW) < C / Cn{a)
i 1

i rI ,. 1

;= : == .. ttdt dt VmVml

VNiy/ml MV-1 = _ £2 _1 f dt- c 2 ,e ^\_ cc2 ' "c 2N9-i ■)■ 2 NO N£2eeJ \< • NJ N* NO -~ N N* = _N _L fa dt- Nc eeyL l\_ N c N9-1 c ' 2 2 2 N eJa N ey NJ N NO - N2'

(3.73)

{

(3-73)

Now, one can obtain the following asymptotic properties for Jacobi polynomials. T h e o r e m 3 . 1 3 . Ife > 0 is fixed, then for Jacobi polynomials the following formula

0 t

a+l,2 \a+1/2 {( ee\

(/

I sii O- ' I( sin - 1 V / IYa + nJ+ 1) TJ S 12J V2n!iVa

12 9~1/2

0+1/2 e\0+1/2

(f cos - 1 ) \ /

PPnn(a, 0) (a, 0; 0; cos cos0)

(3.74)

-- TS£ °<™> + 0 <"- 1 / °>- .«[£,._.] •«[*■-«] & -™+°^ 1

r

holds, moreover, the constant in the estimate of the remainder term is independent of 9. In fact, if 0 € [l/iV,7r - e], then 1 < NO, so integral (3.69) can be estimated by the aid of inequalities (3.72), (3.73), and (3.16) and the asymptotic formula (3.74) is proved. If we substitute (3.71) into (3.74), then +l/2 1/2 { 0y ey+ ff sin ff cos P sin -- JJ cos -- JJ Pnn (a, (a, /?; /?; cos cos 6) 0) V V \ / \ / / an TT\ J( rT(a r(a ( a + n + l) /__. a?r

/

= = =

0

(3.75) 1 \

._

[l

1

c o s + € [^"T ^ ^ j ■ *^ v U ^ V2cos ('(^ [™--TT-~4 IJj +°{> ° ([e^)' [*•—«]■ —

y/7rn\Na+1/2

r— C.m

'2 )>

^From the asymptotic formulae one can conclude the following important weighted estimate for the orthonormal Jacobi polynomials. T h e o r e m 3 . 1 4 ( B e r n s t e i n S . ) . If a > - 1 / 2 and p > - 1 / 2 , then there exists constant c > 0 independent of x € [—1,1] and n G Z + , such that for the orthonormal Jacobi polynomials pn(a,/3;x) the estimate ((1l _-I )x«)/a2/ 2++l1// 4 (( l1

< cc ((_- 1 1 <<Xx< < 1 )1) + x^)/^22++ 1l // 44||pp nn((aa ,^#; xa ). )| | <

+ x

(3.76) (3.76)

holds. Proof. First of all, we prove the relation

/

a+x,2 o\a+1,2 (

(f s i n - J1 («

)'

■ ( •

C0S

0+1/2 o\ e\0+1/2

( c o s 2- 1J

!)'

r

1Il

(G

\Pn(a,0;cose)\
(3.77)

Orthogonal

102

Polynomials

The absolute value of the principal term in (3.75) is less than c/y/n. Consequently, by (3.15) the estimate (3.77) in the interval [l/iV,7r/2] follows from (3.75). Let 0 e [0, l/N]. Then one can estimate Bessel's function above by c(N9)a . Integral An(0) in formula (3.68) one can estimate by the aid of (3.69), (3.16), and (3.70). Then from (3.68) follows the inequality (for 6 € [0, l/N]) a+1/2 { e\

/

0'

ff sin sin -- jj

(

o\^1/2

0' .^L.

ff cos cos -- jj

(

|P |P nn (a, (a, /?; /?; cos0)| cos0)|

1 < c0^2(Ne)a + c^Lo < c-?L=. y/n y/n ~ ~y/n y/n+1 ++11 y/n Thus, the estimate (3.77) is proved for the interval [0,7r/2]. Further, one can use formula (3.4), which can be rewritten in the form Pn(a,/?;cos0) = (-l) n P n (/3,a;-cos0). If the variable 0 changes from 7r/2 to 7r, then the quantity t = n — 0 changes from 7r/2 to 0. Since cos t — — cos 0, then

/

+1/2 +1/2 e0y o {(

)

I sin - J

(

I/

+1/2 o\p+1/2 $Y

if

"(■ f cos - J

v a + l / 2 /,

8

= ( - l ) n ( cos I J •

Pn(a, 0; cos0) v 0+1/2

0

( sin I ) (■

Pn(/?, a; cost), cost),

moreover t £ [0,7r/2]. Thus, the estimate (3.77) is valid at the interval [7r/2,7r], too. The estimate (3.77) is proved. Changing of variable x = cos 0, we obtain (1

_ x r / 2 ++ ll / 4 ( 1 ++xx))^/ V / 22++l /l /44| |ppnn( (aa/ /?? ;; xx ) | <<

_^c_ ^

y/n Vn + 1

By (3.15) we get the estimate (3.76) from the last estimate. Theorem 3.14 is proved. Corollary 3.15. Let {pn(a,/3]x)} (n e Z+) be an orthonormal Jacobi polynomial system. Then the following assertions are valid: 1. if a,/3 > —1/2, then this system is uniformly bounded in every proper inner subinterval of (—1,1); 2. if-l
2.3.

Jacobi

Polynomials

103

The Chebyshev and the Legendre polynomials Among the Jacobi orthonormal polynomial systems {pn(a,fcx)} (n e Z+) there are three important ones, distinguished by their particular simplicity: these are the Chebyshev polynomials of the first and of the second kind, and the Legendre polynomials. (A) Chebyshev polynomials of the first kind: Tn(x)

=pn(

-1/2,

- 1 / 2 ; x)

(-1<X<1; n €

Z+)

or 1 [2 /2 To(x) = —7=, —=, ^n(^) Tn(x) = \/— \ — cos(n cos(narccosx) ^o(^) arccos x) (n = 1,2...) y 7T

V A"

This system is orthonormal with respect to the weight w(x) = l/x/l — x2 in the interval [—1,1]. ^Prom the definition we infer the validity of the estimates |T 0 (x)|<-±=,

| T „n ( x ) | < y |

( z* €e [ - l , l ] n e N ) .

By straightforward calculations we obtain the recursion formula xTn(x) = \T ^Tnn+1 (x) + iT i r „n _!(x) _i(i) *r„(x) +i(x)

(n = 1,2,...)

*To(x) = ^ T i, ( x« ) *To(*) and max |T n (x)|=T n (±l) = ij / ||

—1<X<1

V 7T

(n = l,2,...)

Polynomials Tn(x) satisfy the following equation 2 2 (l-x n ]"-x[y n ]' + (l-x2)[y)[y +n n2yynn = = 0. 0. n}"-x[yn]'

(B) Chebyshev polynomials of the second kind: C/„(x) = p n (l/2,1/2; x) tfn(*)=Pn(l/2,l/2;x) or

^

=

(-1 ( - 1 << Xx <<1 1; ; nn € ZZ+) +)

/2~sin(n + l) l)arccosx /2"sin(n arccos x

V *—VT^—(n€Z+)-

2

This system is orthonormal with respect to the weight w(x) = vT^x y/l - x2" in the interval [—1,1].

Orthogonal Polynomials

104

From the definition one can get the validity of the following estimates

|£/n(*)l
(3.78)

2 y/l - x2\U <J(-1 < x < 1). v/l-x |t/n(x)\ (-1<«<1). n (x)|
(3.79)

It is not difficult to see that *Unn(x) \un+l (x) xU (x) == lu (x) n+l

+ \lJn-xix), + \un-!(x),

tf-l(x) = 0 U-i(x) =

(fl € 6 Z+) Z+) (n

and max |t/ \Un(x)\ (x)| = Un(±l) = «J /ff (n + 1).

—1<X<1

V Y 7T IT

Polynomials Un(x) satisfy the equation 2 (1 Zxy' 4+ n(n n(n ++ 2)y 2)y == 0. 0. (1 -- xx2)j/;'(x) )y„(x) -- 3xy'

(C) The Legendre polynomials: pW(x) = p„(0,0;x) P{n){x)=p (0 0;x) n

1

(-1 < x < 1; n € Z+).

( - 1 < X < 1 ; n€Z+).

The system pn (x) (n 6 Z+, — 1 < x < 1) is an orthonormal polynomial system belonging to the weight w(x) = 1 (—1 < x < 1). The three-term recurrence relation is the following x

Pn

\x) — —

n

rp: rfo

+

1

.v,.

r (Q)

—^rPn+ll. - 5 r Pnn++1 lx,> W ++

V(2n+l)(2n + 3) ^/(2n y (2n + 4- l)(2n + 4- 3) 0 ( n €e Z pl >(s) = 0). Z++ ), pl°>(s) p(_°l(x)

//7T--0B

rP n - lW

V 4n J — 1

v4n J - 1

^Prom a general theory (see Lemma 3.10, (3.7), (3.17), (3.76)) we get the following uniform and weighted estimates /2n+l _max =: v^/ ^ _^ i |pW( ± I II i |pl°)(x)|=pW(±l) af )|=pg»(±l) = V ~ 2 -

((nez n e Z + ))

and z 2 ) 1 / 4 bi 0 ) (x)| < y. / ?^H ± (1 - W\pV>{x)\ p i < const V V

nir

Ti.1T

(n € Z + ; x €e [-1,1]).

Legendre polynomials satisfy the equation (1 - x2)[y )[j/nn}" ]" - 2x[yn}' ]' + 4- n(n + l)y l)ynn = 0.

(3.80)

2.4. Some Estimates of the Orthogonal

2.4

Polynomials

105

Some estimates of the orthogonal polynomials

The Jacobi polynomials are in their totality bounded in every proper inner subinterval of (-1,1). Suitable criteria of the boundedness or of the weight estimates of the orthogonal polynomials play an important role in some convergence problems of orthogonal series. It is not known a much in this field, and we explain different approaches to these problems. These methods are based on a comparison of the orthogonal systems.

Comparison Theorems Let {p n (c^;x)}, {pn(<7d/x;x)}, x e [a, b] (n € Z+) be the orthonormal polynomial systems with respect to the distributions d//(x) and cr(x)d/x(x), respectively. We start from the following simple result. Theorem 4.1. Suppose that \pnn(rf/x;xo)| |p {dfi; xo)\ <
(ne (ne Z+; Z+; aa << xx00 << b) b)

and a(x) is a polynomial of degree m with a(x) > a(x) > 0

(x 6]), > 0, (x € [a, [a, 6]),

then |p C \pnn(crd/z; (adfji;xx00)|)\ <
(n(nG€ ZZ+; < xx00 << 6). b). + ; aa <

Moreover, if the system {pn(d/j,;x)} (n e Z+) is uniformly bounded in the interval

M;

\pnn(d/x; |p (dfji;x)| x)\
(n(n€€Z+; Z+;xx€e[c,[c,d]d\CC[a,[a,b]) b])

(4.1) (4.1)

and a(x) is a polynomial, taking positive values for x e [c,d], then the polynomial system {pn(adfJL; x)} (n e Z+) is uniformly bounded with respect to x for x e [c,d\. Proof. Polynomial a(x)pn(adfi] x) can be represented in the form n+m n+m

= £<

(x € [a, 6]; n e€ Z+), [a,fc];

fc=0 fc=0

where ck = / cr(x)pn(adfi] (atf/x;x)p x)pk(dw x) rf/x(x) dfi(x) (0(0<
^From the orthogonality of pn(crdfi\x) to pk(dyL\x) (k = 0,1,2,.. .,n - 1) with respect to the distribution a(x)dfi(x), we obtain cfc=0

(fc = 0 , l , . . . , n - l ) .

Orthogonal

106

Hence

n+m n+m

k=n fc=n

Polynomials

CkPk(dw (x G€ [a, [o,6]; 6];nn €€ Z+). Z+) ckPk(d^;x)x) (x

Furthermore, by the Schwartz-Bunyakowskii inequality, one gets M < Wk\ <

u:

it

2 2 {| / a(72(x)p (x)p^((rd/i;a?)d/i| ((Td/x;x)rf/zj

|{ / p22(<7
"if.-

M 1 / 2 ! / a(x)p2n(adfjL;x)dfiy/2

= M^2,

where M = max0<x<6 |^(x)|, and, consequently, n+m n+m

^E

\\Pk(d^xo)\ x0)\ <<(m (m++l)VMO(l) l)VMO(l) ==O(l). O(l). ^
The first part of the Theorem 4.1 is proved. The second part (the proof of uniformity) is now easily obtained. Theorem 4.2. Let 0 be a ^-measurable function in [a, b] and suppose that 2
y/

> \

\

rb /

<J-\t)pl{dp,t)dlJL(t)
/ Ja Ja

a-\t)pl{dn;t)dp{t)
((neZ+) n G Z+)

)

furthermore that \pnn(dfi;x)\ [c,d\ C [a,6] (dwx)\ < ip(x) 0 (independent ofxe | ag((gj )) -
l

I\

-

xX — - t t

(4.3) (4.3)

[c, d\) such that

l
(4.2) (4.2)

(4.4) (4.4)

< L

\~

for x e [c,d] and [x — 6, x + 6] D [a, 6]. Then for the orthonormal polynomials pn(adfjL;x) (ne Z+) 2l \pnn(adii',x)\ < [c [ci
holds, where the constants c\ > 0 and c2> 0 depend only on C, L, 8,a(x). Proof. In consequence of Schwartz-Bunyakowskii's inequality we have from (4.2) «<*nk n i k = =/I / Ja Ja

~ vla

\pn(^d^t)pk{dfi;t)\a(t)dfi(t) \Pn \Pn(^d^t)p k{dfi;t)\a(t)dfi(t)

Y Ja

= =

tiWntMQMtW

IT-

m J

y Ja

P2k(dnt)a{t)dn(t)

1l 22 \lj\Ud Jj%l{dr,t)v(t)d»(t)
2.4. Some Estimates of the Orthogonal Polynomials

107

and

-jfw

Pnk |Pn(
ff

p^(.(Tdfi;t)a(t)d^t)Jjf pl(adpt)a(fyd»(t)Jj^

< C11 ''22
= J£pKdfi;t)
pl(dfi;t)a-i(t)d»(t). pl(dpt)a^(t)dfi(t).

in functions Pk(dpx) (see (II. 2.4)) gives

'f

ppn (adp x) = pn (dp x) II pn (dp t)p t)pn (adp (vdp t) t) d^(t) d^(t) pn (dp n (crdp x) = pn (dp, x) J< n Ja bn-1 Ja Ja bn-1 ^ l Pk (dp x)pk (dp t)pn (vdp t) dn(t). / Jfe=0 ^2 fc=0 Pk (dp x)pk (dp t)pn (adp t) dn(t). / fc=0 CCOU On account of

f

-i(t)
Ja Ja

{ad(i;t)Tln-1(t)
= 00

Ja

for IIn_i G 7rn_i, this relation implies bn-l [bn-1 frbn-l

j^Pk Ja Ja Jfe=0 fc=o

(dp x)pk (dp t)pn (adp t)a(t) d^(t) = 0,

J a

whence, taking into consideration the Christoffel-Darboux summation formula one can obtain the following important representation pn (adp Pn (crdpx) x) ==ppn (dp x)x) II ppn (dpt)p t)pn n(crdp (crdpt)t)dfi(t) dfl(t) n(dp n (dp /Ja: > a(x) - a(t) + an(T(T'-11(x) (x) /I aW " ^-^1—^-\p \pn-i(dp t)pn(dp x) n-i(dpt)pn(dpx) xX-t —t Ja Ja

- Pn(dp t)pn-i(dp x)]pn(adp t) dn(t). - Pn(dp t)pn-\(dp x)]pn(adp t) dn(t). However, for x e [c,d\ (4.4) gives However, for x e [c,d\ (4.4) gives | a(a(x) k (j*( )x)\
Orthogonal Polynomials

108

while in view of estimate (2.22) we have the following inequality \p ip(x) + ar-^x) \pnn(adfi;x)\ (adfi;x)\ <

rb Ja

l {L {L + + S~ S~l[a(x) [a(x) + +
Ja \p„(dfi;t)\]\pn(adfj,;t)\dfjt(t) \pn(d^t)\]\pn(ad^t)\dfjL(t)

+ Ctr-\x)
+

0nn)

_1 a + lJ_1V
1 < - V1*- ^1^) x) + < c[l c[l + + 5(T 1 + + J tf+ L aLa-iixflAfWtfx) -1^)]^172^)

in accordance with our statement. Remark. By the aid of Theorem 4.1 and Theorem 4.2 one can obtain the Corollary 3.15 and a less precise weighted estimate for Jacobi polynomials than (3.76).

Estimations of the Christoffel function Let {pn(dfi;x)} (n € Z+) be the orthonormal polynomial system with respect to the distribution
Xnn(dfjL]x) (dfjL;x) == [^^(rf/xja:)! [J2 \pk{d^x)\2}~ ]-11

(n (n ee ZZ++ ;;ii G G [a, [a,6]). 6]).

(4.5) (4.5)

fc=0 fc=0

Christoffel function plays an important role in some problems of convergence and summability of Fourier polynomial series. The obvious consequence of Theorem 2.7 is Xn(dp,x)= A n(
f6 2 2 min / \U(t)\ \n(t)\ dfx(t). d^t). n67rn-i,n(x)=iy Jaa

(4.6)

Theorem 4.3. Suppose that //(x) = «;(x) w(x) > wo w0 > 0, fi'(x)

(4.7)

holds almost everywhere in the interval [c,d\ e [a, b], then A " 1 ^ ; x)
((nn = l 1,2,...) ,2,...)

(4.8)

i.e. (4.9) - yn)-pl ^ ( d / i ; x ) < C (n = l,2,...) n -J2Pk(d^x)
2.4. Some Estimates of the Orthogonal

Polynomials

109

Proof. Let us put 2 ~i \ _ / wo *°r x e [c'^> W-\ o toTxe[a,b]\[c,d\

{

w

Then in view of Corollary 2.8 n-l

X~l{d^x)

= fc=0 £*>fc=0

n --ll

< X^iw'^x) = Y^Pk(™',x) = 5^Pfc(S;x). fc=o

fc=0

By the linear transformation, carrying [c,d] over into [—1,1] the polynomial Pk{w;x) can be expressed by means of the normal Legendre polynomials pn {x) (see the end of §3).

0

pi ) (-i (-l ++ 2|^). 2^). P f c (-^)=(^o) «(»;«)= (^».)""%r (• _1/2

0)

■ ) '

!)•

On account of the estimate (3.80), one can deduce

Iftfl*.)! < ( ^ o ) " 1 / 2 |P10) (-1 + 2 | f | ) | '

C l . V^{/(d-i)(i-c)

y/WO ^ / ( d - x ) ( x - c )

and thus (4.8). The first part of Theorem 4.3 is proved. The second part (the proof of uniformity) follows in the same way. Theorem 4.3 is completely proved. Suppose that the condition (4.7) isn't satisfied, and lim w(x) = 0 0 (-1 < x 0 < 1).

(4.10)

x—yxo

Theorem 4.4. Let {p n (w;x)} (n € Z+; x € [—1,1]) be the orthonormal polynomial system belonging to the weight w(x) on the interval [—1,1], and let the condition (4-10) is satisfied. Then n n

a; 1 Ji™ ^iEpfcK o) = oo n—>oo n0 + 1 £ T" -#

(4.11)

fc=0 *r=0 fc=0

/iotas. 2

Weight iu(x) is not positive almost everywhere, but one can consider the orthogonalization process for the weight IU, which is nonnegative, w € Ljy[a,6] and j w(x)dx > 0. In this case Corollary 2.8 is valid, too.

Orthogonal

110

Polynomials

Proof. We consider an arbitrary polynomial Rn(x) with an expansion *»(*) =

nn na

E>^ajfePfe(d/i;a;), *

fc=0 fc=0 fc=0

which satisfy

i?„(xo) = ll (( n Z+ + )) .. /*n(xo) = n€ €Z In view of the Cauchy-Bunyakowskii inequality n

n

J«(*o)<[X>a 1l = #(*»)<[ ^(x ^ O f c [£>&(«<« J L i L M ^ ^*»)]• J0 ) < t £>2][f>2Mw*o)]. fc=0 fc=0 fe=0

Consequently, by virtue of orthonormality of {pn(dfi; x)} (n G Z+) nnn

1 1 _

1

1 1 Y,PHAH;X > ^n - = = Y^Pk(dfi;X) ) > „ E* ll^nl|2,iy X>i(<^; *>) * 2-k=0 v * —fc? OLXT JPJ II « n II 2,«; Q0

n

fc=0

a a

||/?

(4.12)

(412)

where, as usual, \\Rnh,w=[f \\Rnh,w=[f ||ft.lk w = [/ /

2 2 1/2 l/21 \Rn{x)\ \Rn(x)\ w(x)dx] w{x)dx] ^(aOlM*)**] "

(n ((n€Z+). n € Z ++ ). ).

Let us introduce the polynomial I °0Qn(x) .\o), _ n„ Xh " ((XX)) === ^^n 0) (0)

n

E: ELobi E*=oK (xo)f (xo)]2 fe=o

"I < 1), 2 < XO *0 << I)' M2 X>1 E^^0)«(*0)PJ1 ^ («0)Pr(*) ) ^ 0) (Z) (*) (("I ("I 1),

m2

fc=0

where {pj^ (x)} (A: G Z+) is the normed Legendre polynomial system. Applying the properties of Legendre polynomials (see §3), we have

<#(*) QI(X) < and

r.

0)

2

0)(*)] y\pf\x)} f>L :Ebi EE-olpfWi 11 L

ELo^o)] 22

& ** ELo^o)] ' fc=0

< < cc (( n n+ + ll ))

(( n x € [-1,1]) n€ € Z+, Z+, X€ X€ [-1,1])

1

c

2 L. // Q Ql(nX(x)dx= )dx= L 0) 0 ZLOIP^M? '-* £Lo[pi W ■'-i ELO[P1 (XO)P

(4.13) (4-13)

°— ( n(n€Z+). ^— €Z+). n+1 » ++ Ii »

(4.14)

(n2n (x 00 )) = = l) 1) 2n(x (n

(4.15) (4-15)

Now, we introduce the auxiliary polynomial r in >I 2 n „(x) = = i1 - l(x - ( x -- xo) x0) n22„(x) and put R33n„(x) (x) = n 22n„(x)Q„(x) # (x)Q n (x)

(i? (i?3n(a;o) 1). 3n(a;o) = 1).

(4.16)

2.4. Some Estimates of the Orthogonal

111

Polynomials

By (4.12), in order to prove the relation (4.11), it is sufficient to establish that there exists a sequence {en} (n 6 Z+) with en -¥ 0 (as n —► oo) such that the following inequality is valid

f.

/ w{x)R%n(x) dx<— 7-i n 7-i n

(n € N).

(4.17)

Let 0 < a < 1/2. Denote

4

{•

A n = | x, x € [-1,1], |x \x - x001| < — \| and en = [ - l , l ] \ A n . i,From (4.14) and (4.16)

>L

•L

J(A n ) = /

ii;(x)i?3 R% w(x)R2n(x)dx R2n(x)dx n(x)dx 0 is independent of n and Jn = max te ^ n w(t) —► 0 (by (4.10)). Further, on account of the inequality (4.13) and the definition (4.16) we find 2 2 J(e„)= J(e n )= / Rln(x)w(x)dx= If UUl„(x)Q (x)w(x)dx 2n(x)Q n(x)w{x)dx

L

L

Je„

Je„

I

< maxIl2 n (x)maxQ n (x) / w(x)dx < encnmaxll2 max Il2nn(x), (x), x€e n x€e n

x€en x€en

JCn JCn

ae€en acGe n

and in view of the formula (4.15) one obtains r

-12n

/

\ 2n

r / I maxnLW < max |l - i ( z - x0)22j12n < ^1 - i - L\ j2n

(■

maxnl„(a;) < maxa I|^1 - |(o: - ar0) j

<< ee Xx■pP(

< ^1 - ± ^ J =)'

22 nn == (-|i) (;J (-^)■■ (^j (

0

ll -- 22 aa

n 7 0, -b ~bn>' 7n-^ ^°'

whence the inequality (4.17) follows. Theorem 4.4 is completely proved. Let the measure fi be an absolutely continuous in the interval [—1,1], //(x) = w(x) (x e [—1,1]), and ixr/n\ ( w(cos0)sin0 V> = {\ W 0

W

{

for 0 € [0,7r] for0e[O,7r]. for«6[0,ir].

112

Orthogonal

Polynomials

Theorem 4.5. Let the weight function have the property that for sufficiently small values ofh W(0 + / i ) - W ( 0 ) rl (4.19) eLl

wje)

holds, and let

i, Jo

■\W{0 +

h)-W(O)\ rf

ww

'-°^( >

VV 1 ( 7 )

]S[J .)

(4.20)

be satisfied for an a > 1; then the sequence {l/nY^kZoPk(w'ix)} ( n € ^0 is bounded for almost every x € [—1,1]. Proof. We begin with some notations. Let x = cos 0, y = cos v, 0 < 0, v < n 1 nn -_l1 Il = cosA;0cosfct; n-i(x,y) n„_i(a;, y) = - + + y^ ^ J cos A;0 cos kv n _1 |"l n_1 ~* I1 Ti 11 11 fiT l nn_1 11 cosk 0cosk< v + ev v = 9 + 5 1 « * * ( * ) + 9 9 + Z > s f c ( 0 + ) = + Jl ( ) V = 1Z9?\i o\o Y^ < n lZ\r *«i J ^ 1r / fc=i fc=i JJ jb-i \\{0 - v) sin (n \\{0 + v) sin (n - ±)(0 (n -- ±)(0 =

+

4sin^

44ssiinn^^

*

Hence |n n _i(x,y)| < 71, n,

|IIn-l(*, |IIn-i(*, v)| V)| < 0o ,l A.,,, . V ^ < 7^-7 jx^—r• 22|sin^2 | s i n ^ 2 JJii || |0 —v| —v| |0

Consequently,

0-

|n„_i(z,y)| <2minf |IIn-i(*,v)| < 2min•(■ Ln ,, — ^— — j \.

(4.21)

_TT /. ,N 1lf/ 1 1 sin(2n-l)0\ > W ( ■^ - i + " . ! ( « . « ) 2 2sin0 J 4 : !!.-,(.,«) = - ^» - - + \siae j > jn.-

, A . (4.22)

We show that

" )

(4.22)

In fact, if 0 < 0 < 7r/(2n - 1) or 7r - 7r/(2n - 1) < 0 < 7r, then the second term is nonnegative, and 1 sin(2n - 11)0 )0 ^ 1 2 " 2 2sin0 Suppose now that n > 3, consequently ir/(2n — 1) < 7r/4. In the elementary inequality 7T\ 22\/2 v^ // 7T\ sin x > x 10 < x < --) 7T

V

4/

we substitute x = ir/(2n — 1): ss m m

TT

« T:

^

2>/2

7r > « > ^

2n 2n — — 11

7r > >

\/2 \^

— •• 2n — 1 n 2n — 1 n

2.4. Some Estimates of the Orthogonal Polynomials

113

For 7r/(2n - 1) < 9 0 < ir - n/(2n 7r/(2n - 1) we obtain 1l // 22\^ \^

_ 1

2 2

n

ss i n ^ ^ ( 2nn--llj)f
2sin0 2sin0

+

J)

1/

_ 1_

2

2

~ i \\

1

\

2sin

2

2 f f l2n^5 tTT /i /

--2^-2-i^ 2 ^ - 2 - i ^ nnJJ^4 ^ 4 nn-Thus if l/

_ 1

ssiinn(( 22 n - l ) f0l\\

1

,

, ^

n I ^ - >J > ~n n (n 22 \^" - 22++- W ( ">^ 3, , - -11< <X x< < 1 )1), , 2sin0- j J 4

i.e. the estimate (4.22) holds. By the extremal equality (4.6) and the definition (4.5) of Xn(w;x) infipr infer n-l l ^2p ^ p |2(k(w;x) w ; ; x ) = A-1\~ (w;;a:) {w;x) k fc=o =° = =

^

Ul_l(x,y)w(y)dy

= /

dt

T

.

/ i x 11* ^ ( x , y)W(2/) dy

2 II _-.x1(x, cos v)W(v)dv IlJ[ (a:,cost;)lV(t;)dt;

= /

n ^ ^ x ^ o s v ) ^ ^ ) - ^ ^ ) ] ^ ^+^WiO) ^ ) [/ nl^facasvftWW-WWdv

= /

n n2 _ 1 ( x , c o s v ) [ T y ( v ) - ^ ( ^ ) ] d v + -2W ' ( ^ ) U2 + V c o s

./o Jo ^•^

(4.23)

n*_i(x,x) n^-x(x) „,„ Ig-ifr) !!»_!(», x) > njl-i(*,*) max —i n;U(s) . : > —5 n max —i > — 5 /i n»_ (t) (t)dt ) r f 2dy /. 1 w £ 1 n^ ^(iMt) dt //_iit nI I- i^( ^x2,/ M y)wy(y)

fl, Ill^tMt) The The denominator denominator of of the the last last formula formula is is /

one can

s f chO 0 c ocos s f cfcv J dv \ i\ + V Vc ocos dv

./o J° L L

2

I L

2

t^i i^i

J J

2

fc=1 fc-i

^

I J

= | n nn^ - il{x ( xJx)W(0) f n ^ycosv)[W(v) ^ c o s t ; ) ^ ) - -W(0)]dO W(0)]dO Jx)W(fl) + =^U + PUl^(x and and thus thus by by (4.23) (4.23) Tl-1 n-l

n - i ( a : , xx) 4 24 (4.24) gpiKx) > WW _^ ( - ) w ) + +_n ^r f _c j/ J; ^ nn jr. ^ *^, coB cosv)mv) )]dv )[iy(t,) w(0)]dv' V n

fc=0 Now we apply the inequality l / ( a + b) 6) > (a — b) 6) / a 2 = 1/a — 6/a 2 , whose validity is evident for real a, 6 under the hypothesis a + b > 0 n-l

]T}pfc(w;z) >n nn_i(x,x) _i(x,x) Y2PI(W]X) fc=0 fc=0 X

[__2 1 4 / Ill^facosvftWiv) 2 \w{0) " 7 r W ( 0 ) I I n - i ( x , x ) ./o

-W(6)]dv

J

.

Orthogonal Polynomials

114 Furthermore, in view of (4.21) and and (4.22) it follows that

W(e)J2pl(w,cosO) W{e)^V\{w,cos9) fc=0 k=0

* ! n "-^ x >" ^ ) i L f 4min k(n2' J * ^ <)F ) | W ^ " wmdv n

1l( /

and therefore l

1

1

dv ( ' i ^ FV j Wl * F0 ""Y'jr^pJ—me)—

SI] sin(2n-l)fl\ \sm(2n-l)0\\

+h — sin* --*-^ * - * 1M | sin*

, /f * . / , 2 m mn j nn

1

)- 44 /o |Jy

\\W(v)-W(9)\ \\W(v) \W{v) - W(0)\ W(0)\ , M{0)

Jo

n_1

- -n - V vl(w: cos 0)W(6)

* t!>

I)

0

D

\ |W(v) , |sin(2n-l)*|\ 4 f . f , * V ^ W -~ WW)| ^ W l _... \ sin* \)+nJ0 / * m m ( B ' ] r ^ p J W?(*) *" ry ww (• Jo Recall that (see Notations) 1 / ^™{1

+

+_ / "\0

c+ C

c

for c > 0 for for c < 0.

-{s

Then we obtain from the last inequality n-1

Rn = j f {I - \ J2ptiw;cos0)W(9)}+ / fc=0

M

«u dB

L + 1 / 7 " min (n>; -JL-) 1™ ^ 2 l FF 0 n ™

nn JO Jo I 7T^

nJo Jo

sm

sin #0

(■ \

I\

\0-v\2)

W{6)

^*.

Jo Jo

for the The first integral in the the last sum is the the Lebesgue constant for the trigonometrical system. It is easy to see see that

i

r s m ( 2 n - l ) 0 ,. ^ . , rtX , . ' U<*0> 0 is independent of Jn. So

4r r . ft,
JO

nJo Jo

\

»

mv) m Ww-^wu,.. ' ) <» **• w ^ ) w(#) !

\9-v\2J

W(6)

2.4. Some Estimates of the Orthogonal

Polynomials

115

In the last integral we introduce the new variable h = v — 0 instead of v. Since, by its definition, W(y) outside [0,7r], it results that \y) vanishes van

ff rr-l-^^i^-

\w(9 + h)-w(e)\ \ y m^mi^^ WW) J =L : JO JO

J-n

W(0)

\JO

2 22 9 \ min(n h~2) dh l±de\iam(n ,h-, )dh

and by substituting this into the preceeding formula we find

^^u^r^^^-h^-^ i:{[ e\u

Now we take into consideration that -1lI n n -- 1ll

fit - 7 r nn-l -l

V n, // YlpUw,cose)w(9)de ^2 Pk(w>cos 0)W(0) d0= ^2 pl(w; pl(w; X)W(X) dx dx == n, JJo / J~lk=o 70 - ° fc=0 fc=0 fc=o

and therefore

Jf{= J ° r

V fr'{-\ ~ -

;»)}*

-- Y,PUU>,COS0)W(6)\de = o, Y\PU«>,COS0)W(6)\de

n

t!>

J

fc=0

= o,

whence we obtain finally

Inn== = ffr \\-- -- -- Vp?KcosO)WW Vpl(w,COS0)W{6) <W d9 I/„ y"rf(w,oo8tf)W(») d*

II

=2 ft r./o (-*"- »rri - y^P?K cos 0)^(0)} ^ = 22£U-^Tpl(w,cosO)W{0)\ r\±-±y2pl(w,cos9)W{0)\ J

°V

dB M

"to

J

^{f^n^-b^ o(te) ++ °(^)

(4.25)

> \h

J

It is only now that we make use of the hypopthesis (4.20). The integral on the right-hand side of (4.25) is and sid

°^)l'^"wrT^r if (•

« ■

i E} -A^C^^C^H^> ■to {• f '(

)

Orthogonal

116

Polynomials

and thus

'"-"(i^)' '(. :) whence, in view of a > 1, oo

oo

rr = = ll

= ll rr=

f£> = f;o(r-«)=£o(r-)<=»Hence it can be inferred from B. Levi's Theorem I. 2.3 that the integrand of fc tends to zero for almost every 0 e [0, ir] and thus 22 rr -_l1

12 r _ 1

7T

cos( IPI(W; COS 0) = 'tFE* ') iwW) ^' r-HX) 2 fc=0 W(9) o lim -— T

v

holds for almost every in [0,7r], since almost everywhere W(0) ^ 0. Consequently there exists a function G(0), almost everywhere finite in [0,7r], such that 2r-l

12 r _ 1

= £|£(u;;co80)
0e[O,7r]

** = 0

holds. If n is an arbitrary natural number, then we choose a natural number r with 2 r _ 1 < n < 2 r - 1 and obtain n-l

22 rr -- ll

^ ^ ( w j c o s f l ) < ]5T3 pjfc(w;cos0) < 2 <7(0) G(0) < 2nG(0). E r

fc=0 * = 0 fc=0

*=0

This inequality shows that the sequence A~1(ty,cos0) is almost everywhere bounded in [0,7r] with respect to 9. Since by the differentiable mapping x = cos 6 the Lebesgue zero sets of [0,7r] are transformed into the Lebesgue zero sets of [—1,1], also l/n Ylk=o Pk(w'ix) k almost everywhere bounded in [—1,1] with respect to x, in accordance with our statement.

The estimations of the orthogonal polynomials defined by a recurrence relation By Theorem I. 2.14 the three-term recurrence relation (2.17) completely determines the orthonormal polynomial sequence {pn} (n € Z+). The important problem is to find properties of {pn(dfi; x)} (n € Z+) in terms of the coefficients of the underlying recursion formula. Recall that the Chebyshev polynomials of the second kind Un(x) have a simple recurrence relation with a n = 1/2, un = 0 (n e Z+) (see the end of §3). A system of orthogonal polynomials for which the recurrence coefficients satisfy lim a n = - ,

n-*oo

2

=0 lim un =

n—>>oo n-¥oo

(4.26)

2.4. Some Estimates of the Orthogonal

Polynomials

117

will be said to be a perturbation of the Chebyshev polynomials. In a similar way one may consider orthogonal polynomials with recurrence coefficients that satisfy lim an = - > 0,

n—too

lim un = b

2

n—>oo

and these orthogonal polynomials are perturbations of Un((x—a)/b) and are called belonging to the class M(a,b) introduced by P. Nevai [1979a]. As long as a =^ 0 one may, without loss of generality, limit the investigation to A4(l, 0) = M, which corresponds exactly to (4.26). In this section we show that if the polynomials pn(x) are perturbations of the Chebyshev polynomials Un(x), then one may wonder whether the polynomials pn{x) have similar weighted estimates. We start by considering a comparison equation, which is constructed as follows: define pn(d/z;x) = p n (x) = 2n(aoa\. . .a n _i)p n (d/z;x), then p n (x) has the same leading coefficient as Un(x) = y/ir/2Un(x) and satisfies 2xpk(x) = pk+\{x) +i(x) + 2ukppk(x) k{x) + 4a^ 1 p ib -i(x),

(4.27)

whereas the Chebyshev polynomials of the second kind Un(x) satisfy (see §3) 2xUn.-kk-11(x) Multiply (4.27) by Un-k-i(x) equations to deduce 0 = Un-k-i(x)p -i(x)pk+i{x) k+i(x)

= Un.k(x) + UnU -2(x). -kn--2k(x).

(4.28)

and (4.28) by pk(x) and substract the obtained -k{x)pn--k{x) - Un-k{x)p k(x)

+ 4a|_ 4a£_11C/ £7nn_fc_i(x)pib-i(x) _fc_i(x)pjb-i(x) -

+

2ukU 2u kU n-k-i{x)pk(x) n-k-\{x)pk{x)

UUnn-k-2(x)pk{x). -k-2(x)pk{x).

Summing from k = 0 to n — 1 gives the following important equation (a lot of terms cancel out): ~

n-l

_

£•

_

2 Pn(x) = Un(x) + ]T{(1 - 4a -lfc(x)}p Pn{x) 4a£)l/ 2ufc£/ _i(x)}p k)Unn--fc k-_ 2(x) kUn-nk_ k(x)fc(x) 2 (x) - 2u

(4.29)

fc=0 fc=0

which shows that pn(x) is Un(x) a remainder which is small (because of (4.26)). An important tool in analyzing (4.29) is Lemma 4.6 (discrete version of GronwalPs inequality). Suppose cn and dn (n G Z+) are nonnegative real numbers such that n-l

Cn
(4.30)

fc=0 ik=0 fc=0

where A is a positive constant, then n-l

Cn < A e x p | ] Pdk c k}'J . c
fc=0 fc=0

(4.31) ( 4 * 31 )

Orthogonal Polynomials

118

Proof. We use induction on n: if n = 0 then the result is true (as usual, an empty sum is deifhed to be zero). Now suppose the result is true for CQ up to c n _ i , then we infer from (4.30) n-l

Cn

k-i k-l

dk d dk exp I ^2 < A + [AE^2 ^2dk }' fc=o j=o fc=0 j=0 k=0 3=0

Use the inequality dk < exp(dfc) —1, the right-hand side becomes a telescoping gives (4.31). It follows immediately from (3.78) and (3.79) that \Un{x)\ < n + 1 for - 1 < a: x < 11 and

\y/l - x2Un(x)\

< 1 for - 1 < x < 11.

(4.32)

(4.33)

Now we have all the tools need to obtain the analogues of (4.32) and (4.33). T h e o r e m 4.7. For - 1 < x < 1 one has n-l

P P

|ft,(x)| ( *k + 1)(|1 - 4fl 4afcl3k\++2 2|u \Pn(x)\ < (n + l)exp 1) exp { £J^( W)}> fc |)}, fc=o fc=0

4 34 ((4.34) - )

fc=0

and

n n -- ll

2 V l - x 22lfti(*)l | p n ( x ) | < 11 + + 2|t* 2|t*fcfc|)} |)} Vl-* + {{ ££ (( kk + + 1)(|1 1)(|1 -- 4a 4a2| k\ +

fc=0 fe=0

(4.35) « ■ » >

xx ee xx p p {{ X ] T) (( ** + + ll )) (( || ll--44oaJl || + 2|u*|)}. + 2|« t |)}. p{X>+ fc=o fc=0

Proof. Using the inequality (4.32) in the equation (4.29), one finds n-l

^ <
Gronwall's inequality with ck = (\pk{x)\)/(k + 1), dfc = (* + 1)[|1 - Aa\\ + 2|ti*|] then yields (4.34). In order to obtain (4.35) we use the inequalities (4.33) and (4.34) in the equation (4.29). Theorem 4.7 is proved. Corollary 4.8. / / OO (X)

]T(fc 4al\ + 2| 2\u f > + 1){|1 1){|1 - *4\ |} == A
(4.36)

2.4. Some Estimates of the Orthogonal

then

\Pn(x)\<(n + l)eA

and

119

Polynomials

(-1 < rr < 1)

y/l \pn(x)\ < 1 + AeA y/l-x-2\pxn2(x)\

(-1 < x < 1).

Theorem 4.9. Suppose that there is a positive constant A independent of n such that n-l

^£ ( f c + 1)(|1 - 4a 4ag|\ 4-+ 2|u 2\u |)\) <
fck

(n (n == l,2,...) 1,2,...)

(4.37) (4.37)

fc=0 fc=0

then there exists a constant C > 0 and an integer m such that for — 1 < x < 1 \pn(x)\ < < C(l C(l - x^*)-(™+V/z |p«(x)| J-^+D/2

( nn £e NN). ).

Proof. ^From (4.35) we find y/l - zx22\p |p„(x)| < 1 ++ A(n A{n + + 1) 1)AA log(n log(n ++ 1) 1) << A^n A^n + + 1) 1)AA log(n log(n ++ 1), 1), n(x)\ < where J4I is a positive constant. Use this new inequality in (4.29), then we obtain n-l

)|p n (x)| < 1 + A \og(k + 1)(|1 - 4a2k\ + 2|u*|). 2|ufc|). (1 - x 2 )|p„(x)| Atx $£ (>* + 1)^ 1)A log(k k=0 fc=0

If A < 1, the sum on the right-hand side is bounded and the Theorem follows. If A > 1, then (1 - x22)\p )|p„(x)| + Am AxnAA~-1x log log22(n (n ++ 11)) << AA22(n (n + + l)^ l)^14 " 1 log log22(n (n ++ 1), 1), n{x)\ < 1 + with A2 a positive constant. We can repeat this procedure and get m (VT^x*) \pn(x)\ (y/l^r\pn(*)\

A+1 mm m m < Am(n + l)A+1 ~~ loglog (n (n + 1)

as long as A + 1 > m. If A < m, then one can insert this in (4.29) once more and find that (y/l — x2)m+1\pn(x)\ is bounded, as stated above. Remark. By (3.20), (3.21), (3.22), (3.24) for Jacobi polynomials p n (a,/?;x) it is evident (4.36) is valid only for the cases (a,/?) = (±1/2,±1/2), where the plus and minus signs can be taken independently, and (4.37) holds for all a,f3 > — 1.

Chapter 3

Convergence and summability of Fourier series

inLl 3.1

Fourier series in an abstract Hilbert space

Let an orthonormal system {xn} (n € N) be a given in a Hilbert space 'H. Further, let / e H. The numbers ck = = cckfc(f) = (/.**) (f,xk) c* (/) =

*€ €N N )) (( *

are called the Fourier coefficients of the element f with respect to the given orthonormal system, while the series oo 00

£

x £ >] *c**x*> * , cc*fc = = (/> (/,**) X *) ((k*€€ N N))

(1.1) (1.1)

is called the Fourier series of the element f eTi. We form the subspace %n = £({xi, £2, • • • > xn})> the elements of which are thus all the possible linear combinations of the first n elements of the given orthonormal system. We have Theorem 1.1. The partial sum n n

sn = ^2c Y,ckkxk

(n£N)

(1.2) (1.2)

fc=l

of the Fourier series (1.1) of the element f €% is the projection of the element f on the subspace Hn' sn = npunf«n

121

Convergence and Summability

122

of Fourier Series

Proof. Since Sn ++ ((// ~~ *n) Sn) f/ == «n

and sn € rin, it is enough to show that / — xn ± Hn- But it is clear (f-s xn,x (/ ni kk) (f-s-s ) k) nix so that

= = 0 (fc = (fc = = l,2,...), l,2,...), f/ — —ssnn A. _L ririnn--

This proves the theorem. We get from this, on using Theorem II. 1.1. Corollary 1.2. p(f;H n) = \\f p(/;ftn) | | / -- ssnn|\\. |. So, sn is an element of the best approximation for / in ri. We construct this element. It is very important fact for applications. Further, since

ll/H2 = K|| 2 +||/-Sn|| 2

and

n

E

2 \\Sn\\ IKII2 == XJ2cl >2 lfe=l

we obtain

n

llZ-Snll^ll/f-fX ll/-s«ll 2 =ll/ll 2E -E4

(1.3) (1-3)

fc=l fc=i

Then we get from (1.3) Corollary 1.3. The following BesseVs inequality holds 2 E^(/)
(1.4)

AA :: == ll

If we have the sign of equality in (1.4) oo

E

2 E f >^£((/ ) = II/II Il/H2

fc=i fc=l fc=l

(1.5) (1-5)

for a certain / €ri, the closure equation (Parseval's equality) is said to be satisfied for/. Remark. The Theorem of Pythagoras asserts that the square of the length of a vector starting from the zero point on n-dimensional Euclidean space is equal to the sum of the squares of its projections on the coordinate axes. Therefore, if we denote by
3.1.

Fourier Series in an Abstract Hilbert Space

123

scalar product of / and (fk (the projections on the direction <£fc)2 expresses the contents of the Pythagoras Theorem. Parseval's equality (1.5) is nothing other than the generalization of this relation for the infinite dimensional space. Corollary 1.4. For an arbitrary f eH the relation Cn = Cn(f) -+0 "► 0

( n (U^OO) ^ OO)

holds. Theorem 1.5. Let f be an arbitrary element, f eH, then the following statements are valid: 1. Fourier series (1.1) of f is always convergent; 2. the sum of Fourier series (1.1) being the projection s= = np-Hnf, npn0f, Ho = = C({x C(ixnn}); \): 3. f = s if and only if the closure equation (1.5) be satisfied for f. Proof. It follows from Bessel's inequality (1.4) that the series YlkLi ck ls convergent. Once again denoting the partial sum of the Fourier series by s n , we have n+m n+m

-fc=n+l £< fc=n+l

\\Sn+m ~ ~ SSnn\\ \\22 = = J2 J2 44 -- > > °° \\Sn+m

(n->00) (n->00)

fc=n+l

(i.e. {sn} is a fundamental system in H), whence the convergence of the Fourier series (1.1) follows. (Note, we don't use that c*, A: £ N, are the Fourier coefficients). Let oo oo oo

ss == E y^cfcXfc. y^cfcXfc. fc=i fc=l fc=l

Since s € Ho and x = (x — s) + s, we can confine ourselves to showing, as in the proof of Theorem 1.1, that / — s A-rio. By Theorem II. 1.1 s = n p ^ 0 / . If, finally, we use (1.3), the last part of the Theorem follows easily. Theorem 1.5 is completely proved. Corollary 1.6. If the system {xn} (n G N) is complete, the Fourier series of any f €ri is convergent to f. In fact, if the system {xn} (n € N) is complete, then Ho = H. So, the projection of any element / G H on Ho is the element itself. The system {xn} (n € N) is said to be closed, if for an arbitrary f eH the closure equation is valid. Corollary 1.7. The system {xn} (n € N) is complete if and only if it closed. For, by Theorem 1.5, the closure equation (1.5) is satisfied for / G H if and only if / G Ho, so that the closure equation holds for an arbitrary / G H if and only if Ho = H, which in turn implies the completeness of the system.

Convergence and Summability

124

of Fourier Series

Corollary 1.8 (generalized closure equation). Let the closure equation (1.5) be satisfied for the elements imenti f,g eH. Then 0oo 0

c (/>0) ^2Ckdk kdkl> ^Cfcrffc, (/-5) (/>0) = == Yl

dk (/,Xfc), dk === (9^k) {g,x {k fc€N). N). cfck(f,x (0,2;*) (A;€ k) (k k), <*(/>**)> ee N).

fc=i

(1.6) (1.6)

If the orthonormal system {xn} (n G N) is complete, the generalized closure equation (1.6) is satisfied for any f,g eH. In fact, by Theorem 1.5 oo OO 0 0

OO oo 00

,ddkXki kkxxkk,fc> s.d , 99'«E< == 1 22

ckxki /f = = 22CkXki

9 =

fc=i fc=l

JJffee==l l

hence n

n

x n

= {f,g)== lim (J2ckxk,y2dkxk^dfcX/b) ) == lim y2ckdk (/>
fc=l 1

fc=i =i fc=l

n—>oo«* = 11 *=1

n

J2c d . = fc=l£< 1

=

k k

fc=i

The second part of Corollary 1.8 follows from Corollary 1.7. Theorem 1.9. Given a numerical sequence {an} (n e N), for which 0oo 0

< OO. ][>n 4 <00

n=l

There exists a unique element f 6 ri such that is Fourier coefficients ck(f) = ak (k € N), the closure equation (1.5) being satisfied for f. sure equatit a n De s n o w n t o De oo akXk dkXk ccan be shown to beconvergent convergentas a in the proof ofof Proof. The series E J^fcLi the Theorem 1.5. Let its sum be / . Then Let its sum Cn(/) Cn(f) Cn(/)) == = U

m m

==, lim |((Y] N) (/, V ]^ajfeXifejXn) afcXjb, akxklxnx) n0) == =a = aannn1 (n( ne€€ N) ( / , *xn n))) = m—»oo

*—■*1

since Efcli a * x fc> x n),) == a n for m > n. The fact that the closure equation is satisfied for / follows from Theorem 1.5. The uniqueness of / also follows from Theorem 1.5, since an element for which the closure equation is satisfied must be sum of its Fourier series.

3.2

Fundamental Theorems on a Convergence of the Fourier series in O.N.S.

Let $ = {
;z + ).

Convergence of Convergence of the the Fourier FourierSeries SeriesininO.N.S. O.N.S.

125 125

The numbers rb

Cfcfc == c ffcc((/)= C (//)) ==■ // ■ f(x)ip f(x)ipkkk(dwx)dfi(x) (dwx)dfi(x) Cfc f(x)tp (d/z; x) d/z(x) Ja Ja

(fc€Z+) ( f c6€ Z +) (fc Z+)

are the Fourier coefficients with respect to the system $ = {
c d x (xe[a,b}) x€ a 6 £E< **(/)^*( ( / ) * * ( * */*; ; * )) ((*€[a,6]) E<^CkU)fk{dn\x) ( > '])D

(2.1)

fc=0 fc=0

is the Fourier series of f(x) with respect to the given system $ = {
E<

S (x):= Sn(dw //;; x) := (o>; x) Ck(f)* Cfc(/)Wb(
We formulate some Theorems which follows from the results of the preceding paragraph. Theorem 2.1. Let n 6 Z+ 6e a /ixed integer, and a o , a i , . . . , a n arbitrary real constants. If we write n n

00*0 == ^2<*k
rb..

, [f(x)-g(x)]2,d\ lt2i(x) l\f(x)-g(x))*d»(x)

(X)

Ja

becomes a minimum if and only if ak — ck(f) (k = 0,1,2,..., n). The minimum itself is n

oo

<£(/), /V(*)«W*)-£<£(/) E 4(f), 4(/), E<14(f)= E< ["f(x)drix)-J2cl(f)= [ f2(x)d»(x)-

,./a /o

i.e.

A

f)

fc=0 fc=0

n

*=n+li k=n+l

r*>

2 1 1/11/2 1/2 2 2 /2 ^} /22 == {{ /f\f inf{ aa la/b^ife] ^ l 2 ]^2 dM} }!ul inf { /l\fV - -EEa d»} V - «~n [\f-s (*n(f)) / ) ]n2(f)] d Md»} } 1 l/2 fcV?fc

S<jf"--E *^ 40)<*(/)• £ '<&/ )• --= E o* °* "k

Ja Ja ja oo

fc=0

fc=0 fc=0

Ja Ja Jo

fc=n+l fc=n+l

Corollary 2.2. o^ For every f € ££[a, 6] we have oo

<E<)

x 2

{BesseVs inequality), inequality), ( / ) } 11/ '/22<<^| | /H/ | |H 2a.,M (Bet { E ;c4fc(/)} (Bcwrf'« M fc=0 fc=0

126

Convergence and Summabiiity Summability of Fourier Fourier Series Series

where

,6

2 a 2 1,/a /8 II/II2.M /{ |/(*)| ll/lk == { /j\f{x)\ |A / ( dn(x)) x ) |d/«(x)} d*(«)} ^ ) } 1 / 2 -. M■

Ja Ja Ja

b) For every / € L2[a,6]

c<*(/)-► 0 f c (/)->0

(*->oo).

The next assertions play an important role in the theory of L^-spaces. Theorem 2.3. Let $ = {
E £<4 < oCooo..

(2.2) (2.2)

n=0 n=0

Theorem 2.4. Let $ = {<£n} (n £ Z+) 6e an O.N.S. The following statements are equivalent: a) $ is complete in L*[a,6]; b) $ is dosed in l£[a,6]; c) Fourier series (2.1) for every f € L 2 converges to f in the metric L 2 . It follows and 2.4, ows from Theorems Theor is 1.8 (n € € Z+) Z+) is is complete, i.o ana z.% nif ^$ == {n} (n then oo OO

r6

x 1.2 ^ ck(f)Ck(9) / f(x)g(x) f(x)g(x)da>(x) :(/)Cfe(5) l ( / , P €:L 2 ) = ^2 L*.M), fc=0k(f)ck(g) / f(x)g(x)dfi{x) = ^2c (f,g e L^[o,6]), Ja

/ f(x)g(x)dfi{x)

6

= ^ck(f)ck(g)

(f,g e L^[a,6]),

f(x)
^ c ]c (d/z;a;), /z;a;), n ^nn^(nd ^2cn(f (dfjL;x), n=0 n

(2.3)

(2.3)

n=0

tu/iose coefficients satisfy the condition oo

/n, + 2 ) < o o , L : == ^£ c42;lclo22olog gg1 2^2 (22(n 2)
£<

n=0 n=0

(2.4) (2.4)

Convergence Convergence of of the the Fourier Fourier Series Seriesin inO.N.S. O.N.S.

127 127

converges for almost all x £ (a, 6). Moreover, if S$(c,x) majorant of the partial sums of (2.3)

(c = (en), n 6 Z + ) is the

n

:, x) == sup supP I||V]cfcV? s;(c,x) 5'*(c,x)= SJ(c,x) 'Y'ck;a0|, n n*€z+Z5 ++

fc=0 fc=0 fc==0

then under condition (2.4) we have inequality 1a 2 r.i/2 r1 >1/a l|S»(c *)||23,„ . M *)|| ll'SJ(c,*)H2.M,,

(2.5)

where C is an absolute constant. The proof of Theorem 2.5 is based on the following L e m m a 2.6. Let $ = {ipn} (^ £ Z+) be an O.N.S. The following inequality is e* of 0/ number {{ck} {k = 00. ,11,,......,, nn)) satisfied for every set

'(i>2*

rb

n

l 2(x)dp{x) 52(^ + ++2), 2),2), // *6282(a0 dn{x) <>l)log*(n E C4D) l°g|(n °d(n (x)dn(a 0dMx)
Ja

/o •Ja

fc=0 fe=0 fc=0 fc=0 fc=0

iJj J 6{x)= max ##(( X <5(x)= max ||]]TTCcQJ<<w(*)|> X) | , 5(3 S(x) = max I > 0 < j < n '/=0 £

w .IE' iy^ w(»)i»

where C is an absolute constant. Proof. We may suppose without loss of generality that CQ = 0 and n = 2 r , r = 1 , 2 , . . . (by setting c n = 0 for n < I < 2r when 2 r _ 1 < n < 2 r ). For each j j = 1 , 2 , . . . , 2 r we write j in the binary system: r

i == E^ ]

r fc j = 5^ 3£ef cf c22 r-- f c,, fc=01 Jfe=0

£^fc efc fc ==

eefkfcc{j) y) rr l 1, , **A:== 00, ,1l l,,,..... .r.r. r. . (j) == o0ooor

fc=0

It is clear then that every sum of the form YH=i h can be represented as follows 3

E

2E >^>-- £EE i>1.1

6 ( 2 -(2.6) )

*«■ *6i.■

*«*/0E»:o>c.2P-.<|
^From this we obtain, by using the Cauchy-Bunyakowskii inequality 2 IEM << ( r ++ i»D) ;EE ( ( |f>| al2<(r

i=l i=i

fc:< k^0 fc:< ^o

k:

1=1

2 <) •■)' 2r

j:^«- 2r -'<'sE;-o«- -

5

££«-2r-'<'SE;.o"2r-

k

r1

\2

6

Ec

2 f*c - l

((pp + l ))221r --**

E ^^ + D DDEE( E E EE (( J=p2»E E fe= <

fc=0 p = 0

„

0.) *)• *)••

fc + ll1 Z=p2rr~- fc + Z=j

128

Convergence and Summability of Fourier Series

"s|Efc=lcfcV? We apply this inequality J X3fc=i cfc^*:(^)|> where fc(x)|, wh ^ucuii/jr in iix order v^iv^ci to estimate the sums = x x € : 61: j(x) is chosen so that | Yjk=l Cfc^fcWI £(x), &( )i x € [°> [a,&]

•tlEille

( p + l ) 22rr~- f c

r• 2 f*c - lil

<* <* (x) (x); r) < ((rr

-|2

-DEE : E [| E «*

+ 1) lI ))^Ej E; :[ [[ ^ W
E E

r fc fc=0 pp= 0 Z + ll fc=0 / ==pp22r ~--ff c +1 + fc=0 =0 p p ==000> |Z=p2

Q<#(x)J .■ cWI(*)J m{x)\

(2.7) (2.7)

If we integrate (2.7) and use use ithe orthonormality of {
f s\ ) M*) <('-+i)EE •)E5 ? ( +*) >EX>' 2 2 log 2*) = (l ( l ++ llog oogg22n) n )2E £^ cc ?2 .

J=00

This completes the proof of Lemma 2.6. 6. Proof of Theorem 2.5.. We first SJshow that the sequence we nrst 2 fc*

5 = C IWi(*)> #2* (c; x) = S22*(C;X) fc(c;x) = 5E^w (^rI r( X ) >) ,

A: l1,2,... , 22 ,, .. . . (co A (co===0) 0) 0) *;== 1,2,... (co

c) = E < W

converges for almost everywhere x e [a, 6], and estimate the L^-norm of S'(c, x) = su •vwhere x € Po b ? Let 9k(x) = E w i ^ ^2 i W . * € Z +- TThen II**II2,M 2L 2L. Consequently, (2.4),.Er=oll*fcHU* E£Lo I I ^ I I U * + !l))2 ^
.-E2Si-V«

oo

OO

1

^E

c ^. ==cf;ii» cf)||**iM*E ll»*lli^
fc=0

fc=0 fc=0 Jb=0 k=0

fc=0

fc=0

OO '2 1 // 2 ~ 2 2 1/2

,

o2 2X21/2 11/2 /2 1/2

; 2 1/2 L ^1.^. < + 1)"))) <
x

2 g (2.8)

(5> + i)-y

fe=0 fc=0 fc=0

fc=0 fc=0 fc=0 .«

iri's Theorem that the It follows from 2.8i and B. Levi's the series YlkLo l**( x )l converges for almost so does S,2fc(c,x), A; € Z+. In addition, rot all cui x j/ e *z [a, |^u, 6], i/j, and euiu consequently V/Uixac^i s c x x € a '( > ) ^ Efclo l*fc(*)l> ( >*0> whence (see (2.8)) we obtain <2Xol**(»)l.*€(o,6),i X.

«• 11

f

A

rt

.1

l - n - r

mi

.1

.

.

V-^OO

.^Ell**^)!

;|IEi»* fc=0 fc=0

l.-r.

/

\l

| | ^ ( 0 , ^ ) |2
fc=0

Now consider the function n 5 , ; (c,x)= ;) = sup 6 Sk(x) S"(c,x) 0
I== **(*) 5 x) =

"(

max

»/(x)|, c . .IS' I E «W(«)|. |

3

* << j <^ 2 ++ ' 22 * / = 2 ffcc fc

/=2 +11E /=2" 1

/=2fc

^

fc€N.

* € N.

(2.9) (2.9)

Convergence of the Fourier Series in O.N.S. O.N.S.

129

By Lemma 2.6 0 oo

rb

2

OO

EZ *

*E

fc+l_1

5l(x) dn(x) < C £ ) f b5l(x)dn(x) i2k(x)dfi(x) ++l )l) 22 £ fe=o fc=0Ja•Ja/ o

2

(2.10)

fct

fc=l fc=l fc=l1

=Q fc=0'

- J.

2CL. < 2CX. » E;
2

J=2 i=2* J=2*

^From this it follows that \imk-^ooh(^) = 0 for almost all x G [a, 6] and this, together with the convergence of S2fc (c, x), A; = 0,1,... (proved above), guarantees the convergence almost everywhere on [a, 6] of the series (2.3). Moreover, since £&(c,x) ■S£(c, x) < then by (2.9) and (2.10), we obtain 1

f 7L 1/2. <||5'(c,x)|| ||S"(C,x)||
Theorem 2.5 is proved. Given sequence {o;n} (n G Z+), 1 = UQ < u\ < UJ2 < ... is called the Weil multiplier for convergence almost everywhere of O.N.S. $ = {
oo

^ c2 gCjfcWfc u ; f c << oooo ^C kU)k<<X> fc=o fc=0 fc=0

impUes the convergence almost everywhere of (2.3). Corollary 2.7. Sequence 2 wnn--= k>K a; log \og2(n (n + 2) (n ( n e Z+) u

is Weil's multiplier of an arbitrary O.N.S. { (t, 2>nn(*;<*A ($; >; dfi; d/i; t,t, xx)) == X> 2>nn,(rfM;t,x) (d/ £>nnn(t,x) (t, x) x) 2> n n (2.11) n *<^fc(d/x;t)(f a, 6], n 7 1G G Z+) Z+) = ^2(p (t,x G e[a,b], (dfjL;x) (*,x [a, k(dfi;t)(f kk(dfjL;x) = ^2

axe also called the Lebesgue functions of the O.N.S. $ = {„} (n € Z+; x € [0,6]) be an O.N.S. and let Ln(x) (n = 1,2,...; x e [a,b]) be its Lebesgue's functions. If the following relation (x)
n B = 0,1,2,... 0,l,2,...

(2.13)

130

Convergence and Summability

of Fourier Series

holds for all x from the set E, E C [a, 6], \E\ > 0, where {Xn} (n = 1,2,...) is a positive, nondecreasing number sequence, then the finiteness of the sums oo 00

^Yl >C*cl*n<
(2.14)

n=l

implies that the series (2.3) converges almost everywhere on E. The proof of Theorem 2.8 is based on the following Lemma. Lemma 2.9. Suppose that the condition (2.13) is valid on some set E C (a, 6) of positive measure. Then the partial sums Sn(x) of every series (2.3) for which (2.14) holds, satisfy the inequality CCxXl/ \2lJ/222' (((nnn===lll,,,222,,,.......))) \Sn(x)\ < CX' xAi

(2.15)

for almost all x € Et where as in (2.13) Cx is a constant that depends on x but not on n. Proof. Let nx be the smallest ch maxA^" maxAfc ' Sk(x) is attained, allest index ind < nIV for 1 U 1 which w 1111,1 i.e. S (x) - 1 / 2 =s:(x) = max -k - = s„,( (2.16) < ^ 0
(2.17)

f

b 6Q(x)dfi(x) < 00. I/ 6+(x)dii(x)
Let us show that

lim 6+(x) < 00 < lim 6+(x) *-

(2.18)

for almost everywhere x e E.

(2.19)

For this purpose, it is enough by (2.17) and (2.18), to verify that V6lim / #8+(x)dn(x) 8+{x)dii{x) < OO. 00. ( * )) rf/x(x) <

(2.20)

n ¥00 n_)>00 n—>oc

- *°°JE JEJE

Using the equation 5Sn{x) ( x ) = Snn(dfj,;x) =■(dfj,; x)> =

b

{t)Vnn(dK (dp, t,t, x) x) dfi(t), dfi(t), /I SSkk(t)V

/ ■

Ja Ja

(k > > n) n) (k

Convergence of the Fourier Series in O.N.S. O.N.S.

131

where the kernel Vn(dfjL\t, x) is defined by (2.11), and the Schwartz-Bunyakowskii le Schwartz-Bu: inequality uality one can ca] find 22

l 2 2 = /jX~l' A"]/ [/ 6$(x)dii(x) 5+(x)di (x)dn(x)= 6+{x)diA(x) Jl' [/( xK

JE JE

JE JE

l/j,(t) dfi(x) Sk(t)T S)V t)Vn.{dll nx(dn;t,x)d(i( k(t)V nx(dn;t,x)dn(t)dti(x)

la Ja Ja

I>

2 2 = Jf Skk(t){ {t)[ jf VnAd Vcnx (dmt,x)X-^ (dr,t,x)X^ Vnx(d^t,x)X-^d^(x)}d^(t) d, ' d»(x)}d»(t) dfi(x)\ d/j,(t) 'E 'E Ja US (t){J VnA< C E 1/2 1/2 <{£sl(t)d»(t)} Sl(t)d»(t)} sKQdnit)}1'

Ja

r

x { Jj

2

-.2

( 2 2 1 ) (2.21)

^1/2

2 \v t,x)X-y x)X~y <^(x)]2 2 dfi(t) } 1 2 . [l>„. x)Kl'2 d^(x)] dn{xj\ ni [Vn n.{dix;t,: x (dp, t, ;)] d»(t)} .

£From the orthonormality ofJathe system tern $

{j"s (t)dn( )y Ja

ii=0 =0

If we use this and represent the square of the integral as a double integral, we obtain from(i (2.21) \6.4i.) that in x)dfi(x) <<{ [If" If /I 5+{x)dn(x)<{ #(x)d/x(x) [( fI vPn V{d (rf/x;t,x)P (dn;t,x)T> (d/x;t,y) (dn;t,y) x(rf/x;t,x)P nx ny nv(dp nv nxlx t K y

JE

^ Ja Ja JE JE JJE IE JE }1-\ 1/2 1/2 1 2 2n2 2/22 xX- J X-y d^(t)d^x)d^y)} ifi(t) dfi(x) rf^(2/)j .

xxX-y Kl'X-y K dKt)dfi{x)d^y)} Kl d»{i.

.

Integrating first with respect to t andI using both Doth the monotoi] monotonicity of {A n } (n = 1,2,...) and the relation (which follows from the orthonormality of the system * = {„„„ 2 >V x , 2 /y), ), /I VVnnx ,(d/i;t,x)V (dfi; t, x)V {dfi;t,y)d/j,(t) (x,y), nx n xny,(y(x, nvny ny (dp,

Ja

e r one where3 nXyV denotes tie of the numb num tlViia s m a l ller s nnxx and and nny,y, w( co m c o i u a i i d KJIIX: v. where nXyV denotes the smaller one of the numbers nx and ny3,, , we we nna find tnat that ^» 1/2 1/2

/2 Jj 6^(x)dn(x)
JJE 8t{x)dn{x)
I JEJE J

\VnxJdti;x,y)\X-lydn(x)dti(y)y )dfji(y)}

[ f |£>n \Vx(d/x;x,s ;xyy)\\-^dfi{x)dfi( ni{dn;x,y)\\-ldn{x)dn{y)

JE J E

\Vni{d^x,y)\X-ldli{x)dn{y)

[ JEf Vn JE \V \Vn,(dit;x,v)\)£dit(x)diitoJy (dfi;x,y)\t,y)\KyMx)My] +

/h + / / \v + J[E Jf E

ny

n

\Dny{dti;x,y)\X-ldn(x)dti{y)\

^1/2 /2

+ / Ln JEJELnAx)K i/i(x)+ Lx {y)\ )Y1 2 . n*i?)Kl-Mx) d»{* + --CCUE UE LnAx)K -Mx) JJJEEE L n^^-lx -Kld^y)} l Wy)} Wy)}1''2--. JE J E

ny

E n JEJE 3) i:in the set .E implies the estimate e The validity of the relation (2.13) (2.13) (2.20), and he rrelation fore (2.19). therefore Ltion (2.13^ i

132

Convergence and Summability of Fourier Series

If we consider the series £JJLi insteadofof (2.3), we find from 3S■ E~=l -Wn{ £ ~ ;1-Cn -oo limi 6*S'(x) > —oo - 0 0 (for almost all x € E, x) > O n—foo n—>oo

where

W'

m *» («) == min

.

\AT

l
Relations (2.19) and (2.22) show that / - ( * ) = Hm *-(x) <.*sW<: ^ < lim tf(x) = /+(*) v n-»oo V ^^n n-N» almost everywhere on E, where f~(x) and f+(x) are finite almost everywhere on E. Consequently, |5 n (x)| < {|/"(x)| + |/ ++(x)|}A n /22

\sn(x)\<{\r(X)\ + \f (x)\}\}/

for almost everywhere x € E, in accordance with our statement. Proof of Theorem 2.8. First of all we notice that our assumption (2.14) entails the existence of a positive, steadily increasing reasing number sequence {fin} with oo

oo /Xnnn = = 0 0, CfcAn//n < 00. Km^L /x /x oo, ^2 cfanVn < 00. oo. lim /X 00, V )^ Cfc^Wn n-»oo *—' n—»oo n = l-i °° n=l

(2.23)

n=l

Let then r n (x) = rn(d/z; x) denote the nth partial sums of the orthogonal series 00 OO oo

£■

(2.24) (2.24)

^Cny/KHnVnttK J ^ ]cny/KV>nn
In order to estimate the partial sums Sn(x) of the series (2.3), we consider the evident equation n+m n+m

,

1 c „(x) - 5S5„(x) (x = = V S5n+m( (x) /T =CkV^k^k(dfJi' cfcky/>
-£:

n +m m - l // n+m—1

E,l£ J2 \S 0 - nSn(a \Snn+m(x) (x)\< £ I5 (x)-5 n++mm(x)-S n (x)i<

.

1

.S,(v \VA*W

/i h^ rf == ~- A

fc=n+l

fci^i

,,

M*)l 01 l r n(x)|

\ A n + l / Mn+1 i V^n+lA'n+l V%i+lMn+l

+| |

1

\

:+l/*fe+l /

| Tfc r f e (x)| X fc((*)i >l i|T

A/^+iMfc+i/

lr (x)| \Tn+m -m(*)l |r wn+m + m (a)|

((2.25) 225)

V VA^»»+»»A n +•m^n+m m tn+m

V^n+mMn+m

estimateof ofthe thepartial sums of (2.24), we have. Using Lemma 2.9 for anLestimate have, for almost all x e E, the inequality 1/2

A m, |r„n+m C:x\)!^ |r„+ (x)| <

I-\Tn(x)\ \rn(x)\ < C |r„(x)| CCxxS y/>^. fK. xy/K. I

Convergence Convergence of ofthe the Fourier FourierSeries SeriesininO.N.S. O.N.S.

133 133

Therefore it follows from (2.25) that n+m—1 n+m—1

l^n+mjx) < (x) |\S5n+m ( x ) -- 5„(X)| Sn(x)\ < n+m s

Y, E

k=n ib=n fe=n

11/2 /2 1/2

11 72 2 (Afc+iAijk+i)/'-]^]\r |7Tfc(x)| -" ((Afc+i/zfc+x)A f c + i / i ftc)+-O M * ) ! + o01,(1) ^k(x)\ x(l)

1/2 [(**M*r [(^Mfc)" [(A fcMfc)fe)- [(■

as n -> oo, for almost all x € E, where o x ( l ) depends on a: and n, and tends to zero, as n —► oo. To prove Theorem 2.8ititremains remainsonly onlytotoshow showthat, that, for almost i 2.8 all x e E n+m—1 n+m-1

) - V '2/ -_ (A r v 'a]i iM . x)| = '[(Afc/ifc)[(Afc/z*)(Afc+iWk+i)" =* [(Ak«k) i/x ■i)|7i(*)| o .((ll)) (A*+iWk+i)]M*)| E I fc=n+l fc=n-hl - I11/ a22

E

fc+

fc+

11/ 122/ a

0 xx

as n -+ oo. For this purpose it is enough, by B. Levi's Theorem, to verify that oo

* = =E / k=oJa k=0 K

1 2 //22 )\dfi(x) (Afc+i/Zfc+i (A i)-111/2 |rf cfc (x)|
(2.26)

Using the monotonicity of the sequence {A n // n } (n G Z+) and (2.23), by by Schwartzsvskii's in Bunyakowskii's inequality, ty, we derrv derive

K

;

sfr

b *(*)}m1/22 { frl{ )dn{x)} 1/2 2 ==){f d^xy (*)} *E f-^rr ~ A * ) <[ M*)} { Jaf ri{XX)dn{x)Y' yAfc+iMA lWt+1 / Jo fc=0 \ 1

1

b

—V

1/2 21 11 2 1// 2 < <

fc=i

This establish iblish (2.1 (2.26), and consequently also Theorem 2.8. 2.8. We consider another application of the Lebesgue functions. Denote 2.26), and
where where

*»» ri aa<x<6
n n)n) , (x) Pn(x) = £ o]a>k<

=ffc=0 E< P»(*) = >L W*) fc=0

(x (xe[a,b]) € (x€[a,b\)

(xe[a,b])

fc=0

>ns, tc with real a^'. Let H denote a class of iun< functions, then nonnegative number ss oi pn(n)K) := suprfd'n{f) := sup su Pn(n) n(n{f) fen fe£ R is said to be the best degree of approximation attainable for the entire class 1Z with arbitrary Pn{x). We prove the following T h e o r e m 2.10. If Sn(f;x) = Sn(dfi;f;x) f;x) (denotes the n-th partial sum of (2.1) t and Ln(x) is the n-th Lebesgue function of the system $ = {„(#)[!++ M*)1

Convergence and Summability

134

of Fourier Series

holds. This estimate is called the Lebesgue estimate. In fact, let e > 0 and let Pn(x) = P n ( / ; x) be a linear combination for which

\m -

sup |.\f(x) \f(x) --Pn(f;x)\
a<x<6 a<x
Thena

\f(x)-S \f(x) -- P P»(/;x)| + |P„(/;x) |P„(/;x) ;x)\ n(P \f(x) |/(x) -n(f;x)\<\f(x)Sn(f;x)\ (f; x)\ < \f(x) (/;x)| x)\ + |P n (/; x) -- S„(P„;x)| 5 nS(P n (/; nn; z)| + |S \S ;x)-S„(/;x)|. + + \Sn( | 5 n ((P P nn(P ; xn);x)-S - 5 (n/(f;x)\. ;x)|. n

n

n

Taking into account the orthonormality of the system {(pn}, we deduce that P n»((//;;xx))--SS„n ( P „n ;;xx))= 0 . . Pn(/; On the other hand, one obtains ~ . - ~ . ,

w ~ ~

^ ~ » ,

|P„(/;i f;t)-f(t)\\V {t,x)\d(i(t) ||S„(P„;x)S „n ( P n„;;;xxx)))---SSSnnn((//;;xx ) | < /f \P |P„(/;t)-/(t)||2> nf(t)\\V n(f;t) n(t,x)\d»(t) n (t,x)|
< |P„(j < sup sup ||P„(/;*)-/(*)! Pn(f;t)-f(t)\[ „ ( / ; t ) - / ( t ) | If \vn\V < sup (t,x)\d»(t) n(t,x)\dfl(t) a
Ja Ja

Ja

< (dn(f) <(d ee)L n(f) +he)L n(x), n(x), <(dn(f)

+

e)Ln(x),

whence we obtain our assertion from the inequality for■ | / ( x ) - 5 „ ( / ; x ) | . whence we obtain our assertion from the inequality for \f(x) —

3.3

Sn(f;x)\.

Behaviour of the partial sums of Fourier series in orthogonal polynomials

For / e L^ [a, b] its orthogonal Fourier series S(d/z; / ) in the orthonormal polyno mial system P = {pn(dfi; x)} (n e Z+) (O.N.P.S.) is written by oo

= £<

S(d/x, / ; x) := Y,c^Tc ]ck(f)Pk(dii\x), S(dfjL,f\x) S(dp,f\x) k{f)Pk{dvx), k(f)pk{dn;x),

(3.1)

fc=0 fc=0

where the Fourier coefficients ck are given by rb

C Cfc fc == Cfc(/) C f e ( /==) = /I f(x)p f(x)p (dfj.;x)dn(x) Ck ck(f) f(x)pkkk(dfx; (dfi] x) x)dfi(x) dfi(x) Ja Ja

(k e€e Z+). Z+). (k

(3.2) (3.2)

The partial sums of Fourier expansion[i (3.1) \o.L) are are n

d/x; //;; x) x) := := V (df, 5Sn{f) ( / ) =^S 5»n((f; (/;ar) (d/x;/;x) X >]cfcckk(f)p ({f)p / ) pk(d/jL\x) (<*//;*) x) / ;r\x) xx) = S5Snn(dr, f c{d^ h fc=00 t!> fc=0

;; .S 5 _- i ( / , * ar:)))s=O0)).. ( ni eGZZ++-;

(3.3) (3.3)

Partial Sums Sums of Fourier Fourier Series Series

135

It is easy to see that Snn(f;x)= (f\x)= S (/;*) = where

f(t)V {dii]t,x)dn(x), fI f{t)V f{t)V{dwt,x)dii(x), (dwt,x)diJi(x), ■A

(3.4)

n n

Ja Ja Ja

n n

= £*>

P n n(d/x;t,x) £> (d/x; t, x) = ^p p(d/x; (dfjL;x)x) (£,(*,xxGG[a,[a,6],6], nnGGZ+) Z+) fc(d/x; t)pk(dp,; £>n(d/x; t, x) = ^p ff cc (d/x; t)pk(dfjL; x) (t, x G [a, 6], n G Z+) fc=o fc=0

(3.5)

(3.5)

fc=0

is Dirichlet's kernel of the system P. By orthonormality of the system {pn} (n G Z + ): / VVnnn(dii;t,x)d/jL(t) (dfi;t,x)d/jL(t) (dfi; t, x) d/z(£) === 1, 1, 1, (nE ((n (n n E€ Z Z+; ++ ;; xG xx G € [a,6]). [a, [a, 6]). 6]). (3.6) ./a Ja Ja

Hence

-/v.

Hence

(nG Sn„(f,( / ,x) t, x) dn(t) dfi(t) (n x € [a,6]). [a, 6]). (3.7) S x )-- /f(x) ( x ) = I [f{t) [/(*) - /(*)]P„(d/i; /(«)]2>n(d/i; *.») (e Z + ; xG S n (/, x) - f(x) = Jaf [f(t) - f(x)]Vn(dfi; t, x) rf/x(t) (n G Z+; x G [a, 6]). (3.7) Note (see Ch.II, §§1 §1Ja and ^2), that polynomial system P = {pn(dfi;x)} (n € Z+) is Note (see Ch.II, §§1 2), b]that polynomial P = {pn(dfi;x)} (n G Z+) is complete and total inand L^a, (—oo < a < 6 <system oo). By virtue of the results of §2 complete and total in L^[a, 6] (—oo < a < 6 < oo). By virtue of the results of §2 we have the following wc uavc i. lie luiiuwiiig we the3 .following T h ehave orem 1 . Let f e■Ll\i L^[o,6]. T/ien £/ie following statements are valid: Theorem 3.1. Let f G L^[o, 6]. Then the following statements are valid: 1. Fourier series of f (3.1) converges to f in the trie quadratic quadratic mean:

1. Fourier series of f (3.1) converges to f in the quadratic mean: |||5 | 5n»(/) ( / )-- / |/|| | aa.,„ M --+ ^ 00

(n -+ —>• oo); oo); ((n

||S„(/)-/||a.M->0 2. the Parseval equation is valid r■ » OO (r °°oc

x 11/2 1//22

r,

(n^oo);

r -6

1/2 i} 1/2

fb

2

2 |jfV(*) **(*)}i" ,. :4a) (£ *(/)} = 2 / (x)dM^)} { £ <*(/)} ={Jaf (x)dn(x)j , J

=0

in. particular, ular, puincuiu

in particular, , oo ( °°

^ X ^50 ^ f

U 2 v /1/2 x 1/2

fc=0

■^67106

f

1/2 1v 1/2 1/2

,i

6-63 2 2 ( *a ; r f x x -i{j[ 1 / / //(*)d/i(*)} ( ) / ( )f •

jf<

inequality). 0(BesseVs {BesseVs inequality).

22 lim cckfc(f) (/) = ( / €e LL = 0 (/ £ 2 )));;

n—»oo

(3.8)

(3.8) (3.9)

^

2

< oo; [a,&]< Ll\a 3. //€L 5. €€LJM L j M **££*<£(/) £ £ o <£(/)<«>; <°°; Aii.rp ^. i/ we introduce

i.{/>>UUA 6b

)= == inf i n i <\ i K"U(/) = E?HdKf)

£

( 22))(

n€7Tn [

:)-U(x)}2d^x)\ Hx)

x-I>>l1/2 l// 2

\

,

(3.10)

[[/(x)-n(x)] / ( ? ) - n ( * ) ]2adM^)| ^(*)|

,

(3.10)

136

Convergence and Summability

of Fourier Series

where the infimum is to be formed over all polynomials U (x) of degree at most equal to n, —then ""7

-"•"-

(\m-s (

2 22 (f;x)} dn(x) [E<»{dn-J)\ dn(x) [EtfHdK f)}22 = j f [fix) [f(x)-S - nSn(f; x)} n(f;x)]

Ja JaOO oo

- EI = E

fc=n+l *=n+l fc=n+l

cC^if) *W) <(n ((nn= - 11,,00,,11,,22,,......)) *(/)

(3.11) (3.U)

i.e. the infimum in (3.10) attains for•II(x)=S U(x) = nS(x). n(x). Observe that the sequence.*£>■ En (d/x; / ) is nonincreasing and lim£< En22>(
(3.12)

n—>oo n->oo n—>oo

holds for feLl[a, b]. First we consider the problem of pointwise convergence of the Fourier series (3.1). The following simple Theorems immediately follow from Theorem 3.1 and the results of a preceding section. Theorem 3.2. Let the O.N.P.S. P = {pn(x)} (n € Z+) be bounded at the point xo and for f e L*[a,b] let 2 Ht)-f(x0)) \ d/x(t) < oo t -—Xo Xo J /' \

jfP^)'***-

holds, then S(dfi;f;x) Proof. We put

(3.13)

converges to f(xo).

ftofr) ==

/(*)-/(*o) X — Xo

In view of the Christoffel-Darboux formula (II. 2.17) and (3.7) one obtains In view of the Christoffel-Darboux formula (II. 2.17) and (3.7) one obtains SSn„(f;x [/(*) - r(xf{xo))V ( / 0;)x 0 )-- //(xo) ( x 0o ))=== / j\f 0)]Vn(dn;t 1x0)dfi(t) n{dn;t,x 0)dn{t) Sn(f;xo)-f(x0)= f [f(t) f(x0)]Vn(dii;t,xo)dii(t) Ja i(F (F (^)[Pn(^o)p Xo0)) -p X = an(d/ Po)Cn+l(-Rt lPn+l(xo)c ( X00)c ) Cn n (nf{F » ooXo)], ))]] yi = Ja a>n(dn)\pn(xo)Cn+i{F ~- Pnn++i(x Xo) = an(dti)\pn(xo)cn^i{FXo) -pn^i(x0)cn(FXo)]y where Ck(FXo) (k = n,n + 1) are the Fourier coefficients of FXo. On account of where Ck(FXo) (k = n,n + 1) are the Fourier coefficients of FXo. On account of (3.13) we have F*n e L?.[a,M, and thus bv (3.9) (3.13) we have FXo e L*[a,b], and thus by (3.9) lim Cn(FXo) = 0. n-¥oo n->oo

Since an(d(j,)
(n€Z+) (neZ+)

we get S5„n(dfi; -¥->■ oo)oo) (dji; //;; xo) xo) -¥ -*■f(x/(x 0) 0 ) (n(n

(3.14)

Partial Sums of Fourier Series Series

137

as stated above. 21 Theorem 3.3. Let f e=L*[a,6]. £>,&].If// OO

r^logfc. 2a 2 ££ ^ ^^f\E?\f)} ^ W> ] a](/)] <
E

(3.15)

fc=i

then the Fourier series (3.1) converges almost everywhere. Proof. We apply Theorem 2.5 with (pn(dn;x) = pn(dfi;x) (n 6 Z+). Using (3.11), from1 the t n c equation uion

and Cn — cn(f)

c*/) (/) == <£(/) of ^ log A:-1 E^[^(/)] =E< £)«K/)[£nrl E ^ i ^ w -== EEE i r^ E EE«K/)=E4[i:nrl

*[£f f (/)i2i22=:

^ log A::

OO

fe=l 1 fe=l fe=l

^

fc=l=i

fc=l fc=l

and

OO

OO

"5-1

5=2 5 5= =2 2

L L ff cc = = ll

2

J L 555=fc+l = = ff cc + + ll

n-l

^ log A;

J J

1 c log n + 0 ( l ) E^-^»+o(i) E 2 fc

fc=i fc=i

2

>er i

^

it follows that the condition oo

2 22 A f> f>*(/)log (/)logoe k
fc=2 fc=2

is equivalent to (3.15), and our statement follows from Theorem 2.5. 'he function f satisfi Corollary 3.4. Suppose that for a S > 0 the satisfies s /(x / l ) -fc)-/(x)=0 / ( x ) = o( o ( llog" o g--1i1"* - * i±A iij f{x + h)-f(x)

\h\

uniformly vrith respect to x for h —► 0, x,x + h G [a, 6]. Then the Fourier series (3.1) converges to f(x) almost everywhere. Proof. By Jackson's Approximation Theorem (I. 2.18) we infer the existence of a sequence of polynomials {E[fc(x)} with

fix) - nkfc((x) =o nx)-u x) =

uo^-±^. _ 1+s

^\\og

Consequently,

kj' :)

2 i_\ =o o([*fC(/)] .J K °^log>ga+M JbJ' )

&Ut'°{i&*i)'

■ ) ■

So that

g ^ -[E^if)f W ==o[o(g^)
i g^fr^(£^)
M

fc=l :1

\fc=2

< OO,

138

Convergence and Summability

of Fourier Series

and our assertion infers from Theorem 3.3. Theorem 3.5. Suppose that the measure /x(x) is absolutely continuous dp/dx = w(x) and satisfy

£ -. (a<x
C C

( x )<<•<- 7—= ££ = 0<w w(x)

((-1<X< ), , ((--11<<XX<<11)1),

(3.17)

Vl Vl —-x* x* where C > 0 is independent of x. Next, we prove the inequality 1

r 222 n = l,2,...) [ ^£2() (/)l C ^--JI^-1 (n=l,2,...) (n [4 )]i :<2 C /£) ]W "'n n^+ 1l <'

(3.18) n+1 where C > 0 is a constant independent of / and n. For, denote note S*(f,x) Dn\Jix) is the urier-< partial sums of f(x) for Fourier-Chebyshev series!S n s;(o;) s;(/,x) S;(:r) = S' hTk(x), n(f, *) = ;£ ^&A.Tfc(x),

ifc=0o

6bfck = =

/

/ ~;1

/(x)Tfc(a:)/ ( ^ ^ ( x ) ^ ^ , r> 1

%/

fc=0

v

1

x

firs kind (see Ch.II, §3). where Tfc(x) is the Chebyshev polynomial of the
2 2

~

VL

X

oo

c±bl

fe=n+l fc=n+1

Now, we show the validity of the estimate 3stima1 ,1 (fc = l,2,...) \h\ 0 doesn't depend on k. In fact, using of function Value Theo:tion / , we have by Mean Valu >ing the monotonicity monotonii rem for monotone is (see Ch.I, §2) me functions

bk

{x) f{=l «& l)= /Tk{x) dx )dx J\ x )*)-%&= nJ 4 ^= 'f(-

=Ij ^VlVl ——=xx - L /, x

—i

l

f T (x) dx )fm**, 7r^ ' h Trrp-1l

r +//d)j w h

T.k (*>\

T-, / \v

cfcr dx

vrh 'VT^x* -l

Tk(x)-==

Partial Sums of Fourier Series

139

where f e [—1,1]. Changing of variable t = arccosx

(—1 < x < 1)

one can obtain fa r /.arccos^ h = J-\f{-l) / cosktdt4-/(1) " ^ I «/«•

,.0 >l / cosJttcftL /arccos^ J

Integrating the last expressions, we get the estimate (3.19). Consequently,

[ ^ ( / ) ] 2 fc=n+l
\Vn(d^t,

x)\ dfi(t)

(n e Z+; x e [a, b}) where Vn(d/jL]t,x) (n G Z+) are the Dirichlet kernels defined by (3.5). Our main aim is to estimate of L n (x), which plays an important role in some problems of convergence (see Theorem 2.8). We start from the following simple assertion. Lemma 3.6. If the relation (II.4-9) holds at the point x € [a, b], then Ln(x) < o / n + 1 (n e Z+), where the constant C > 0 is independent 0/ n € Z+. Proof. By the Schwartz-Bunyakowskii inequality and (II.4.9) one obtains

Lnn(x) (x) <{( <{j

U U

dfi(t)\ dn(t)\ bb

Iif f VV22nn(dii; {d^tt,1x)d^(t)\ x) dii(t)\

)\l/21/2 f(r nnn ^

1/2 1/2 ..^ 1/2 fc=0

}

Mt)> M*)j {&*(*)} {£*£(*)}

and this was the statement to be proved. Lemma 3.7. Let [c, d\ be a subinterval of [a, b], and let P = {pn(d^\ x)} (n e Z+) be a polynomial orthonormal system with respect to the distribution dp, satisfying in [c,d\ condition (II.4.I). Then for Lebesgue's function (3.20) the estimate Ln(x) = Ox(\ogn) (3.21)

140

Convergence and Summability of Fourier Series

holds almost everywhere in (c,d). Moreover, if the measure \x is absolutely continuous dfi/dx = w(x) and w(x) is bounded in [c, d\ : sup w(x) < C < oo, (3.22) x€[c,d]

then the estimate Ln(x)

= O(logn)

(3.23)

holds uniformly in every inner closed subinterval of (c, d). Proof. Let x G (c,d). Then for sufficiently great n the interval [x — 1/n, i + l/n] belongs to (c, d). Therefore the points a,c,x — 1/n, x + 1/n, rf, 6 divide the interval [a, b] into five subintervals, non overlapping apart from their end-points. The Lebesgue function Ln(x) may be written in the form Ln(x) \Vn(dn;t,x)\ dfx(t) == IInlnl ++ JInn22 ++ / Jn n3 M * ) := • = // !^n(rf/i; *, X)\ dfx(t) 3 ,, ./a Ja where re

rb

Inl = / + / , ./a

r-x—1/n

/n2 = /

^d

rd

+ /

./c

rx+l/n

,

/n3 = /

Jx+l/n

Jx-l/n

The estimate of 7n3 is simple, since from (II.4.1) one obtains 7

/ • x + l / n j T*

n3 = /

I

/»x+l/n

^ P f c ( ^ ; t)pk(dii\ t)pk(dK x)\ dfi(t) = 0(n) /

Jx—l/n

I £_()

I

Jx—l/n

L H) V ) '(•■-)]• \ )\

= O(n) /* ( x + - 1 - /* I x n n

1.

Taking into consideration that the relation

y(x + fi(x + 1/n) - p(x /z(x - 1/n) _ 22Ai ^ ~~

xx( Wj

holds at every point x at which the derivative /z'(x) exists, we find that for almost every x € (c, d) Inz = O x (l). (3.24) To estimate In\, let us consider that by the ChristofFel-Darboux formula

\Vn(dn;t,x)\ = Ox 1 _ ^ 1 [|pn+i(djj;*)||p„(djj;x)| + |pn(d/i;t)||pn+i(d/x;x)|]. |£>n(d/r,t,x)| = Ox I

j [\pn+i(dwt)\\pn(diJL;x)\ + |p„(d/x;*)||pn+i(d/i;x)|].

Because of (II.4.1) we have also

|Pn(d/x; t)!+|Pn+i(d i; t)\). |2> (dM; t,x)\ == O Oxxj I ^— | ^ i j (\pn{dfi;t)\ (|p + |p |p„+i(d/x; \Vnn(dn;t,x)\ I). n(dM; t)\ + n+i (<*/*;*)

° (jrbf) (

' *)D •

Partial Sums of Fourier Series

For x - c , d-x>h>

141

0 this is equivalent to

|X>nn(*«;*,x)| (<*/z;*,z)| = (d M ;t)|). n(d/x;t)| + n+1i(^;*)l). |D = oo fTM M (|p (M«>.*)l + |p | Pn+ Since

r

b

/

i»6

rb

_ 2/2

^^^^^^______

|Pk(a>;t)|a>(t)<{y «^(*)y j£(«>; *)«>(*)}

= VM(&) - Mo)

\pk(dti;t)\dn(t)<[j dfi(t)J p2k(dKt)dti(t)} (fc = 0,l,2,...)

= vV(&) - /i(a)

we immediately show (* = 0,1,2,...) we immediately show

|2> (d/ ; ) Mt) = -T^i = (>Ja jT° +-*-0jT^ i )V'M'O-Ma) ^/MC*> — MC«> = «>C1>- (3.25) (3-25) ") |X>*TOC<<M;*'*. X«r>l ' «*MC*> °VrW = O(l). /nl = (

Let ns finally proceed to the estimation of Jn2. We firstly obtain in the intervals [c, x — 1/n] and [x+ 1/n, d] by consideration (II.4.1) and a n < const (n € Z+) (see (II. 2.22)) the estimate (we use the Christoffel-Darboux formula again)

v

"^-°-{whi)

and consequently J»2 In2 == (( / T K v Jc Jc

"++ //

)) |©n(d/i; ) | dfxit) |2>n(<*/x;*,*,**)| d/l(*) Jx+l/n' ./x+l/n'

-oMr+om^

We next infer by partial integration that

r*-l'n Mfi = _ r~l/n d^ = [ _ /x(t)-Mx)1 a " 1/n Jc \t-*\ - X JJ 7c I* — 3^1 JcJc X-t X - t[ [ t t-X

-r^™-

t«^(«).

But we have almost everywhere

M(«)-M») M(«)-M«) = 0 > ( 1 ) t —X X

and therefore

r^c°^™r^--°^

Ox(logn).

142

Convergence and Summability

of Fourier Series

The integral // Jx+l/n Jx+l/n

\Vn\V (dfi;t,x)\d/jL(t) n(dfi;t,x)\d/jL(t)

can be estimated exactly in a similar way. Thus In2 = Ox(logn). I«2 /n2 = O (logn).

(3.26)

x

^From (3.24)-(3.26) there follows the estimate (3.21) almost everywhere in (c,d). The proof of the second part of our statement follows immediately from the above consideration. For the boundedness of w(x) in [c, d\ implies the relation rx+l/n

lJn3n=0(n) 3=0(n) / In3

w(t)dt = 0(l),

Jx-l/n

while as to In\ we have the estimate (3.25). Furthermore, one obtains for In2 the estimate /«2

O

o(i)l

rX—1/n rX—l/n

rd

.

J±

+ Jx+l/n / ))hr h'\t-X\ rJ^ = 0°(( lolossnn)'

c

since these relations hold independently of the situation of the point x € [c+e, d—e] (0 < e < (d — c)/2), our statement concerning the uniformity is then also proved. Lemma 3.7 is completely proved. In view of Lemma 3.7 and Theorem 2.8 we immediately deduce the following convergence statement. Theorem 3.8. Let the O.N.P.S. P = {pn(dfi;x)} (n e Z+) satisfies in the interval [c,rf] C [a,b] the condition (II.4.I), then the relation f)4(/)Iog(fc + 2 ) < o o fc=0

implies the convergence almost everywhere in (c, d) of the orthonormal polynomial series (3.1). Corollary 3.9. / / 00 00

^c£log(fc ^c£log(fc + + 22 )) < < oo oo ,, k=0 fc=0

then Fourier- Jacobi orthonormal series ff^cMpkfaftx), ; cf'Vpkia, /?; *), k=o fc=0

c<£"•« i 0 ^ == // f(t) (a,(a,frt)(i-t)«(i 0; t)(l - t)a(l + dt (k(hez+) € Z+) f(t)PkPk + tft)edt

converges almost everywhere.

•J'~- l1

3.4. (C, 1)-Summability

3.4

almost Everywhere

143

(C, l)-summability almost everywhere

We consider two approaches to the investigation of the (C, l)-means. One is based on the estimate of the (C, 1)-Lebesgue's functions of an arbitrary O.N.S. The second approach is used the strong summability of Fourier series in orthogonal polynomials. Let $ = {
fc=0

are called the n-tfi Fejer's kernel of the O.N.S. {
(n € Z+; x G [a, 6])

where Sk(x) = Sfc($;d/u;x) (A: G Z+) are the partial sums of the series (2.1). Functions L\{x) = L\(Q\dp\x)

:= /

\Fn(t,x)\dfi(t)

(n € Z+; x € [a, 6])

are called the nth (C, 1)-Lebesgue functions of the O.N.S. $ = {^n} (^ G Z+). (C, 1)-Lebesgue functions play an important role in some problems of the (C, l)-summability of orthogonal series. Theorem 4.1. If {An} (n G Z+) is a positive, nondecreasing number sequence for which relation Li(x) = 0(A n ) (4.1) holds on a set E C [a, 6], |JE| > 0, then under the condition

£\£
(4.2)

n=0

/or tfie (C, 1)-means of the orthogonal series (2.1) the estimate (Tn(x) = Ox(y/K) holds almost everywhere in E. Proof. Denoting, as above, by nx the smallest < n for which the relation (?nx(x) —7-^—- =

y/\Z

°< < V*h

(4.3)

144

Convergence and Summability of Fourier Series

is valid, we obtain in the same way as in proof of Lemma 2.9 gn,(*) JE

y/Anx

ifi(x)

11 Mj25^* w * w * w } , / *.

=O(D{ f

W a JE JE

y/*nxy/*nv

Let (as above) n x , y denote the smaller one and nXiV the greater of the numbers nx and ny. It then follows from the definition of the kernel Fn(t, x) by orthonormality of the system $ = {
1 OT

1

nXV

{nx-k

. lOT ■ iik=0 H(n* -

+ l ) ( n y - k + l)
k

+ 1 )( n v " * + % f c W ^ M

71* + 1 7ly + 1 j ^

is correct. Since

©*(«,*) = (* + l)Ffc(t,x) Vk(t,x) = (* + l)Ffc(t,x) (F_i(t,a?) = 0, t,x € [a,6], (F_i(t,a?) = 0, t,xe [a,6],

*F fc -i(t f x) *F fc -i(t f x) A; € Z+), A; € Z+),

then by Abel's summation formula we have

(t,x)Fnyny(t,y)dn(t) {t,y)d^t) J FFninx(t,x)F =

nnxx,,yy - nXiV + 11 r^ ^ i g ^ o " 1 ^ 1 ) * * ^ ) XtV + rn nx,j, + l - . » ^ ^ " h Z ((nnxx,vi / + i)(fi l)(nx>1/ x,y + 1) *

Therefore I/ ^

F n . ( t , * ) F n , ( t , y ) d M ( t ) ^ FnA*,y)\

a

1 l*k(*,v)l. n X y ++ l1 Xi 'V fc-0 "*•» fc=0

^From the assumption (4.1) and the monotonicity of {An} we find

/ / ^-' I r 1/

.

/ / - 7 = ^ = 1 / •Fn« (*,Fx)F x)F (tiy)d^t)\dfi(x)d (t,nyy) dn(t) j dn(x)dfi(y) nx(tiny fjL(y) My) JE JE v Ar»x v Anv Ja

JE JE

+C /

A

n« n + i

^nx nxx + 1

[

JE JE

=c

A

nx

]\Fk(x,y)\dn(x)dn(y) ^3

\Fn T-\F x(x,y)\dn(x)dtJ,{y) n.(x,y)\dn(x)dfi(y)

0 10(1). /J,obr)|°< /J,obr)|°
3.4. (C, l)Summability almost Everywhere

145

Hence, immediately,

//^MdM*) ^ d M x ) == o(i), 0(l), 0(1), An

JE

v

*

and from this, as in the proof of Lemma 2.9, our assertion follows. Theorem 4.2. / / {An} (n € Z+) is a positive, nondecreasing, concave number sequence such that the relation (4-1) holds in a set E C [a, 6], \E\ > 0, then under the condition oo

^ cC2nAAn«<°°
n=0 n=0

(4-4)

Using Abel's summation formula twice, for an arbitrary {Xk} and {u n } (n € Z+) one can obtain the following representation

£ i1 - ^i)'Xkuk=2 (* - ;£l)<*+1)(A2*fc)'fc fc=0 fc=0 fc=0 fc=0

fc=0

n_1

2 n_1 + — r> > (fc — TT ffc + l)(AA i)a fk l)(AAfc+1 i)*fcfcfc + ^nAn, fc+)<7 Ac=0 fc=0 Jk=0

Afefc -- **+i. Afc+i, A222A Afefefc = = AA(AA where s n = E L o u*> ^n = l / ( n + 1 ) Efc=o«*. E L o s**> *> AA *k ^ = *k A A ( A * *fcfc)) fc+i, A (fc = 0 , l , . . . , n ) .

146

Convergence and Summability

of Fourier Series

On account of this formula we find find
-an(x) ,,

...

(t+1)< " (( " ;I)a2 ;I)a2 (^) ('-^r) (t+1)<

-- E E ('-^r) "" (^) --E('-;^i)c+'^^*(^) £('-^T)(*+')^^!(^) -E('-;^i)c+'^^*(^) +

v

K=0 K=0 9

n + P _ 1 n + P _ 1

9

rr4TT £ 71 + P + l _ n—:

j^jj

(k +

v

/ /

1 1

\ \

l)ak(u;x)A(—=)

^V^+l7

-^ n +B l f'- + 'M-K^) y/Vk+\'

<J (l/]x) n{l/\x) C 7n+ np+(l/',x) p ( ^ 0 _ (7(7 n (^;a:)

\[Vn

yJVn+p

By reason of A„ = o(un) it follows from Theorem 4.1 that the last two terms on the right-hand side tend to zero almost everywhere in E. As to the two terms immediately preceding the last two terms, we notice that — — V *<7 kaf ck(«/;*)A(-==) (v;x)A(—=)7

m+ 1 ~

^xA^n

is the 2m/(m + l)-fold difference of the (m — l)th partial sum and the (m — l)th arithmetic mean of the series jries oo

-

]ak(u;x)A[

).

^Jvk+i'

We see that the two terms in question converge to zero almost everywhere in the convergence set of this series. We intend to show that, apart from a set of measure zero, this convergence set covers the interval (a, b). For, by the SchwartzBunyakowskii inequality we obtain that

i A ^)jr^* ) , < w x ) = 0(1)±A(-±=) = o(i)f;A(-l=) £^c^=o(i)f;A(c ^ = 0(l)f>(-^)
v

VA*+i/\j-o

ifco

V**+i'

According to B. Levi's Theorem this implies the convergence of our series almost everywhere. Now we have to show only that the difference of the first two terms

3.4. (C, 1)-Summability almost Everywhere

147 147

tends almost everywhere to zero. Since this is the difference of two (C, l)-means of the series

ff^(k > + l)<7l)* )A2 ( - 2!(-^), =), fe(*k;a;(v;x)A it is sufficient to show that the latter converges almost everywhere, but this is (as above) a consequence of Theorem I. 2.3 since applying the Schwartz-Bunyakowskii inequality we find that f>

+ 1)A2 ( - = ) OO

jf" \ok{jr, x)\ «fc(x) dn(x) jf" \a {u; x)\ r

1

a2

fc

-, .11 // 2

-od)E(*+i)A = o ( i ) B f c + i ) A (-^) - L £«$* E c U Jb=0 fc=o

V v

*

L

j=0 j=o

JJ

= 00((ll))ff>> ++ l)A l)A22(-^)
22 A f > + l)A i = - (n 2)A(—Lz) -J= ++°(!)o(l). l)A (-^=) (-^) = 4= 4= - " —r= (n ++ 2) (-7^=) == -7= We We have thus proved the relation ao-n+p {x) - 0; then under the condition (4.2) the series (3.1) is (C, l)'Summability almost everywhere in E. This statement is an immediate consequence of Theorem 4.2 and of the following T h e o r e m 4.4. Let the O.N.P.S. P = {pn(d/J.]x)} (n e Z+) satisfy the condition (II.4-9) in the set E, \E\ > 0, and let L\(x) = L\{P\dyL\x) be the nth Lebesgue functions of P. Then the estimate L\{x) = 0,(1) Li(x) Ox(l) holds almost everywhere in E.

(4.5)

148

Convergenceand Convergence andSummability Summability of ofFourier FourierSeries Series

Moreover, if the measure fi is absolutely continuous dfi/dx = w(x) and (3.22) is valid and the condition (11.4.9) satisfies uniformly in [c,d\, then the relation (4-5) holds uniformly in every subinterval [c + e, d — e] (0 < € < (d — c)/2) of (c, d). Proof. Let Xn(t,x) and Xn(t,x) denote the characteristic functions of the set, in which 2 £ = 0 ^Md/x; £, x) > 0 or < 0,respectively. From the definition of the Lebesgue functions L\(x) the representation

ii(«)

n

1 n_hl

fb

fc=o

y

Xn(t, x)Vk(dw t, x) dfi(t)

°

2 dM**w (t) ==#>(*) + /<*>(*) n(t, x)V k(dK t,*.X)*> + ^ T T £ f X*»(<> *)*>*(*; #>(*)+4 >(x) n+1

fc=o«/a immediately follows. We intend to show that every one of the sums standing on the right-hand side have the order of magnitude Ox(n) and therefore L\(x) = 0 X (1) holds for almost everywhere x € E. Let x € E. Then for sufficiently great n the interval (x—1/n, x + l / n ) C [a, 6]. We estimate the sum / ^ ( x ) , the term j£ 2 ) (x) can be estimated exactly in the same way. We divide the integrals Xn(*,Xn(t,x)V x)Vk(dfJL] t, X) rf/i(t) k(dn;t,x)dii(t)

j into the following two parts: , v

rx+l/n

fx—l/n

t v

Jfito-/

rb

42fc>(*)=/

.

Jx—l/n

+/ •

Ja

Jx+l/n

To estimate \J^ (x)|, we use the Schwartz-Bunyakowskii inequality. Then from (II.4.9) and x £ ( t , x ) < 1 follows [J$(x)}2<

2

X n(t,x)dfx(t) ^(t,x)d/i(t) rlh\2n(t,x)d»(t) f'vKt^dfiit) Jx—l/n Ja

+

1

x

1

k

x ;K* £)-K -£)]I>?< > *K* »)-' ( -»)]£*<*> +

i x

*=0

)[ M+ (x + I)- M (x-i)]. ==OO )[^(x i)-,(x-i)]. x ( fcl(fc Since /z(x) possesses almost everywhere a finite derivative //'(x) (by Theorem I. 2.5), the relation u(x + 1/rc) — /x(x — 1/n) __ H'{x) + o x (l) /x(x + 1/n) 2/n - /x(x - 1/n) , or, in other words,

2Ai

,+

= "(*) + ^(!)

x

K,+£)-K—£)=<*■(;;) £)-K -£)-<*■(£)

3.4. (C, 1)-Summabiiity 1)-Summability almost Everywhere

149 149

holds for almost every x. Therefore almost everywhere in E n

J(n + l)£>&>(x)|»2 *=J(»+i)E*o-(i/»). J(n + l)$>O x (l/n), £i4*(z)i < J^+iJEiJiite)! fc=0

\

A:=0

\

fc=0

^

A:=0

^

fc=0

and thus

El^(*)|=0,(n).

(4.6)

i2\J$(x)\=O fc=0 x(n).

(4.6)

fc=0

Now we proceed to to estimate \J%}(x)\. |«/^fctx;|. Putting rutung f Xl //(* ( t "- xx)xn{t, x) )Xn(t, x) n (t r\-f 9n{t,x) 0 n l t , x:j - \| 00

for **€€ [a, [a,xx -- 1/n] 1/n] U U [x [x ++ 1/n, 1/n, 6], 6], for otherwise otherwise

we then obtain by the Christoffel-Darboux summation formula the following rela tion

£>$(*)! < z | / fc(*,*)pfc+i(*)**(*)|afciP*wi fc=0

IJa

fc=0

I

+ 5 3 / ^n(*,a?)Pfc(*)d/i(*)kfc|Pik+i(*)|. fc=o I ^ I Since an = O(l) and (II.4.9) are fulfilled, we infer application of the CauchyBunyakowskii inequality that

fc=0 0

IJJJWI n

[ fb

"|2^ 1/2 1/2

<*(*)£[ ]PK )Y1 \ /fc(*,*)pfc+i(t,*)*i(t)l gn(t,x)pk+i(t,x)dn(t)\ }} X

fc=o

{

fc=o

Lya

n

n

J

T T

6

12-*l/2

£ p0 k i ( * )k=0 £ / *('.*>p*«<w*> } • lJa

J

The integrals under the summation signs are expansion coefficient of the function gn(t, x) which is surely L^-integrable on [a, 6], since \t—x\ > 1/n implies | x)\ < l / | t - x | x n ( * , z) = 0(n), i.e. the function gn{t, x) is bounded for a fixed n. Therefore one can apply BessePs inequality (3.8) and obtain in this way (i = 0,1):

(t)d»(t)\
OU.' [

rb

Ja i

ft w ♦JL> + <, l)"?F3»* d ( )

l, / > ' <^F * '

MT+U&

Convergence and Summability of Fourier Series

150 Hence integrating by parts

_1/ ^(t) / r/•*-!/» " prb \x Mt) W. UvJ{t-x)2

1_1/n L(t)-/x(a)l U<)-^)1 L

(t - x)2

v

+2(

y

\ Jo

/'-"%/*

V«

6 U(f)-p(a;)l U*)-M*)1 L

(t - X) 2 , ' Ji+l/n

v

)*tkjp*

Jx+l/n'

it'*)3

Taking into consideration that at every point where / / ( # ) exists, i.e. for almost every x, we have the relation

M(«)-M») M(«)-M*) ==Qa(1)Qa(1)We infer that

+

,+

(jT 0^-*«^(r jL)^-«* This proves the estimate 2

Ox{y/n){Ox(n)y'1/a = Oxx(n) £ I ^ ( ^ 1 = 0,(V5J){0,(n)}

(4.7) (4.7)

A:=0

for almost every x £ E. It follows from (4.6) and (4.7) that n

b rb

Xn(t, x)Vk(dn; t, x) dfi(t) = Ox(n) ] T / xn(t,x)Vk(dfi;t,x)dfi(t) = Ox(n) k=oJa is true for almost every x e E in accordance with our statement. It is clear from the proof, that the second part of Theorem 4.4 (the proof of uniformity) is valid. Corollary 4.5. If the weight function w(x) satisfies the condition (II. 4-V almost everywhere in the interval [c, d] C [a, b] and the orthonormal polynomial system P = = {pn(w;x)} (n G Z+) belongs to the weight w(x), then the Fourier series (3.1) of a function f G L^ is (C, \)-summable almost everywhere in (c,d). Proof. According to Theorem II. 4.3 the validity w(x) > wo > 0 in [c, d] almost everywhere implies the estimate (II.4.9) fulfilles in [c + e, d — e] (0 < e < (d - c)/2). On the other hand, since / e I?w, then (4.2) fulfilles. Corollary 4.5 follows from Theorem 4.3. We shall say that the series (3.1) is strong summability at the point x to f(x) (briefly: (H,q)~ summability), if the relation

UP ;TTT £ l5*(d« /' *) - fW = ° («> °)

n—>oo 7 1 + 1 *—■* k=0

(4.8) (4-8)

3.4. (C, 1)-Sununability 1)-Summability almost Everywhere

151 151

holds, where Sk{dfx] / ; x) are the nth partial sums of series (3.1). Obviously (ff,^>l)4(fl,l)4(C,l). Theorem 4.6. Let f € £*[a,6] and let (II.4.9) be satisfied almost everywhere in E C [a, 6], |JS| > 0. Then the relation 1

n

n->oo n + 1 f—*

fc=0

\Sk(dn;f;x)-f(x)\=0

(4.9)

/lo/ds almost everywhere in E, i.e. the series (3.1) is (H,\)-summable almost everywhere to f{x). Proof. Put

Eo = x € E

{ > h. [-

and

m /(x)|2d/i(t)=o(1) {h

"

~*0)

i M = o,}.

By Theorems I. 2.5 and I. 2.15 the set Eo is situated almost everywhere in E. We prove (4.9) at the point i G ^ o Let, for x € EQ, denote /n = £

0

n ( x - - ^ , x + - ^ ) , \

71+1

71+1/

rn =

E0\In-

It follows from (3.7), that

f(x)]Vk{dnt, 5fc(dM; /; x) - f{x) = // [f(t)[/(*) - f{x)]v x) d»(t) k(dn;t,x)dti(t) Ja

= [I Jln

[f(t)-f(x)}V [f(t)-f(x)}V k(d»;t,x)d»(t) k(dfi ]t1x)dfx(t)

+ f(x)\Vk(d + /f [f(t) - \f(t)-f(x)]V (dn;t,x)dp{t). ti;t,x)dfi(t). To order the estimate of S£\dfr, f;x)= ff \f(t)-f(x)]v [f(t) - f(x)\D k(dfi;t,x)dfi(t) k(dnt,x)diJL{t)

(k e € Z+; x e E0) {k

it is sufficient to apply directly the Schwartz-Bunyakowskii inequality \Sk1\dKf;x)\<{J Ja

J,y

1 2 Vk(dfi;t,x)dfi(t)j {dlx;t,x)dn{t))1'2{Ji2{^[/W-/(x)][f^-fix^d^t)} rfM*)}1/2'*. -

Convergence and and Summability

152

of Fourier Series

From the orthogonality of the system {pn} (n G Z+) k

rb

Y^P%dii\x) j V2k(dfi; *,x) dfjL(t) = ^pjidw x) ■^

(k e Z+; x 6 [a, 6])

i=o

hence /ix

U / 2i

rJU «

1) 2 / \si (dn;f;x)\<{f^p j(d^x)y ^ |5<1} (^;/;x)| < {£i$(**;*)} l

j=0

.x+lAn+l) /-*+l/(n+l)

1

Jx-l/(n+l) W«-l/(n+l)

J

p5

r

{{ //

.1/2

2

[/(*)-/(*)] 2<*M*)} [f(t)-f(x)] d„(t)} >>

whence in view of (II.4.9) fX+l/(n+l) i.x+l/(n+l)

\S^(d^,f;x)\
O'

112 1/2

[f(t)-f(x)}2dn(t)}

l)^{ /

+

Jx-l/(n+l)

2

■ »

[/(*)-/(*)] (W*)} •

Since the point x belongs to the set EQ,* -then l / ( n +we l ) obtain Since the point x belongs to the set £Q, then we obtain l) l) ]imSi BmS£(dr,f;x)=0. (dto/;*)=0.

J

K—^OO Ac—^OO

So, by T-regularity of the Fejer means (see Theorem I. 2.21)

]\si1\dti;f;x)\=0.

lim

n-too n + 1 ]Jk=00

It remains to consider }Vkfc(dfi; (dti;t,x)d S<2)(d/*; / ; x) == // [/(*) [/(*) -- /(x)JP f(x)]V t, x) d/z(*) (fc €6 Z+; Z+; xx €€ ft). fib). (4.11) k(d^;*,x)dW*) li(t) (* We avail ourselves of the auxiliary x.iiicuy function luuctiuii

(«W

= < fcW-|~t^T"

\[

0

t-X

f o r tforte/ 6/; » /£. outside of I' . n

(4.12)

By the Christoffel-Darboux formula (II. 2.32) one can find from (4.11) and (4.12) &{****) pfc(d^,x)cfc+i(d/x;px)], = a>k(diM)\pk+i(dwx)ck(d}i\gx) -pk(dp,x)ck+i(dii;gx)], where 9x(t)Pj(dw t) d^(t) dfi(t) Cj(dfi;9X) = // 9x(t)pj(dfi\ Ja

*+ U k,k + 1) 1) (j =fc, *,

(4>13)

3.4. (C, l)-Summability

almost Everywhere Everywhere

153

are the Fourier coefficients of the function gx(t). Because of / € L^[a,6], the function gx(t) belongs to L 2 , too, so Bessel's inequality gives oo

c d Y^, )( K9x) f>?(^;
j=0

and by (II. 2.22) we have an(d/Li) <\ const (n G Z+). In consequence of these relations and the Cauchy-Bunyakowskii inequality, it follows from (4.13) and (II.4.9)

^|5f(^;/;x)| n

fc=o ife=0

n

; Y^ n \Pk+i(dn;x)\\ck(dti,gx)\ - ^n |pfc(d/z;x)||cfc+i(d/i,0x)| fc=o +1 fc=0 nfc=0

fc=0 fe=0
j/2

-.in

TI+1

[5>fc(d,*;
fr=n {^|c

f c (^;(/ x )|

2

}

CV^TT{J {gx(t)}2d»(t)}1/2 [gx(t)]2drtt)}

Accountfc=0 of the defining relation (4.12) fc=0

•

Ja

/(*)-/(*)] RP^M"

Account of the defining relation (4.12) n

cv^ni

|S<2)(
(

V

£— X

-|2

-£\S^(dn;f;x)\
\l/2

W)

where the constant C > 0 is independent of (n € Z+). For x 6 Eb we introduce the positive integer N = N(n,x) such that

,

2 N 2" -. r < 6 < Xx + r. nn ++ 11 n+ 1 To obtain the estimate of integral on the right-hand side of (4.14), we consider

X+

2

N-1

(4.14)

[/(«)-/(*) l x

«/*+l/(n+l) [

I — X

-

r1/(n+1)

f(t)-f(x)

M*)>

J

because the integral

Ja

L

can be estimated in quite a similar way.

\mzm]2Mt)

t —X

J

(4-14)

Convergence and Summability of Fourier Series

154

We have without difficulty

jr./*

"/(*)-/(*)] teal'*,, dfi(t)

Jx+l/(n+l) fe |_ t~X J rx+2 /(n+l) [" fe N /.x+2 /(n+l) r f m - f ^ l '7x+2fc-V(n+l) [

'/(*)-/(*)]

c

2

dfi(t)

.x+2fcfc/(n+l) J ^ (n 4- lV2 //-*+2 /(n+l)

^ EJ1ir-

2

fr[

™

2 d//(t)

[/ / lf(t)-m) k l Jx+2 - /{n+l) y*+2*-v(n+i)

d»(t)

and by the definition of EQ oo

1

X Jn = C(n + l ) o x ( l ) 5 Z 5fc = °o*x(n ( n ++1) ) fc=i fc=i

( n "> °°)-

Hence ^ r E l ^ 2 ) ( ^ ; / ; x ) | = o xo(x(l) l)

(n->oo).

fc=0

The combination this estimate and (4.10) yields (4.9), in accordance with our statement. Remarks. 1. It is possible to estimate the integral

Ji'n\

'/(«)-/(«)" dn(t) t— — Xx

by partial integration, as in the proof of the Theorem 4.4. 2. It is not difficult to see that Theorem 4.3 follows directly from the Theorem 4.6.

Chapter 4

Fourier orthogonal series in L^ (1 < r < oo) and C 4.1

On a divergent Fourier series of the continuous functions

Wide generalization of the Fourier operator Sn(f) is the concept of polynomial operator. Let C2TT is the space of 27r-periodic continuous functions, and write Hn for the subspace of C ^ consisting of all trigonometric polynomials of order not higher than n. A linear operator Vn on the space Ci* is called a (trigonometric) polynomial operator of order n if 1- Vnf £ Hn whatever the element / G C2n\ 2.iffeHn,thenVnf = f. In the other words, a polynomial operator associated each 27r-periodic func tion a trigonometric polynomial of order not higher than n, the polynomials them selves being invariant. An elementary example of a polynomial operator is provided by the Fourier operator Sn : f — ► Sn(f), where Sn(f) = Sn(f,x) is the partial trigonometric sum of Fourier trigonometric series n n

S5nn(f,x) 6 [—7r,7r]), (/, x) = —+ —- +- S^(ak S^(ak coskx coskx bk sinkx) (n EEZZ++; ; xxG[—7r,7r]), y^(dk cos kx + -f+bk & fc sinkx) sin kx) (n fc=i fc=i where

1i rr f(t) cos kt dt -f(t) cos cosktdt ktdt (k(ke Z+ ak = - - / f(t) € Z+) Gfc

and

bfkc = &

i- /[ f{t)sinktdt f (£) sin A;t dt ^ J-n

155

(fceN).

156 156

Fourier Series in L^(l < r < oo) and C

Then « /,

, , x sin(n + l / 22))((*t - x ) _, ![ r ,/ r J-n 2sin((t-x)/2)

x

If /(£) is some function from f(t) by translation

C2TT,

, lx (n € Z+; x G r[-7r,7r]).

we shall write Thf(t) for the function got from

Thf(t) :=/(* + h).

It is clear that T h € C2TT for any /i. It may also be noted that, in view of the uniform continuity of f € C2n \\Thf - f\\ := max \Thf(t) - f{t)\ -+ 0 for h -+ 0.

(1.1)

Some general propositions of great important can be established for polynomial operator and for sequences of such operators. All of these depend on the following lemma which relates any polynomial operator to the elementary operator 5 n . Lemma 1.1. IfVn is a polynomial operator of order n, we have the identity

sn(f; t) = ^J"

T-hvnThf(t) dh,

(1.2)

where Sn(f\t) is the partial trigonqmetric sum of Fourier series. Proof. To start with, let / e Hn so that also Thf{t) € Hn. Thus VnThf(t) so

T-hVnThf(t)

= Thf{t),

= T~h(VnThf(t))

= T~hThf(t)

= f(t).

Since Sn is a polynomial operator of order n, we also find Sn(f;t) = /(*), which proves the identity (1.2) in this case. Now let f(t) = cosrator f(t) = sin rat, where ra > n. If we confine ourselves for the sake of definiteness to the former case, we obtain Thf(t) = cosra(t 4- h) = cosratcosra/i - sinratsinra/i = f\(t) cosra/i — f2(t) sin rah. Hence, putting 9i = Vnfi,

g2 = Vnf2

(gi and g2 are elements from Hn),we have T-hVnThf{t)

= T'hgi(t) cosmh - T-hg2(t) sinra/i.

Divergent Fourier Series of Continuous Functions

157

But T~hg\(t) and T~hg2{t) are the trigonometric polynomials in ft, of order not higher n; they are thus orthogonal to the function cos raft and sin raft. Therefore

LJ7rT-hvnThf(t)dh

= o.

The left-hand side of (1.2)is evidently also equal to zero. We have thus proved (1.2) in this case; and hence, since both sides are additive with respect to / , is automatically established for any trigonometric polynomial. We now consider the right-hand side of (1.2) and show that the functional <£*(/) a meaning for every fixed number t. We do this by proving that the integrand is a continuous function of ft. We have \T-h-TVnTh+Tf(t) / ( * ) --

T-hVnThf(t)\

< \T-h-TVnTh+Tf(t)7(*)-- T-h-TVnThf(t)\/(*) + \T~h-TVnThf(t)7(t) ,h < \\Vn(Th+T - Th)\\ + \T-h-TVnThf(t) /W -

< \\Th+T - Th\\\\Vn\\ + \T-h-TVnThf(t) */(*) -

T-hVnThf(t)\

h T~hVnT/(*)l f(t)\

T-hVnThf{t)\.

In view of (1.1), the first term here is as small as we please for sufficiently small r, and similarly for the second, in view of continuity of the function VnThf(t). The functional
M/)| < ± f

\T-hvnThf{t)\dh< ±£||K,||||rh/WIIdh< 7WII

IIKJII/II.

If, finally, we write ipt(f) — Sn(f]t), identity (1.2) will be implied by the identity of the functional ip and ip. But it was shown earlier that these functionals coincide on the set of all trigonometric polynomials, which is dense in C2n- Since ipt and ^t are linear functionals, it follows from this that they coincide throughout the space C2TT, which is what we have to prove. Lemma 1.2. The following estimate of the Fourier trigonometric operator \\Sn\\c2„-+c2„ >C\n(n + l) (n = l,2,...) holds. Here the constant C > 0 is independent of n. Proof. It follows from (I. 3.2) lie ll

sin(n + l1/2)* 1 f" lsin(n+ /2)t|A

\\Sn\\c^C2, = -J_^

2sin(1/2),

dt

L

7 \ (nt net zZ+)

\ = n (n€Z + )

Ln are the Lebesgue constants. Put JIlOLUUIlld. XT U t IIra = 2n + 1 and z = t/2, then II\\Sn\\c C IIn-+C2n \\Sn\\c 2

r S[nmZ r \|sinra;z| \ Ay 7T J0 sin z X

158

Fourier Series in L £ ( l < r < oo) and and C C

We now use the equality kit

sinmfzH sin m ( z H JII = |sinra;z| V m/ and the familiar inequalities . . 2 | s i n z | < | z | , , ssin i nz2>> -— zz |sinz|

(0
IT

Then we obtain sin rat i r\ —: if at - E

- m—1

II $n || C2,r->C2,r

:

* mJo i suit n—: -1 . r/2m 1

■~7

£

~ 7r 0 *7r "^-f £^[ yyJo 0

s i n

/2fc7r/2m

*

|smmt|

/ 7

m-1

H2fc+lW2m|sinmt|^

; — ^ — r "*

n(t + k-K m) ^ (2/7r)mt ^, sin(t + kir/m) kit/m) sin(t ++ K-K/rn fc7r/m) sin(t

fc=i ' - m—1 2 771 4171 1 m' ^- 1 (2/7r)m(7r/4m)

-

1y * ^ 7U—1 -

r / 2 m (2/*)mt ^ .Ar/4m * + W ™ -

-

7r > - V i /*) ^/ ) * > ^J— _ y>" T i >>^ — —In— — IInn, nn l n — >>- — — ^ Jkfc ~ 8TT ~ 7r ^-^ ^ (7r/4m) + hir/m kit/m Am ~ 8TT 8ir £? 8ir U 2 ~ 8TT Sn nU'

-Tfc=l

x

fc=2

'

in accordance with our statement. Lemma 1.3. i*br every polynomial operator Vn the following estimate

||V n ||>Cln(n+l) is £nze. /fere C > 0 is a constant independent ofne In fact, by the identity (1.2) ||S„/||:=

max |5„(/;*)| <\ -

— 7T
max max

Z+.

f r|T-V \T~hVn nT"/W|dft
7T -7T
So

IIK.II > ||5„|| IIKII and by

||S„||>Cln(n ||S»||>Cln(n + l). i).

Lemma 1.3 is proved. We introduce the operator Vn acting on the space C ^ and possesses the following two properties: 1- Vnf € Hn, whatever / 6 C2
Divergent Fourier Series of Continuous Functions Functions

159 159

2. \ifeHn,Hn, then Vnf = f. The following assertion follows immediately from Lemma 1.3. Lemma 1.4. For every operator Vn the estimate \\V Cln(n + 1) \\Vn\\ n\\>Cln(n+l) holds, where C > 0 is a constant independent ofnE Z+. Let f(x) be a continuous function in [a, b] (f G C[a,6] or / 6 C). Function

■p

f(e) = f(^cose f ( ^ c o

+ ^+ ^\

(o<e<*) {0<0<«)

s 6

*)

is called an induced function (for f(x)). Evidently, (b - a)/2cos6 + (a + 6)/2 belongs to [a, 6] for all 0 G [0,7r]. SO
y/->*>, which every function / G C[a, b] corresponds an induced function (p. The next four properties is obvious: 1°. For every / € C[ay b] the induced function if belongs to C ^ , where C^TT is a subspace of CI-R consisting of all even functions. 2°. y is additive and homogeneous operator, that is, for all / i , / 2 £ C[a,b] and Ai, A2 € R 1 the equation y(Ai/i + A2/2) = A 1 y / i + A 2 y / 2 holds. 3°. For every
Vv

~ ~

))

Thus, the operator y, y : C -> C ^ has inverse operator y~l : f

-+y~lip.

4°. For any function / G C[a, b] the following equation

ll/llc = ny/iio,. \W\\c2„ is valid, consequently, ||3^|| = H^-1!! = 1. Properties 1° — 4° show that y is a linear isometry between C[a, b] and C ^ .

160

Fourier Fourier Series in Z L £ ( l < r < oo) and C

5°. If Pn e 7rn, then 3^n belongs to Hni and conversely, if Tn e Hn, then y

1

^ € 7Tn.

Proof. Since the operator y has inverse operator y~l and dim/J n = dim7rn = n + 1, then it is sufficient to prove the first assertion only. Because y is a linear operator, then it is sufficient to verify of our statement only for Pn(x) = xn, i.e. yPn = (cos0(b - a)/2 + (a + b)/2)n e Hn. But this relation is evident. The property 5° is proved. We can similarly consider, in the space C[a,b), (algebraic) polynomial oper ator, i.e., linear operator r n such that 1. if / e C[a, b], then rnf e 7rn; 2. if / e 7rn, then rnf = / . Theorem 1.5. For every polynomial operator rn the following estimate |||r | r n ||>Cln(n | | > C l n ( n + l) is valid, where the constant C > 0 is independent ofne Z+. Proof. Let r n be a polynomial operator. We form the linear operator Vn = yTny~l, acting in Ci*- By the definition of operator r n and from the proper ties 2° and 5° of y follow that Vn is polynomial operator from C2n in Hn. By Lemma 1.4 and property 4° of y we have Cln(n + 1) CIn(n l) < ||V„|| ||Vn|| = Wyrny-'W < ||3^||||w,||H^" IMIKIIP'- 11!! 1| = ||T„||, ||r n ||, which proves our statement. Corollary 1.6. Let {r n } (n € Z+) be a sequence of the polynomial operators. There exists a function f e C[a, b] such that the sequence of polynomials {rnf} (n € Z+) is not uniformly convergent. Proof. By Theorem 1.5 ||r„||->oo

(n-»oo).

So, Corollary 1.6 follows from Banach-Steinhaus Theorem (I. 3.7). We consider the operator Sn : f{x) -► -* SSnn(d/z; (d/z;//, ,x), x), 55nn :: CC -¥ -¥ 7r 7rn,n, where 5n(d/z; / , x) is a partial sum of Fourier polynomial series of / (see (III. 3.3)). ^From Corollary 1.6 one can immediately deduce Theorem 1.7. Whatever the orthonormal system of polynomials, there exists a continuous function such that its Fourier series with respect to the system is not uniformly convergent.

4.2. Estimates of the Lebesgue A-Function A-Function

4.2

161

Estimates of the Lebesgue A-function

We consider the trilinear T-regular method of summability defined by the matrix A = {Aln), A; = 0 , l , . . . , n , n + 1; n = 0 , 1 , . . . ; A
(2.1) (2.1)

For every function / G l J one form A-means n

t/ (/) =l/ Un(f) =Unn(/;x; (f; x; A) A) == $>j, £ n)cX^c fc (/)j> fc (;r) k(f)p k(x) fc=0

(2-2) (2-2)

(nGZ+; XG xG [a,6]), where Cfc(/) (A: = 0 , 1 , . . . , n) are the Fourier coefficients of / (see (III. 3.2)). We study the following problem: to investigate the A-summability of a given Fourier series (III, 3.1), that is, to obtain conditions for the O.N.P.S. P = {pn(d//(x)} (n G Z+) and the entries of A-matrix (2.1), for which the relation lim C/ £/ nn(/;x;A) (/;x;A) = /(x)

(2.3)

n—>oo

holds at the point x G [a, 6], uniformly in the set G C [a, 6], in Lr-means and almost everywhere in G. The Lebesgue functions (A-functions) defined by Ln(x; (x;A)=L A) =Lnn(d^;x;A) (dfj,;x; A) := / \Kn\K (dwt,x)\ dfi(t) n{dn\t,x)\dii(t)

(2.4)

(n G Z+; x G [a,&]), [a,6]), where the A-kernels are defined by n

n) ir (t;x;A) pfc(^;*)Pfc(^;x) Knn(t\ x; A) =Knn(dfi;t\x]A) (dfi; t\ x; A) := := ^£ ) A^ ^Pkidfi] t)pk(dw x)

(2.5)

x e [a,6]) (n€ Z+; xG [a, 6]) play the important role in the study of the convergence of such processes. Our goal is to obtain the estimates of Ln(x; A). It is easy to see that the representation n

K^ x; A) = £ ^AX^V (d^; t; x) Kn(t; *; W kn fc=0

(2.6)

(*,xG [a, 6]; ne nG Z+) (t,xe [a,b]; n) is valid, where AA^n) and Vk(dn\t\x) are defined by AA[ = A A^«n) - AJ^ AAW A « and (III. 3.5).

162

Fourier Series in L£(l L^(l < r < oo) and C

Theorem 2.1. Let P = {pnn(d/j,(x)} (dfi(x)} (n €G Z+) 6e tfie O.N.P.S. Then the following estimate

1/aV 2 l/a 1/2

Ln(x; A) = 0,(1){ £ |AA 2 (dp; *)] o.(i){f:iAAi:)|[x:rf(^*)] fc=0

}}

Z=0 1=0

(27) (2.7)

x { ^(AAl £ ( A A inn) )))22b^(dM; [ p £ ( ^ ; x) + pirf+1 (d„; x)]} x)]}V1 /22 +1(d/x; fc=0

holds almost everywhere in [a, b]. Moreover, the relation (2.7) is true uniformly in [a, b], if the measure fi is absolutely continuous and uniformly bounded in [a, 6]. Proof. Let x e [a,b]. Denote hn = hn(x) (n € Z+) an arbitrary sequence of positive numbers, more precise definition will be given below. First, we establish the following estimate

2 Ln(x; A) = Oxx{l){h\!\x)

V2 x)]V3} } o (i){^/ (x) ££ |AA< IAAI| [£>?(d/x; [J2pf(^*)] fc=0 fc=o

^

i=0 J=O

(2 8) (2.8)

2 11/2 /2 0 Ix(l)/ - 1(x){ / 2 (x){ ^(AAi +O (l)Kl1/2 £(AA
fc=0 k=0

for almost all x €:{a,b}. [a, 6]. We consider only the integral L„(x;A):= L n (x;A):=

Ja Ja

fXfX\K \Knn(t,x;A)\d»(t), (t,x;\)\dfi(t),

because the integral / |# \Knn(£, (t, x; *; A)I A)| dfi(t) (n e Z+; 1 6 [a, 6]) Jx can be estimated in the same way. We treat two cases 1) a < x — hn < x; 2) x — hn < a < x. Let a < x — h/inn < x. x. Therefore, for L'n(x; A) one finds X — hn

L n (x; A) = f / Ja

" \Kn{t,x;A)\dn(t)l (t, x; A)| dM(«) + /f |tf„(*, x; A)| d^*) |ff„(t,x;A)|dfi(t) « / ! —/l iz-h„

= L; i (x;A) + L; 2 (x;A) and we estimate every integral separately.

n

,. n,

(2.9) (2.9)

A-Function 4.2. Estimates of the Lebesgue A-Function

163

By Schwartz-Bunyakowskii's inequality and by the orthonormality of the system {pn} (n e Z+) one can obtain n ) n) |^AAi PPf cfc(d//;t,x)|d/xW IJ2 AA[ (dM; *, *) I <M<)

£;n2 L' (x; A) = /fX 2 (*;A)

Jx-hnn Jx-h

fc=0 fc=0

AA n)n)i [ j f iAAi

2 1/2 < j^^ I i I [ [ h^ dn(t)\ dM(t)]1/21/2 [[£p2 ( d M . t >t,x )x) d / 1<*„(«)] (t)j ^ f Vl(dn;

=0(1)] X>A<">|[£ P L L

fc=o fc=o

ii=o =o

2

(d M ;*)] 1 / 2 [f JJ

lJ

h

' ■*~ • ' "n- A -

d^)l 1 / 2 JJ

Since

Ox(hn), dlt(t) = 0«(hn), ff dn(t)

. / x - /ni n Jx-h

then the estimate 1/a

^ n2 2(x; (^A i; A)) ==:0,(l)Ai/»f:|AAW|[X:rf(*.;«)] O x (l)Ai/ 2 £ I AAin) | [ £ > 2 ( ^ *)] V 2 fc=0

(2.10) (2.10)

z=o

holds almost everywhere in [a, 6]. To order the estimate of the term 142 ( x ) denote by Xn(t, #; A) the characteristic function of the set of t G [a, x — hn] for which Kn(t, x; A) > 0 and x n (t, x; A) is equal to 1 — x n (t, x; A). Then (x\A) = / L'n(x;A)--

Xn(t9x\A)Kn(t,x;A)dfjL(t)

J'a ai ,6

+ / x»(*,*; Xn(t,x;A)K A)JM*.*; A)dp(*) = /;,(x) + n(t,x;A)dp(t) Ja

I'^x).

Using Cauchy-Bunyakowskii's inequality and Christoffel-Darboux's formula, we obtain

&i(*)= KxV)

rX — hn

[/

Ja Ja

E

fc=0

Xn(*,*;A) \n(t,x;A) AX(n)

Pk+i(d&t)p -pk{d^t)p k+l{d^x) pfc+i(d/z; t)pkk(diJ,;x) {dfi; x) - Pkjdwt)p x) k+1(dfi] t—x

d/i(*)

164

Fourier Series Series in L£(l < r < oo) and and C C 22 11/2 n Ai Ain >p W^;*)] } /2 f c (^;z)] } ]fc=0 fc=U

xx {{ V Pt+1 (^t) £ II PjX'hn" Xn(t,a:;A) *.(«,«;AW4M)

- a-^ a i / a^

n) 2 1/2 + c{^[AAl c { ^ [ A A i nPW ( ^ ^ ) ] 2 }}1 / 2 f c +ii(^;x)] fc=0

* { E 1 /*"*" *"(''*;/_)p;(dM) ^)| 2 } 1 / 2 Put for tte[a,xe [a,x - /ihn] /.x -- I Xn(t, x; A)9x,n{t) r—x ( outside of of [a, [a, xx — — /i/in],], 00 outside n then gx,n{t) is a bounded function for everyfixedn. Therefore, Xn(^x;A)p (d/x,t) Jmm (d^t) n(t1x]A)pkfc

I

rX — hn _

/ Ja Ja

73 J. t — XX

O/X = = Cjk(p X|n)i dfJL Ck{9x,n),

where ck is a Fourier coefficient of the function gXjn. Hence, by Bessel's inequality one finds Xn(t,x;A)p £\rh"Xn(t,*;A)p^,t) iy> k(d/jL,t) Mt) l

k=o J"

x

*/*(*)f'

0(1)<

J

Ja

M*) \

it-*)2*

Integration by part gives

/*-*»
Ja 70

it-xY (t-x)»

=

M»)-/*(«)!*-*. 9 /*"*- M*)-M*) ,,dt. 2 ((*-«)» x - t ) lo I.

+

U

(*-*) 3

The relation

fj,(t) -- n(x) //(x) H(t) "<«>-*«>=o.(D o-(i) t—x is valid almost everywhere ini [a, 6]. Therefore f*-h" dfi(t) d/j,(t) Ox(h~l). Ja (*-t)2 Thus l a I'nl(x) COr) = Ox{\)h. ^ ^'\x)< x ) / f>A<">) J2(A\P)*\pl(dr,x) |pftetax) fe=0 ik=0

Pk+i(dwx)j +pl +pl+1 (dr,x)}1/2. +1(d„;x)} '

'

4.2. Estimates 4.2. Estimates of the the Lebesgue Lebesgue A-Function A-Function

165

The integral 1% can be estimated in a similar way 2 22 /2 ^(AAin))n) [p (d/z;x) +P +p22fc+ (^;x)} 11/2 /;' 1 (x) = = o Oxx(i)h-^(x){ (l)h" 1 / 2 (x)« £(AA< C(x) ) b^;*) -. + 1i(
Consequently, the relation ) 22 2 L'nl(x; A) = : O f > A < n n) ) )[ p[p2(^;*) (d,z;:r) oxs{l)h-V\x){ (i)K1/2{x){ f>A< k=o

(2.11)

1 1/2 +j>iUi (<*/*;*)) Pfc+i (<*/*; *))

holds almost everywhere in [a, 6]. The bound (2.8) now follows immediately from (2.9)-(2.11). Let x — hn < a < x, then x — a < /i n . Now use this and Cauchy-Bunyakowskii's inequality and by the orthonormality of the system {pn} (n e Z+), one obtains L'n(x; A) = f

I j ^ AAi n) ^ fc (d/x; t, x)| rf/x(t)

= 0 , ( l ) ( x - a)1/2 £

|AA 2 ( d / r , x ) ] V 2

fc=0

a

z=o

B

=o,(i)^ (x)x;iAA{! >i[x;rf(dWx)]1/a fc=0 fc=0

1/a

+ ox(i)/c (x){ ^(AAin))2b^(^;x) +P2+1(dWx)]}1/2. fc=0

Thus the estimate (2.8) is established. Put

hn(x) = { £>A< n ) ) 2 b^;z) +rf+i(^;*)]}1/2 fc=0

^ElAAWl^rf^.)]17*}-1. fc=0

/=0

Then we obtain the estimate (2.7). The second part (the proof of the uniformity) is now easily obtained.

166 166

Fourier Series Fourier SeriesininL£(l L£(l<
The following statement plays an important role throughout this Chapter. Lemma 2.2. Suppose that for some 2 < go < oo

U

}

1 i/go l/9o

d

[J \pn{dfi;x)rdfi(x)}

Mn
(2.12)

(ne Z+) (2 < g0 < oo),

\pn(dii; x)^ o>(x)J < Mn < oo (n € Z+) (2 <
(2.12) (2.13)

(2.13)

x€[c,d]

w/iere d] belongs re £/ie interval [c, [c,rf] belongs to to [a,b]. 2 and define s by Let q'0 = qqo/(qo 0/(q0 - 1) < r < 2 9o + 2 - ^ o rr s5

(2.14)

= 1

Then for every / G L J [ C , 4 vanishing outside [c, d]

1/r r {f;Mr (^;/)r} 1/s < {^ {f;Mr22icicnn(d/z;/)r} { /d|/W| i/(x)r^w} d^)}1/r.,

(2.15) (2.15)

W c

n=0 n=0

tuyere c n (d/z;/), (n € Z+) are £/&e Fourier coefficients of f with respect to the orthonormal polynomial system P = {pn} (n G Z+). Proof. By Holder's inequality one obtains

I rd

MdM;/)!

\cn(dfi;f)\ =: /

U

Hence

1 I f(x)pk(dfi;x)dfi(x)\ /(x)pfc(dM;x)d/z(a;)

d

*\ l / ? o ( ]

\f(x)\«>dn(x)j \f(x)\«odfi(x)j

M-VnC^;/)!

U

d

On account of Bessel's inequality

rd

\ I/90

\Pn{dKx)\<*> d>i{x)\

|jfe |p„(dM;x)l*dM«)}

•

, ^ / 9 o (n € Z+). \f(x)\«°d»(x)j

(2.16)

| / ( x ) | * d/x(z)j

(2.16)

1

(neZ+).

.1/2 \f(x)\2dnj . On account of Bessel's inequality 2} {J2\cn(dn;f)\ n=o ' Wc J n=0 00 ^ 1/2 ( rd 1 */ 2 We introduce the measure i/, which assigning 2 is an additive measure 2 (

~

j

(2.17) value M~ 2

{^ICnCd/.;/)! } x<=| n, j [n |/(x)| (2.17) to the sets consisting of a single point = 0 , 1 ,d/,J . . . , and. vanishing for sets2

We i/, which not introduce containingthe anymeasure such points. If is an additive measure assigning value M~ to the sets consisting of a single point x = n, n = 0 , 1 , . . . , and vanishing for sets c(x)points. = cnM not containing any such Ifn for x = n, n = 0 , 1 , . . . c(x) = cnMn for x = ra, n = 0 , 1 , . . . and is arbitrary elsewhere, (2.16) and (2.17) can be written in the form

\\c\h, < ||/||2>l. 1 \\ch,u < ||/||2,M Moo.* IMIoo., < ||/|| 9g > . ]/

4.2. Estimates of the Lebesgue A-Function

167 167

This is the linear operator c(x) = Tf is simultaneously of types (2,2) and (g 0 , oo) with norms > 1. It is defined for functions which are not necessarily simple, but we confine our attention to the latter. We use the Riesz-Thorin Theorem with c*i = / ? i! = - , a C*l <*2 1,, #2 # 2= = 00.. 2 = l In this case

1

2 " ;2• - 2^ 2 Then T is also of type (r, s), where r = 1/a, s = 1//?, and satisfy (2.14). Hence, we have

Ml..., llclk* < 11/llr.M ||/||r,M which is exactly (2.15). Remark. Lemma 2.2 and its proof are valid for an arbitrary O.N.S. Corollary 2.3. Suppose for some 2 < qo < oo

U [J

d

*\ i/go

|pn(dM;x)|*>d/z(x)J

C
( n e Z+) (2 < 90 < oo),

9o

|p n (d/z;x)| d/z(x)J
(2.18) (2.18) (2.19)

xG[c,d] x£[c,d]

where [c,d] C [a, 6]. Lei a 0 < r ^ 2 and define s by (2.H). vanishing outside [c, d]

( £ | c n ((d a 7 z ;/)r) ;/)r) (^|c n M

l/s

Then for f € L^[c,d\,

=c{y i/(*)rdM*)}1/r..

< C { / |/(x)rd/i(x)}

(2.20)

J c

n=0 n=0

/fence (2.21)

) -> 0 0( a (as s n -n > —oo). cn(rf/x; (dfj>]/ f) —> >> oo). Corollary 2.4. Suppose that the condition (II. ^.1) is satisfied: (d/z;x) < C C |p n (d/x;x)

(2.22)

(are ( x E [ c , d |],,nn€e Z Z++))..

T/ien / o r every re (1,2] and every / G L£[c,d] ; vanishing outside [c,d\, 00

(Y, n\cn{dn f)\rr)') (^|c (d M ;/)l n=0

1/r'

1/r

r \f(x)\ i/(x)rd/i(*)} d M (*)}

(2.23)

/io/ds, where, as usual, \/r + \/r' = 1, and £/ie constant C > 0 is independent of

f. Now we apply these statements to the estimate of the Lebesgue functions.

168

Fourier Series in L^(l < r < oo) and C

Theorem 2.5. Suppose that for some 2 < qo < oo O.N.P.S. P = ( n e Z + ) satisfies (2.12) or (2.13) with Mn<Mn+i

(n€Z+),

{pn(dfi;x)} (2.24)

and a£ £/ie pointf XQ G (C, d) #&e conditions \Pn(d/i;x0)\
(n€Z+)

(2.25)

Mft)-M(xo)=0jo(1) M*)-M*o) =Qxo(1) t — XQ

(2.26) . 26) (2

are va/id. Then for the Lebesgue functions (2.4) we have the following estimate 0

2(r_1)

,) 2(r 1) L n (z 0 ;A) = O-(i){i+[(EVfcTTiAAi")|) " 1 0 (l){l + [ ( £ v ' £ T T i A A £ | )

)/(

*=°

r (3r 2)

(£
(2.27) (2-27)

fc=0 fc=0

where Oxo(l) is independent ofn€Z+, q'0 < r < 2 and s is defined by (2.14)Proof. Let xo e (c,d). As above, denote hn = hn(xo) (n e Z+) an arbitrary sequence of positive numbers, more precise definition will be done below. We consider the integral only \A)== L'n(x0;A) because the integral

\Kn\K {dfi\t,x n(dii',t,x 0\A)\d^{t), 0\k)\dii(t), JXQ i

rXo rXQ

/I \Kn(dii',t,xQ',h)\dn(t) Ja Ja can be estimated exactly in the same way. It is easy to see that in view of Christoffel-Darboux's formula (II. 2.32) and by (II. 2.22) one finds I ^ A A ^ P ^ d ^ t j x o ) !dfjt(t) ^^)

/ =

n M juP H i f^ e d^^ f cPkf ^ e ^d^ o )x^- ~Pk(dli>;t)p pk(dii;t)pk+i(d/j,)Xo) \ \ f h\X^A\( p A A ( )na) a (^ \Pk+l k+i(d/j,)Xo)

J*\h> -"

k

*-*<>

|fc=0

= 0 I O (l)^|AAi n ) |{|p f c (d M ;xo)| /

Ja

- 70

fc=o

+ |pife+i(d/z;a;o)| \pk+i{dwxo)\l / |pfc(d/x;*)| |p fc (d/z;*)|d/^)J. <*/*(*) J. Ja

\Pk+l(dvt)\dii(t)

I^

4.2. Estimates of the Lebesgue A-Function A-Function

169

From (2.25) and Schwartz-Bunyakowskii's inequality follows ) // \K (l)]T|AAi \Knn(dp,t,xo\A)\dii{t) (d^t1x0;A)\d^t) := O OX0 l X0(l)J2\^

Jd

n)

|.

(2.28) (2.28)

fc=0 fc=0

We consider two cases xo + hhnn >> d. d. XQ + hnn < d or xo #0 4If xo + hn < d, then In{xo) := /

«/xo JXQ

|#\K n(dfx;t,x 0;A)\dfi(t) n (c^;*,x 0 ;A)|d/x(t)

= /

(2.29)

\Kn(dfi\t,x0;A)\dfi(t) |Jf„(d/i;t,*o;A)|^i(t)

JXQ Jxo

+ /

Jxo+hn

\Kn(dfjL;t,x0A)|d/n(t) \A)\dfjL(t) := IIn,l(xo) |tfn(d//;t,xo; + / n ,2(x In,2(xo)n,iO*o) + 0 ).

Applying Schwartz-Bunyakowskii's inequality we obtain rxo+hn

Inti(xo)=

/

\Kn\K (dfjL\t,x n(dn;t,x 0;A)\dfjL(t) 0)A)\dn(t)

'XQ JXQ n

n n

/-XO r x 0 + /Zln in

Aiinn )) || //
rx0+/in

fc=0 fc=0

"^*

J^ *^

t^ \ < °+hn r rxXo+hn

n)n)

< £|AA< £|AA< |{|{//

Jx/x

fc=0 fc=0

■° »

|2>Jk (dM;t,x )|dAiW |2> ;t,x00)|dAiW Jk(dM 11/2

r

r6

n„

>>|i 1/2

dfi(t)\ { / ©2(d **(«)} d/x(')} Vl(d^t,x M;t,*0))dy.{t)} 70 -Ja

and at the point xo € (c,d), for which (2.25) holds, one can find n

2 2 Y^VkTT\A\^\ 0*„(1)># £ VfcTl|AAin)| (n€Z+). (n € Z + ).

InAxo)=0Xo(l)h^ In,l(xo)

(2.30)

fc=0

In order to estimate In,2(xo), as above, we denote Xn(t\ xo; A) and xn(*; xo; A) the characteristic functions of [xo + hn,d\ for which lfn(d//;£,xo; A) > 0 or ifn(d/z;£,xo; A) < 0, respectively. Then Jn,2(zo) ^n,2(x 0)

= /

Xn(*, x00; A)ATnn(d/x; *, x00; A) d/x(t) Xn(t,

«/Xo 'XQ + /ln hn

+ / 'Xo + /ln

Xn(*, c^(t) •= /; Xn(t,xx00;; A)Kn(dK {dw t, t,xx00;; A) ^(^) ^2(^0) + /^fro). 2 (*o)+/;'2(zo).

170

Fourier Fourier Series SeriesininL^(l L£(l<
From Christoffel-Darboux's formula (II. 2.32) ^,2(^0) = /

Xn(t, x0; A)Kn(dfji] t, x0; A) dfi(t) ~j

n Xn(*,xo;A)^AAi

= f Jxo■-o-r»n + hn x

n)

afc(dM)

fc=Q fc—Q

(<*/*; ^o)] rf -p i(dfi;xo) k(dfi]t)pk+ -Pkjdfi]t)p k+i(dwx0) rf -dll(t).

Pk+i(d^t)p Pk+\{dp,t)pkk(dfj,;x (dfj,;x00))

£t — — Xn Xo

We introduce the auxiliary function

{

Xn(t,X00;A)]A)^ /. AX \Xn(t,x G„(t,x0;A)= < *t — - z XQ o ^ 0

for d\ fortt € [XQ [x0 +4- hhnnj ,d] _ for te[xo £lT[xo + /ihn>n,d\. rf]-

Then by (II. 2.22) and (2.25) n) n )

+ |c )|], l£,2(*o)l < C C X0 £ |AA<|[|c|[MG )| + | C fc+1 (G nn)|], l0 £|AA< fc (G nn)| \I'n,2(*0)\ f c + 1 (G fc=0

where C Xo > 0 is a positive constant independent of n e Z+, and /

Ck(Gn)=

Gn(t,xG0]A)p (dn]t)dfj,(t) n(t,x0k]A)pk(dn]t)dfj,(t)

(k = 00,l,...,n). , 1 , . . . ,n). (fc

'a

Applying Lemma 2.2 and taking into account (2.24), we obtain by the defi nition of Gn(t, xo; A) and Holder's inequality (1/s 4- l/s' = 1): AA n)

M s)/s s 2)/s

s ) I0I0(l)^|AAi">|[M( | Cfc (G nn)|)i J;.2(*O) = o (i)Ei i i[ 2r- s )/ s M^^ - 2 >/ic fe (G fc=0

)/s )/8

)/s +
A.-TX

■ — ■ - \

■-/

ij

fc=0 fc=0 fc=0 fc=0

1/ )/(s- 1) 1)iAAir ) i'') ) 1/s (E
= o,0(D(E
7^V)Vr-

\^0+/in i r " x oj y

One can see by a partial integration that the last multiplier is 0(hn ) (as it has been already fully calculated for r = 2 in the proof of Theorem III. 4.4),

4.2. Estimates of the Lebesgue A-Function A-Function

171 171

consequently r)/r 1 / 1/s £gw: i ,£) / (r-)1/)(| A- »A IAA<-> = 0Oxo - . ( (D^l ) * i 1 - r ) / r ( i(: M < " ) | « ' )\°') * ' .'.

U*o)

(2.31)

fc=o fc=0

Therefore, in the case x0 + /i n < d, from (2.28)-(2.31) follows the estimate Therefore, in the case x0 + hn < d, from (2.28)-(2.31) follows the estimate + l'nj2(xo)

In,l{xo)

=

0*.(l){/4/22X;N/fcTT|AA£">|

/n,i(*o) + /; i2 (xo) = Oxo{\)[h]!

J2 fc=0 fc=U fc=0

VkTl\A\^\

i ) ''} fca-)/'(^Mg.r IAAi->|-') +*i -^(s:
l/

jb=0 jb=0

and, consequently, n

/»,i(xo) + /»,a(xo) = O X 0 ( l ) { / # 2 J2 Vk+TlAAJPl *=°

1 r)/r r)/r(i:M&')/()/( 1) ) l/''}. 1/J fcff" -1>|AAW|'')

+*i -

(2.32) (2-32)

(E
fc=0 fc=0

In the case xo + hn > d one finds n

/

) ^AAi nl^AX^Vkidfi^xo^d^t) 2> f e (^;t,xo)|

fc=0

1/2 <±\AxP\[[\»} j(d»,Xo)]1/2 i/2 f:iAAi«)|[/ddj[£i/2P[x:p?(^xo)] k=0 fc=0

Jx

"« ~u

1=0 *=0

2 I AA n)n)|, (i)fty2^v^TT|AAl

= Ox„(i)/ oX0 l y E^+ i Jfe=0

i i.

and we obtain the estimate (2.32) without the second term. Put

^n

{

MJ x 2r/(3r-2)

/s 7ELnM£r)/(=*-VAL"vy 'l / T!

S3-OV*+T|AAW| ELoVfcTT|AAi">|

j

Taking into account T-regularity of matrix (2.1), we establish (2.27). Theorem 2.5 is proved. Theorem 2.6. Let the O.N.P.S. P = {p n (d/i;x)} (n e Z+) is uniformly bounded in the interval [c,d] ([c,d] C [a,b]) (cf. (2.22)) and for the entries of A-matrix (2.1) for some r (1 < r < 2) the estimate i nA) |l )" 2) !( )r 2- ^1 )J2 X >|AAl A l nn)) r << CC ( ££v /vm / f |cA + TAl A fc=0 fc=0

( n € ZZ++)) (n

(2.33)

172

Fourier Series in L L^(l £ ( l < r < oo) and C

holds. Then for the Lebesgue functions (2.4) almost everywhere in (c, d) we have the following estimate Ln(x; A) = 0,(1) Ox(l)

d); n G Z+). {x 6 (c, (c,d);

(2.34)

Moreover, if the measure fi is absolutely continuous and the weight w(x) = j£ is uniformly bounded in [c, d\: sup — = sup w(x) < C < 00,

x6[c,d] dx «£

(2.35)

x€[c,d] xe[c,d)

then the estimate (2.34) holds uniformly in every compact subinterval of(c,d). In fact, the estimate (2.34) follows from Theorem 2.5, if qo = 00. One can obtain in the same manner (see also the proof of Theorem 2.8 below) the second part (the proof of the uniformity). Corollary 2.7. Under the hypotheses of Theorem 2.6 condition (2.33) can be replaced by n 1 (n + If" l ^ - ^^I AlAAJ^r A J ^ r ^
( 1K< r < 00; n 6 € Z+).

(2.36)

fc=0

The proof of Corollary 2.7 follows immediately from Theorem 2.6 and

1/r 1/fi {(n { ( n ++irif>Ai">r} i r ^ l A A ^ r J ^ ^ I C{(n n ++ll)«-^|AAi«)|«} ^-^IAAW^}1^ /b=0 fc=0 fc=0

< 00), (( 11<< r << R R< oo),

fc=0

which is valid by Holder's inequality. Remark. The method of summability defined by {AA[">}, A-«{AAM}.

AA<»> = ( ( * T I F 0 [

f o r 0

^*

for A k; >> nra

(number A one can select from Ylk=o &*£ = *) satisfies the estimate (2.33), but doesn't satisfy the equation (2.36). Corollary 2.7 can be generalized in a different direction. We put for the entries of matrix (2.1)

*S3.(A)- (l + ln^ElAA^I + ^ - ^ I A A ^ r ) 1 7 " , \ V

/

fc=0 fc=0

fc=0 fc=0

(2.37)

where r > 1 is given and m € { 1 , 2 , . . . , n}. The following assertion gives the other estimate of growth of the Lebesgue A-functions.

4.2. Estimates of the Lebesgue A-Function A-Function

173 173

Theorem 2.8. Assume that the O.N.P.S. P = {j>n(d/x;x)} (n e Z+) satisfies the condition (2.22). Then almost everywhere in (c,d) we have the following estimate L nn(x;A) (x;A)=O = Os (l)TW(A). s (l)TW(A).

(2.38)

Moreover, if the relation (2.35) holds, then there exists Cv independent ofn,m and x such that uniformly for all x e [c + n,d-n] (0 < r/ < (d - c)/2) and r e (1,2] we have L (2.39) M*;A)
pc rC

px—l/m rX—l/v

/ = / + / Ja Ja Jc

px—l/n

rx

+ // ++ / / =h+I2 + h + h. Jx—l/m Jx—l/n Jx—l/m Jx—l/n

(2.40)

One can suppose c < c, but < xx — — 1/ra, 1/ra, since since xx — — \/m \/m < < cc < < xx — — \/n 1/n or or xx — — 1/n \/n < < c, . . both . . . one can obtain the estimate /rX* in cases as below. ^From Christoffel-Darboux's formula, (II. 2.22) and from the orthonormality of the system {pn} (n € Z+) one finds |y^

(n)

, , vPfc+i(<*M, t)pfc(d/x, x) - pfe(d/x, t)p +i(rf/x,X)x)

Pfc+1 (
I

Ja

fc=o

U=o

l~ x

^ f 7 T 7 £|AAi n) |[| Pfc+1 (d,x,t)|b fc (^,x)| Ja

x

l

1 Mt)

I

I

k=0

+ |Pfc(dM,*)l|Pfc+i(dM»«)l]dM*)\Pk(dyL,t)\\pk+$dv,x)$dv(t). ^From Schwartz-Bunyakowskii's inequality and the condition (2.22) follows [\pk+i(dn,i)\ dn(t) h < c i £ | A A l n ) | / l\ + \p\pk+i(dfi,i)\] Pk+1(dfi,t)\ + k+1(d»,t)\}drtt) 71

fc=0

Ja

1/2
Therefore, we obtain that uniformly in [c + rj, d — rj\ (0 < r) < (d — c)/2) h < < ^C,I^(A). T^(A).

(2.41)

Fourier in in L£(l r< 00)oo)and CC FourierSeries Series L£(l< < r< and

174 174

Introducing the characteristic functions Xnm(£>#; A) and x nm (£,x;A) of t n e s e t from [c, x - 1/ra] for which Kn(dn; t, x; A) > 0 and Kn(dfi\ t, x; A) < 0, respectively, and carrying out\i the arguments as in the proof of Theorem 2.5, we deduce nxxt; a i g u i i i u u b o a o xxx uxxi^ y x w u x vrx x u w i v u i , x-l/m rx-l/m

/

n

) ^AAi nn) P d/i(t) |£AA< Pf cf (rf/x;^x) c (dM;M)| <*/*(*)

/

^s^rfjr^r

Since the measure // satisfies the condition (2.35), then ) r /r 1 / r r( r1_ 1 )r / r h < cfe\A\£ -(Ei A 4 n\ )yr)m< -- V, '

(2.42)

fc=0 fc=0

where the constant C > 0 is independent of m, n. In order to estimate the term I3 we choose the positive integer N = N(m, n) such that (2.43) 2 " - i I < I <2 " i n

771

n

In view of Christoffel-Darboux's addition formula (II. 2.32) and the hypotheses of our assertion one can find Pk{d t)pk+l(< 1/n n ^ p f c + 1 (rfjz, ,f. h = /,Xs~1/n I E ^Xk^n)^^ ak{dn)Pk+l{d ^t)pk{d*)p ^X)fc(rf/i, g ) ~I Pfc( »' *)fffc+l^'x)(<*/*, g ) IId dy(t) Jx-l/m I * x I fc=0 dfljt) n) AA n | ^E|AAi

'Ei i fc=0 fc=0 i.—n

;CX;IAA«I v n

fc=n k=0 fc=0

X — tt X

Jx-l/m -Jx-l/m 1/m ^ N

- 1 /r~ 2i_1(1/m)Cdfji(t) j l rx-2 - {l/m)

„

2J jri 21 Jx-V(l/m) jr[ Jx-V(l/m)

(

\

S)£

/ fc=0

*

(m = 1l.-".«! , . . . , n ;" n €eZZ+)' ), ("»= +

(2.44)

where the constant C > 0 is independent of n,ra and x. It remains to estimate J4. It is not difficult to see that

f

Vx-l/n

|EAA
k=Q

n) < C-X> c £ ( * + l)|AAi l)|AA
Jx—l/n Jx-l/n

175

4.2. Estimates of the Lebesgue A-Function Hence, by (2.35)

AAl fc=0

where the constant C > 0 doesn't depend on n and x. Then from the last relation and (2.40)-(2.42), (2.44) follows the uniform estimate (2.39). Theorem 2.8 is completely proved. We apply this result to the de la Vallee-Poussin kernels of the O.N.P.S. {pn} (n e Z+): n 1 = — — V]£>i(d/i;£,x) = -——— Vx>j(d/i;t,x) n-fc + il rrt ™-fc + ^ (A; = (k = 0 , 1 , . . . , ra; ra n€€ Z+; t, x £ [a, &]),

V -k(dii;t;x) .k(dii;t;x) Vnin nin-.k(dii;t;x)

(2.45) (2.45)

where Vk{d^\ t,x) (k = 0 , 1 , . . . , n) are the Dirichlet kernels defined by (III. 3.5). For these means we have (n^ I 7 :; AA[ n) = I n-k+l n - k +1 0 [

for fc k <
ra

, (2.46) (2.46)

Note that if k = 0 then the de la Vallee-Poussin kernels become the Fejer kernels Fn(dwt,x) (see (Ch. Ill, §4)). Corollary 2.9. Under the hypotheses of (2.22) and (2.35) for all x € [c + rj,d-rj\ (0 < rj < (d — c)/2) and r G (1,2] we have the following estimate

c 1+ ] J/ l K\V l >ntn n -^kf(dfJt;t,x)\d/JL{t) c(d/i;t,x)ld/x(0
*(

»^ki}j . + l

(2.47) (2.47)

In fact, if we put m = n - k + 1, then from (2.37), (2.39), (2.46) follows the relation (2.47) immediately. R e m a r k . The estimate (2.47) shows that the Lebesgue functions of the de la Vallee-Poussin kernels don't increase faster than In n_rjc+1 on all compact sets in (c,d). The condition (2.36) gives the power growth only:

((n+ir^iAA^r) 1 7 fc=0

f (n + ir* v1/r 1

| (n - fc + l ) ' " j

n+1 {[n-fc+lj \ .

Hence the estimate Tm„(A) < C is the generalization of (2.36). Approximating the A-kernels by the de la Vallee-Poussin kernels, we obtain

Fourier £ ( l < r < oo) 00) and C Fourier Series Series in L 1^(1

176

Theorem 2.10. Suppose that the conditions (2.22) and (2.35) are satisfied. If for valic the entries entries o/ of A-m< A-matrix (2.1) the followingg hypothesis is valid notrix (2../,) (fc l )l)(n ( n - f-cfc++l ) f/ £ ■>a(*(fc+++l)(n-*+l) I 7" 1++ 111 n V \ fc=0

B„(A):=£ - ifcj E5

a 22

n ))

n n+1 V _n ++ l1 \V

(.,n)

— r — r ■W I'|A Ai k fh 7ri-k+lj* |A 1 ln

^nrrrJ n-fc +

Aa2

| < C,

1 //'

n )n)

(ni = l,2 AA<"> A[" A; Ali»> =A(AA[ =A(AAi .(AAi">)), = A(AA< )), ,2,...,A = = ll,2,...,A , 2 ,1....,A (n = i,:

(2.48) (2.48)

theni there J/iere exists an at absolute constant Cv > 0, independent of n and x, such that the inequality Lnn(x]A)
((n ++)) (nne€eZ ZZ+)

(2.49)

holds uniformly in the interval [c + 77, d — 77] (0 < 77 < (d — c)/2). Proof. It is easy to see that T> (k+. Vk{dp,t,x) (dfi; t,x) = = (k (* (k l)Fkfek(dir, (dfi; (d/i;t,x) kFfckfc-i{dn; _i(d/it; _i(d/i;t,x) t,t,x) x) ++ l)F t,t,x) x) -- fcF kF k(d(i;t,x) and V = (n-k (n-k l)V (dn; \t,x)t,*,x) T>k(dfi;t,x) k)Vntn-k-i(dn;t,x). £>fc(d/x; *, x) = (n - A;+++l)V l)K, x)--(n(n--AOVn.n-fc-iOtot.a:). JO^n.n-ife-i (d/x; t, x). k(dn; t,x) n>n-n-k-fc(dji; k(dp;t,x) ntn Then n

= E^

K {d^\t,x t, AAi (d/z; t,t, x) A£n)n)n)PPfcfc(rf/z; (d/z;*,x) Jr»(d/i; t, x) x) = ==E Y^ AAi x) t,x) J2^ n(dn fc=0 fc=0

v1/

= l)F (d/i; t, t, x) x) -A:Fik_i(d/i;t,x)] *Ffc -i(d/i; t, t, x)] x)] A^Kt + l^d/ijt.xJ-fcFfc. fc(d/i; fc-i(d/i; = 55 33 AAJ^K* AAJ^p + l)F -- *F fc fc=0 ib=0 =0 A:=0

n n

E^ +fc=i/+l E -f-i

^ AAi nn) [(n-^ [(n-^ + l)V;, __-fcfcfc(dp + 5 5 33 [(n-* l)V (d/i;t,x) l)V;, (d M ;t,x) nn_jb(d/x;t,x) nffl fc=i/+l ib=i/-»-l fc=i/+i

--(n-k)V ( n - k j-*)V^, V nn ,. nn_-nfc f-ib-i(dAi;t c_i(dAi; - i f d f i ; * *,*)], , *lx)] ) ] ,> -(n-A

where 1/ = [n/2]. sum In view of: Abel's summation formula nw) K /ifn(d|i; t, x) x) = ( *fc+ l)AA22A[ Ain)n)Ffc(d/x; (d/i; t, at) x) + (1/ (y + l)AA(, F„(dji; tt,, x) tfnfe l)AA<, n(dfi;t,x) .JAAWj^dji;*,*) === £53(* E( +!1)

fc=0 Jb=0

+ (n --1/) V ln,,.nn_„_ . ^1i(d ^ ;w t,x) i/JAA^) 1Vn.n.^ifdAi; t, x) •i')AA^

(2.50)

n - l1

I»(n - fc)A AinVV- i, b (rf ^ ^ ^Ai ^ ^ ^; t, xx). ). E (w*)A ._ --_l(d/i;t 5> --" £fc=i/+i < fc=i/+l hi fc=i/+i 22

wn)

nnf nn f cfe 11

/i

>

4.2. Estimates of the Lebesgue A-Function

177

It remains to apply ippiy ithe n e relation (2.47), then we have i/-i

ME'

2 n) nn (x; A) < C„{ (( ** +f l)|A ( „++ l)|AA£ 1)|AAW| >| LL C„{ £ ] £^(fc 1)|A 2Al Ai n >)|| + („ (» l)|AA<, 1)|AA<.">| >| + (n (n ( n - i/)|AA<,">| K)|AA<">| ^i ^l A A ^ I! M*;A)
(2.51) (251)

=0

fc=0 »-° r - ln *,„2±i],A^»|}. i i aA 2 i + nE- l( »( n-_*f c)fe + In=r5||A A<»>|}. 1 + )[f l,[ 2±il, 1 + ln' ^ i f c A'■2 A («)||. i E« n K J + L J) l |A2A<">| + fc=i/+i £ ■l> - k) L 1 + In2±j .

We will prove the the following estimates: folkr

l We will prove the followingn -estimates: .1 . n + 1ll . A2 .2 AAAl "^)'|| + 2 ((n-fc) L 2 n) n f c ]fc=0 [ > + l)|A A< | + fc=i/+i £ (n - fc) 1 + In 2 ± ± |^ 2 A£ TI >| < C£ n (A) n and ndfc=0 fc=«/+l L *]

E(

E(*

[

^ikj '

(2.52)

(2.52)

(2.53) iy + 1)|AAW| + (n - IOIAA^I < CBn(A): and + (n - (2.53) ^ l A Aare ^ I independent < ^ n ( A ) , of n G Z+. (2.53) where the constantiyC+>1)|AAW| 0 in (2.52) and whereWe theobtain constant > 0 in (2.52) and (2.53) are independent of ndeduce G Z+.exactly theC inequality (2.52); the estimate (2.53) one can obtain inequality (2.52); theformula estimate (2.53) one can deduce exactly in theWe same way,the if we use the following in the same way, if we use the following formula n n

t / Aa<») A ^ = 1 - A< \Wn> - £ nid Ai") i/AA^ \W J2(k l)A222AJt Ai!n)n) aanc and Ai"> A<,n) = = £ AAJ^. AA£°. i/AAW >;jfc + l)A Ai fe=0 fe=i/

k=vV

To order (2.50) observe that ior for 0
<-H±I^ d C I< ^^ Y| <<
(fc == i/ 2,...). l,2,...). ti//++ ll,,......,,nn--ll;; nn ===l ,l,2,...).

ifc + 1 -

So

) ^(A: 5 > + l)|A l)|A |A 22.22A<">| A£ Aln)n^n>|
E< fc=0 0 fc=0

and

n-l

+1 n) n) n) ,(">i CCB B nn((A) A) (n -- fc) fc) [[1 + lIn Inn nn££±jU ± jEU |A |A |A2222Ai Ai Ai <^ CB„(A) ||| << ^ (n (n-fc) 11 + — K \ fc=i/+i +i L n — A; are valid, whence by (2.48) the estimate (2.49) follows. Theorem 2.10 is proved. Corollary 2.11. Assume that for the entries of A-matrix (2.1) the hypothesis (2.48) is true. Then Lebesgue's Lebesgn function of the Jacobi orthonormal system

*)[l+ln£±iL

££

-a

0) L
1

1

n n

IS

a 0 a t)(l+tfdt (l +\-tf a(l+tfdt(a,l3>-l) \yip^p (a;/5;t)pife(a;/?;x)|(l (l Pk t) dt(a,0>-l) pfck(a;0;t)p (a;/?;x)|(l \yiPk(a;l3;t) -t)(a;0;x)\(l-t) (o,/3 > -1) k(a;l3;t)p k(ak;p;x)\(l-t)

fe=0 i n

a

178

Fourier Fourier Series in X»£(l L £ ( l < rr < < oo) aand nd C C

satisfies the estimate

a,/?) (n LW\x) 0 is independent of n € Z+ and x G [-1 + n,1 - 77] (0 < 77 < 1). Remark. It is easy to see that Ilcll !(C,a)-means (0 < a < 1) satisfy (2.48). In fact, Aa~2 in this case A 2 A £ ° =- A£lllK (0 < k < n) and using A" ~ na n -> 00 (I. 2.40) we obtain in

4 (*)

a 2

+ 1)( fc++ 1l) 2-2 ^ (n-fc l ) ( n - fkc+++1}1}Q( 1ln +i nl--n ^i±M(iL (nGZ+). IMA) < tn(*i(Jfc "71 +"-1^ lf)fl/l++ln £^n ±+U1U (n<" (neZ B„(A) - -~ B ^ + lKn ^ t7 l1a ) 0)°" " << CC (n n(A) < € Z ++). ;

I

A;=0 For, w e h a v e

a

n-k + lj n-k+\}

1/

n

-0>:

2 i= «(A) + + B( BW(A), 2#>(A), ))=Z#)(A) =BB«(A) >(A),

* n ( A=) =(( £X >+ £ *„(A)

fc=0 fc=i/+l fc=l/+l fc=0 fc=

where 1/ = [n/2]. Hence, Hence, using usir the asymptotic of A", we obtain

4 (A) === ~ ! jfc=0 ^ Bi)1)

l)«-2 )A j/ l ) ((nn- -f c*+++ll))ll( nH +iM) \(n-fc £f (*;t +++ l)(n-* ^l n _^±L\ ( n - * ++ !)-» a 1 + la nn+1 + 11 II n-k + lj \J n n<* I n-fc + iy

E

^<- S aT+ l ■ DD*^ n n n

A:=0 k=o fc=0 fc=0

1

(nE Z+). ) ^ C (n€Z+).

On the other hand, and,

rc n- , *-^ n" , ^^ i

n

22)(A)(A)
n -— A K:++ \1 n— k+1

fc=i/+l 1—V n—1/

7la

'"Mn-fCC s3=1 =l

5

(nt G Z + )

and our assertion follows from the two last estimates. Corollary 2.12. Under the hypotheses of Theorem 2.10 for bounded function f we have the following estimate |l^.(/;*;A)|: t fnB(f;x;A)\
(2.54)

a<x
Here the constant C > 0 is independent of x e [c + 77, d - 77] (0 < 77 < (d - c)/2) and f. It immediately follows from Theorem 2.10 and the following obvious repre sentation b

-f

(f;x;A) = /" £/„(/; Un(f; ar; x; A) A) == j f(t)K f(t)Knn{dy,; (d^ t,t, x;x; A) A) d/i(t) d^i{t) (n (n €G Z+; x €G [a, [a, 6]). 6]). (ne ZZ+; xG (dp,t,XiA)dlA(t) +; x Ja Ja

(2.55) (2.55)

4.3. Strong Summability of Fourier Polynomial Series in Lft(R L*j;(R>> 1) 1)

4.3 4.3

179 179

Strong summability of Fourier polynomial se ries in Z£ (r > 1)

Problem of strong summability Fourier series (III. 3.1) in L^ was investigated in Ch.III. §4. Now we consider this problem in^L£0 (r > 1). Theorem 3.1. Assume that for some qo,, 2 < qo oo, the estimate g0 < oo,

U

, b6

1/90 l/» 1}1. l/9o 1/9

IIPnlUo.M : = '

{ jf'|ft,(d/i;x)|»rf/i(x)| qo \p (dfji;x)\ dfi(x)j \Pnn(dfjL;x)r dfi(x)j

C Mn < oo

(3.1)

<M
(3.1)

holds, and {Mn} (n G Z+) satisfy (2.24). Let f(x) is L^-integrable in [a,b] (q'o < r < 2) and s is defined by (2.14)- If XQ G (c,d) is lr-Lebesgue's point of f, and the point XQ the relations (2.25) and (2.26) hold, then (1/r + 1/r' = 1, 1/go + l/q'ou = 1)/ sj l1/ » /.

n

Ml- \S (dfi;f,xo)-f(x0)\ - /(*o)|*} >™ {4 9q^K^P) 0 / ( /{q, 9 0 ++r,) ^ ) - E * Okf t «« /. *»> >)ISJ == 0,= o, n—>oc lim

1

1A I n n° °

2

s

(3.2)

ifc=0

. ^ nrp where Sk(d[i;f, Xo) (k G ZZ+) aret.hp the partial sums (III. 3.3) and

n / *M <)■ A := n Ann ::= = £E MMM*rrk22-2(( E M i) i) S=0 > ' ,' fc=0 ^ j 0 = o k=0

°E'

nG e€z Z+). ((n +)-

3 (3-(3.3) )

Proof. Let hn = hn(xo) (n € Z+) an arbitrary sequence of positive numbers tends to zero, more precisely definition will be done below. Define the sets /„ and I'n by Define the sets /„ and In by /„ = [a, b] n (x0 - K, xo + hn), /„ = [a, b] n (x0 - hn, x0 + hn),

I'n = [a, b] \ In. I'n = [a, b] \ In.

By (III. 3.7) By (HI. 3.7) Skk(dn; (dn;f;x (d»; // ;; 0xx)-00)) -- f(x -- f(x /(x00)) = = // [/(*) [/(<) /(*»)]©„(dp; rf^W (t)-f(xo)Y 0)]Vn(dfx; t; x0) dfi{t) + /[ [/(<) - /(a:o)]^fc(
T Fourier Fourier Series in L L£(l r < oo) and C U{\ < r

180

3.1) and Holder's inequality inequalit follows Rrom (2.25), (3.1)

I y i/w-/(«o)]'

)dfi(t)\ | jJ [f(t) f(xo))V (dp; [/(*) --■f(x /(«o)]© t; 0xo) *»(*)| 0)}Vk(dn;t;x 0) (*)| Cx„(i) |/(t) /(*o)l•IE M*)l Mt) |p*(t)l«M*) p. = Cx„(l) oxo(i) ■"» \f(t) -- /(xo)| £E IPiWI Mt) 3=0 Jl

» fc

-

jj=0 =0

9i

1/9

= ooxoxoO(i) (i) i/w -/(x ^w) 1/9^°° == (I)J2\\P £J2INI».*.( llftll«,»( // I/W-/C l/W --/(*o)i /(*o)l* dM(*)) 0)|*° <*/*(')) X0 }\I :i)Eii* CII«H».M(/ i=o 3=0 i=o =0 i=o kk

/n ^^ ^^ *'

=o (i)x:M {[y i/w- /(x )iM^)]9i/r[y[y dM*)] ^w] 1 - ^} 1 ^ 9i/r ,i/r

M

= 0O„;o l(l)^M )X > ,>{> {[ j T l/(*)-/(*o)l a:o)rrfM*)] £>*{[/ = X0X0((i)E > n | / ( * ) - / (^(xo)|M^)] 0 j, = ==0 0 0

^^

[/

d^)]1"^"}1^-

*T1

The relation (2.26) gives )]X> (dp [rf|x; t;*;t\ xo) rf^(0 \f(t) - /(x f(x00)]X>fe(dM; )}Vkfe(dfi; || y/" [/(*) *o) d/i(*)|
/1/r r | / (t«)-/(xo)r^(t)] ) -•/(«o)l / ( « o )prdM*)] d M * )1/r ] |(E ((E E( EM^i))i ),' = o„(i)i/»ii/»-i/r[y[y i/(*)-/( |/(

1 „|l/»-l/r[j = O l 0 (l)|J„| /90-Vr^

1/r

M

Jj == 00

r

where |Jn| is the Lebesgue sgue measur measure of Jn. Since xo is L -Lebesgue's then cogue; o point, |

it" £S

)\y 15

M 22 / 1/a {J2 ' { E M k~ r | \I /fy [f( [/(*)-/(xo)]P r0)]2>fc(rf/i;t;x wo-t)-f(x/(xo))^(d^t,xo)dKt)\y 0)rfM(*) fc(^;t;xo)dM«)r} fc=0 l fc=0 i.—ft

=

, JI //

"" '■''»^ i j i / * oXo(l)\Ii in/ \^Al/', I -^n >

'

J

where An is defined by (3.3). Applying the Christoffel-Darboux summation formula, we get I // W)-f [f(t) - /(xo)]%(d/i;t;x f{xo)]Vk(dfi]t)xo)dfi(t)\ [f(t)-f(x )]ak(dfi) {dfi) ) | == /j [/(*)-/(*o)W(^) [m-f(x0)}a 0)d/tx(«)| I\[ l/(*)-/(«o)]^*(4«*;*o)d/i(*)|= Pk+i(dKt)p Pk+i(d(i;t)p (d^x Pfc+i(a7z;t)p (ri/i;so) xxPfc+iO kk{dn;xo) 0)xp) ^x; t)p (afx\ kfe x^

-p -Pk(dfjL]t)p \ kp{dn;t)pk+i(dfi;x k+i(dfjL;xo) 0) --Pfe(^;t)pfe+i(d/i;a:o) k(aw t) <*M*) • tt — Xo — Xo

Put

/(«) - /(xo) J/fc/0*) (MzIM

■c

(t,n) := < g9 == g9x *£ -— xXo X00(t,n) := ^ 0 9 = 9X0(t,n) := I[ t - 00x0 { 0 Then, using (II. 2.22) and (2.25), one obtains [/(*) -- f(x0)}Vk(dn;t;xo)d )]Vk(dfi;t)Xo)dfi(t)\ I / [f(t) ti(t)\ I JIk

I

; forf€7 ff oo rr f*€€//;; for t€l'n. for <€/;. fort€/;.

= Oxo(l)[\c ifo (l)[|cfck(g (
t x (3.4)

(3.4)

4.3. 4.3. Strong Strong SummabiJity Summability of Fourier FourierPolynomial PolynomialSeries SeriesininLf*(R L**(R>>1) 1)

181 181

where Ck(gXo) (k = 0 , 1 , . . . ) are the Fourier coefficients of gXo with respect to the O.N.P.S. {p n } (n € Z+). (2.24) and and Minkowski's inequality So, by Lemma Lemn 2.2, condition dition (2.24) M n

)d»(t)\y ) {£>r | / lf(t) ~- f(x )]V f(xo)}Vk(dvt;x (dfi;t;xo) : )rf/x(*)| E* r 1a1/,!/(*)-/(*

ja^l 1/s

M

{ E

0

/S

0

k

0

fc=0

n 22 1/5 s 22 1/a s 1 / OoX0o(i)[EMr =Ox = O 0l 0(i)[; ( l ) [£M ^ Mfcf c-s -Mffx, | ck(9x | c ffc+ ( 5 x0„)r] )r]1/i f e ( f0f)\'] x „ ) | ]+ V i (l 0 ( l ) [ ^ M r ic c + 1i(p, 'k=0 fc=0

fc=0 fc=0 fr \ 1/r

f /• i/\f(t)-f(x (t) - ■/(f/:o)r wrf,lM |It-sol' *-x0|r [7/;

= Ox.(l)|

We consider only the integral r = Jn

-f

r

r l/(*)-/(*o)| /6 l!/(*)-/(*o)l /W-n»o X r / + h, —i7—ri;— Jxo ft |t -~*o| 0V

...

d,„ dp, ^

n

JXo + hn

because the integral

h r " i / ( t ) - / ( 0x)r o )Mt) rdu(t) . r—i/ft)-/(x r"\m-fM\ rXQ—hhn

(

|t -~ X x 00\| rr \t

Ja one can estimate in the same way.

^ '

We choose the positive integer N = 7V(n, #o) such that x0 + 2NN~1l hn
+2kh

- i/w-zwr^ i/w-z^r^ i (t)

fc 1/.x 0 +2 /i

J„<£/

r I*| *-~ ^^o| 0|r

-2fc-l/ln

1 fc=1^xo+2*- N i

W

M

k /•xo+2 hnn rx0+2 h

k

fc r rr AT 2feft = r,(2 hft»)" |/i \f(t) f(xr0)\ dp(t) nny)" // k 1 l/(t)-/(*o)| dMt) = Ox„(l) o - ( i ) V(2 E( l/W -" /(*»)r<*/*(*) kn il Jxo+2 h k = Ox„ ( 1 ^£r[ )^(2 Jx +2 J i o0+ 2 -- hn„

fe=i

N

«

/»xo+2 /•Xo + 2 hn

f cf e 1 1 r == O OxXl 0.0(((ll)l)()(/(/i»/„i„)n)-)-- ^ 2 - *^'2- 2 - - / fi tn„- -^gZr TTTr -- //

E2-

sir/

fc=l fc=l

h

r

n

r r l/(t)_ \f(t) dp(t). |/(t)) -- f(x0)\f(xo)\ f(x0)\dfl(t). lj

Jxo+2k-lhn

By definition of L -Lebesgue's point oo

r r _1 1r) -)f f1c c) f c o (l) = o x ^ l ) ^rr . Jn Jn = Jn = O O x*O„oX0((l(l)h lKK-n-r-rff>>- -( (r2-( O 0 x*„X. (o( ll )) == Ox„(l)^O x „ ( l ) / l „ - ..

E; Jfc=l

Therefore 1 l r / rr M M 2 22 f(xo)}Vk(dvt,xo)d»(t)\y/3)D 1/S' = o Xoo (l)ft - » .. n n-^ Xo(l)h ; ^rr\ | /I [f(t)I/(t)-/(xo)]i?fc(dM;t;xo)dMt)r} ({flE 'k~ [/«-/(*o)]2>fc(d^^^

ii/« _

fe=0

182

Fourier Fourier Series Series in in Z L y£ ( l < r < oo) aand nd C

Combining this relation with (3.4), we find n

. 1/3 _

2

a 1/S

2 1/S /9 D^An ++ \S*k((f;x {J2M* == o, M„ + { £ A Mr fkr-a \Sk(f;xo) |S / ; * 0o)-f(xo)\ ) -- f(xo)\'} / ( * o}) | s } 1 / S = = OxoCDJ^'Mn O x0 (l){fti 0(l)|'

i-r)/r\ r)/r hV-'V}, kg""*], ht \,

fc=0

where An is defined by (3.3). It is easy to see that _> ^ 0 A2 n11l- 1 -> A' -> 0 (n (n ->• ->• oo). oo).

So, one can put

^-9i/(2-gi+sgi)

o) hn = Anq'0,{2-q'°+ '0+sqsq'Q)

(n e Z+). (n

Therefore n r |- i 1/5 l/s 1/s s 1/a 1V/ S 2 22 . (» 2 --9 o«- f4s + g 0^ - s-) /)( /2 P - q-0 + stg+ i )^ ) | ^ a £ Mr M° \S (dv ; /(xo)| } (l) U « {{ J2 \S (d»; / ; xo) / ( * o ) | ' } ' = o (l) \ g-*+*-)/V-*+*)'\ "\ k k IO k Xo (i)LA itfr l*(d/x;/;*o)-/(xo)r} = Oxo jfc=0

which is coincide with (3.2). Theorem 3.1 is proved. One can obtain (using the local variant of Lemma 2.2) the local variant of Theorem 3.1, too. Theorem 3.2. Suppose that for some qo, 2 < qo < oo, the estimate (2.18) holds, function f(x) is L^-integrable (q'0 < r < 2) in [c,d\ and L\-integrable in E = [a, c)U(d, b]. Ifxo G (c, d) is lr-Lebesgue's point of f and in this point the estimates (2.25) and (2.26) hold. Then nn

|5fc( ; /; X0) /(X 5 s)|S =°=0 | 5)|5fc(d/i;/;*o)-/(a?o)|' f c ( d / X ; / ; Xo) / ( X o ) |)l === tS&> (nn + 1 ) « ^ W / W ^ ) £ " + W 1

^ ( l)(U /(^) g ^ lim

" °

s 1 n—>oo ( n + l ) ( ( + ) 9 o ) / ( 9 0 + ^ ) fc=0

°

(3 5) (3.5)

'

is valid, whenever s is defined by (2.14)Proof. First of all, observed that in this case An = (n +

iy+1

and (s l)<7o. (s+iM>i ^— ±r+:!— Mr - >:>£ 1i. 1.. -

(3.6)

-

g gq'00 + ++ r' r'r'r' 0

K

-

As in the proof of the preceding Theorem, Sk(dfj,',f)Xo) ^(d/i; \f{t) ■- - f{x f(x00)\V )}T> (dfj,;t; t;x 0x)0 )dp(t) d/tt(t) Sk(dn; / ; x 0 ) == // [/(*) 0)d/ji(t) k(dfi\t]x k{dn;

- IE[lf(t)-f JE JE

Vk(dfi' t]Xo)dfji(t) - /(a;o)]2?fc(dM; *; a;o) ^MW [/( )\f(t)-f(xo)]T>k(dp;t;xo)dn(t) +1Jin [/w -

+I- /I

1

+ I [/(*))-/(afo)]l'fc(d«*;«o)dM(*) " f{x f{x00Vk{d^t;xo)dfi(t) )}V )}Vkk(dn;t;xo)dn(t) (dn;t;x0)dn(t)

+ •Jl'n "/A[ / ( * ) 2) 3) 1Jl'n ) = 4 1)(x 0 ) +f 4 j f ) ((x x 00)) + 4 33))(xo). (*o).

-Jl (x0) + .

!

of Fourier Polynomial Series in L*(R > > 1)

4.3. Strong Summability

183

The estimates of integrals 4 2 ) (x 0 ) and 4 3 ) (x 0 ) are the same, as the estimates in the proof of Theorem 3.1. It still remains to consider J ^ ( x 0 ) - By Christoffel-Darboux summation for mula one finds ids

mzJM j(i)1){xo)= /(*) - f {axM(a ° \(^){p ) { p M(dp;i) ( d ^ t ) pPfck ((d dw ^ xgo0)) Ji (*o)= .—~.———akidriipk+iidKtfpkidwxo) {(Xo) k ( d f i fc+1 —: a>k fc •tf^o) = IIf' f{t) JEE JE

ll

xx t ~" — xXQ 0 ~"

(d/x; t)p i {dfi; (rf//; x 0x)} d/i(t). ---PiPk(dfjL\ Pk(dn\ t)p +i (rf//;XQ)} dfi,(t). dfi,(t). kfc+ k+\ 0 )} ppkfc(dfjL\ t)p k+\ {dw x0)} dn(t). Using (II. 2.22), (2.25) and (III. 3.9), one obtains 1) (x )->0 (A; -* oo). 4 1) (*-»oo). 00

Since (C, l)-mean is T-regular, then by Theorem I. 2.20 n

n 11 lim •—L-y^lJ^^oJ^o. lim >l"WI = o. k V n - >»o 1onn+ + l1 *-* " n->oo

k-0 fc=0 fc=0

Hence by (3.6) we have the limiting relation (3.5) in accordance with our state ment. {p{pn(dfi;x)} Under the further condition that the orthonormal system{ P = {Pn n(d^x)} o1 •ir, LI. (n £ Z+) is uniformly bounded in an interval [c, d] C [a, b], Theorem o3.2 is capable considerable improved. Theorem 3.3. Suppose that the relations (2.22) hold, function f(x) is L^-integrable in [c, d] for an 1 < r < 2 and f € L^ is valid in E. Then for every q > 0 the series (III. 3.1) is (H,q) (q > 0)-summable, i.e. the relation n

1 lim dd lim — - //(^i* ( *xo°)i ) | 9 9== -- rrrr £ ]|5ii55fc*( *((rf/i;/;xo)-/(xo)|« ^^/; /;x*<>) °) --'( ==0°°

>nn -j+ l1 zf*—' n—►oo n—¥oo n Ac n—►oo n -H 1 —^ fc=0

3 7 ((3.7) -)

holds for almost all XQ e (c,d). Moreover, if f(x) is even continuous in [c,d] and the measure dfi(x) is absolutely continuous and the weight w(x) = d^/dx satisfy (2.35), then (3.7) is valid uniformly on all compact sets in (c, d). Proof. By reason of Theorem I. 2.5 and Theorem I. 2.15, respectively, one has for almost all points #o € (c> d) nXo + h

//

Jxo

r

|/(t)-/(*o)rdAi(*)=o(|h|) = o(|fc|) |/(«) - /(xo)rdA*(*) )| 4 I|/(*)

(fc->0)

and /i(x 0(\h\) {h ^i(xo) 0(|fc|) (fc (ft-><>). /i(x -> 0). u(x0 + ft) h) --- M*o) 0 ) = 0(|/i|) Consider p, 1 < p < rr then by Holder's inequality, these two relations imply that Consider p, 1 < p < rr then by Holder's inequality, these two relations imply that pXo + h [Xo+h

/ «/xo /

/

Jxo

,p l/(<)-/(«0)l | / ( * ) - / ( * o ) |'d/i(t) <***(*) == o(|h|) < »

'■

\f(t)-f(x0)\pd»(t)

= o(\h\)

(ft->0) (A->0) (h^O)

(3.8)

(3.8)

184

Fourier Series in L £ ( l <
is valid for every xo, satisfying of the above inequalities. This relation holds there fore for almost all points of (c, d), and we can show the validity of (3.7) for all such points xo € (c,d). The tie validity validity of ot (3.7) (3.7 for a fixed value of q implies its validity also for every smaller value value qq of of q. q. In In f
n

rf f;x0) --/(*o)i«}' f(x0)\ } 7' *" *(7^W{I>( n + 1 ) 9 / 9 Ei "/;*°) ' fc=0 fc=0 *

x

r

' 11

g 9/9

fc=0

nn'fc=0 ^U

i Q/Q /

oo). oo) ) - / ( a ; o ) l ( 1) ) (n n ->00 { ; T Tif: T^£| 5| 5i*f (cd( ^/ x;;//;;xxxo)-/(«o)|«}' ~ 7 * ==oo(l) (("-* -><*>)• )fc

fc=0 fc=0

la It therefore suffices to prove (3.7) for large values of q. Accordingly, let q > r' = r/(r — 1), otherwise arbitrary. Let \/q + 1/p = 1, then 1 < p < r and (3.8) holds. The first part of Theorem 3.3 one can prove in the same way, as proof of the preceding theorems, but using Corollary 2.4 instead of Corollaries 2.2 and 2.3 (observe that in this case the exponent in (3.5) is equal to 1). The second part (the proof of the uniformity) one can easily obtain by a similar way. Theorem 3.3 is completely proved. Remark. Orthonormal Jacobi polynomials satisfy (2.22) and (2.35) uniformly on all compact subsets in (—1,1).

4.4 4.4

Linear methods of summability of Fourier poly nomial series in U^ U (r > 1) and C.

We consider the linear methods of summability of Fourier polynomial series (III. 3.1), defined by A-matrix (2.1). The main result about A-summability of the series (3.1) at the point of con tinuity is the following Theorem 4.1. Suppose that the O.N.P.S. P = {pn(dfi;x)} (n € Z+; x G [a,6]) satisfies the conditions (2.22) and (2.35), and for the entries of A-matrix (2.1) the assumptions (2.33) or (2.48) are valid. if rir~ r.2n E = [a, c] U [d, b] (4.1) /€LjMnl£(E), E=[a,c}U[d,b} /€LjMnL»(E), E=[a,c}U[d,b] (4.1) <= ^ / i l ^ and x £ (c,d) is the point of continuity of f, then the relation (2.3) holds. Moreover, if feC[c,d]nLl(E), feC[c,d]nLl(E), (4.2) (4.2) Ll( 1

w

then the Fourier polynomial series (HI. 3.1) is A-summable to f(x) on all compact subinterval in (c,d).

Linear Methods of Summability

185 185

of Fourier Series Series

The proof of this statement follows directly from the representation (2.55) and the following auxiliary assertion is true. L e m m a 4 . 2 . Let

-A

/In(x)= »(*) =

./a Ja

i d/j,(t)I (n (nG € Z+; x xG f(t)Kn(dpL;t,x;A)dii(t) G [c,d]), [c,dj),

Ja

where the kernel ATn(d/i;t,x; A) is defined by (2.5). If the hypotheses of Theorem (4-1) are fulfilled, then the following statements are valid: 1) at the point of continuity of the function / , satisfying (4-1), the limiting relation oo) (4.3) In(x) -> //((xa)) ((n n -— >► o ooo) ) (4.3) /Inn{x)->f(x) W -► (n -+ holds; 2) if the function f(x) satisfies (4-2), then the equation (4-3) is uniformly valid on all compact subsets in (c, d). Proof. It is easy to see that by orthonormality of the system P P = = {Pn(c {pn(dfi;x)} {Pnid^x)} (n 6 Z+) (see (2.1)) one finds b

t, x; A) dn(t) dfi(t) I/ Knn(d/jL] (djj,\ {dn', t,x; cfyj(t) = 11 (n GZ+; xG G[a, [a,6]). 6]). ((ra n €G ZZ+; +; x XG Ja Ja Ja

(4.4) (4.4)

Hence Hence

f n €G Z+; [a,b]). f(x)]Kn(dfi; d»(t)) ((n J n (x) -- /f(x) [/(«) - /(x)]AT Z+; x x€ G [a, 6]). (4.5) /»(*) ( * ) = I [/(*) n(d/x; t, x; A) d/i(t) In(x) - f(x) = Ja I [f(t) - f(x)]Kn(dn t, x; A) dfi(t) (n G Z+; x G [a, b}). (4.5) Ja Since the functionn// is continuous at the point x e (c,d), then for any arbitrary Since the function / is continuous at the point x G (c,d), then for any arbitrary e > 0 there exists S > 0 such that e > 0 there exists S > 0 such that (4.6) \f(t) <£e for for \t-x\< | / ( * )-- /f(x)\ (x)|< |t - x| < S. *. (4.6)

We divide the integral on the right-hand side of (4.5) into three parts

-/w

/n(*)-/(*)

a

= ( / V

'

^|*-*|<4 J\t-x\<8 t—x|<<5

+ //

>1

[f(t) -- f(x))K dd^W (t) + /I f{x)]K t, x; + )) [/(*) f{x))K(dfi; (dir, t, t, x; x; A) A)dp(t) n(dw +/ ^|t-x|>«, t€[c,d] ./|i-z|>*, J\t-x\>5, te[c,d] J\t-x\>8, t€E t€E '' J\t-x\>6,

te[c, t€[c,d\

2 = #>(*)+ IP(X) + lW(x) I?HX)+I?HX) (x) + + I?Hx = /< >(*)+je>M

J\t-z\>S,

n

M

t€E '

?>(«)

and estimate every integral separately. It is not difficult to ;o see that tha by (2.22) and the Theorem 2.6 |jW( ;)|<£Lnnn(x;A) (a:;A) = O xx(l)£ \iP(x)\<eL =O (l)e,) r^>Caa;)|<£L where Ox(l) does not depend on n G Z+. '+•

(4.7) (4.7)

186

Fourier Series in L£(l L^(l < r < < oo) and C

On the other hand, from Christoffel-Darboux's summation formula (II. 2.32) followss n / ( * ) - / ( f{t** ))_ff x)x)f{x)pu (dn,t)dii{t) m\ x) [ f{t M 4 3 ) ( ^ ) = E A A i n ) a f c ( d il*)sPk(dn,x) . t x l~ J Pk+li Pk+i(di*,t)drit) M)(pfc(^,a;) / t— t-x ° JE s ~t-xa; fe=o =o I JB fe fc=0 fc=0

f mm-f(x] {x) ~J Pkid^,t)dp® Z x—Mp Pk(dn, t) dp® J. J. fc ( dAi ,t)dM*)}.

c ^Ml, xx) -p -- ppffcfcc+++i11((d (dp, x)) jj /

m

t

f {x) x

JE

t~x

Since the Fourier coefficients of L^-integrable function tend to zero (see (III. 3.9)) and (II. 2.22), (2.22) are fulfilled, then from T-regularity of the matrix (2.1) we obtain 3 )33 (4.8) 4jrt (>(*)-*0 >(z)->0 x ) - * - 0 (n ( n --> > * •o(oo). o»)).. J< In Ch. V, §1 we will prove that the Fourier coefficients of L*-integrable function tend to zero, if the system {pn} (n G Z+) is uniformly bounded, that is, the hypothesis (2.22) holds. So,i, as as above, auuve ) Ij, J< (x)->0 )->0 4i2)2)2)(2(x)->0 (*)

(n ( n (n-too) -— >►o ooo) )

and relation (4.3) follows from the last equation and (4.7), (4.8). The second part (the proof of uniformity) one can show in the same way. Lemma 4.2 proves. T h e o r e m 4 . 3 . Suppose that the relation (2.25) is valid and for some qo, 2 < qo < oo, condition (2.18) holds. Liei Let for oj noias. jor some some r (q0 < r < 2)

r

\r

| / ( * ) - / ( » o ) I Mt)<00 [d'\mzJM\ (t) < 0 0

Jc Jcc \I

£t — - XXQ t-XQ 0 t-XQ

I|

be true, and in E = [a, b] \ [c,d\ even L^-integrable. Under these hypotheses the orthogonal polynomial expansion (III. 3.1) of f(x) converges to f(xo). Proof. We put

/ ( * ) - / ( * » -) ftfor t € f/(*)-/(*o)

■{'

F < ^x„(t)= ^ -X0o(t)= (*) = ^ {[ and

t* -— —0* XQ t-x oXo 0

{I-

[c, d]JJ

L

XQ xe0€(c,d) e(c,d) (c, d)

for for tG te[c, rf] t e [c, d]

0

for d)] f o r£tt € [ [c, c , dd]

G«„(t) = /(*) ~ /i( *M o) f a = | /(«) / W - for t€[c,d|. *£ - Xo XQ — XQ £— t - Xo an GXo(t) €: L»(E). Evidently FXo(t) € L^[c,d] L^cd] and ^

187 187

Linear Methods of Summability of Fourier Series

It follows from Christoffel-Darboux's summation formula (II. 2.32) that stottel-Darbon

■A

rf / ; x00)) - /(x f(x00) ) = = // ' ([f(t) f(x0)]Vn(dfi; t, xr 00) d//(£) dfi(t) 5n(rf/x; (rf/x;/;x M;*>z / ( * )-- ./(xo)]£>n( Ja Ja = Ja an(d/x){pni(( d / i , X o ) [ C n + i ( d / Zi,F i(rf/x,G +i(dp,G )])] Xo) + cn+ = x00)[c )[cnn+i(
Cn(d/i;F Cn(dfi]F == 0 0 Xo) Xo) = llim i m
n—»oo n-yoo3 n—»oo

and lim^Cn(dAi;G cn(dfi;GXoXo) ) = 0.

n—►oo

Consequently, by (II. 2.22) and (2.25) one finds lim [i [5 (dM;/;a;o)-/(xo)]=o, f(x00)])] = 0, [Snn(dfi; / ; x 0 ) - /(x

n—too n-foo n—foo

in accordance with our statement. One can conclude the following assertion directly from the results of §3. Theorem 4.4. Suppose that the O.N.P.S. P = {pn(dfi;x)} (n G Z+; x G [a,6]) is uniformly bounded in the interval [c,d] C [a, b] (see (2.22)) and f(x) is L£integrable in [c,d\ for an r, 1 < r < 2, and / G ££(#)• / T / o r *^e entries of A-matrix (2.1) for some a (1. < <^ aa <<; oo) n

((nn ++1l ))"°---111^^|| A AAi AA |i!"' ,n)))rr = 0 ( l )

(n ((nne€e€ZZ Z+) ++ +))

(4.9)

fc=0

/lo/ds, f/ien /or almost all lr-Lebesgue points of /(x) (and, consequently, almost everywhere in (c,d)) the relation (2.3) is valid. Proof. It is not difficult to see that n

«E<

AAi £/„(/; Un(f; x; x; A) A) == ;££ ^ AA£ A A n) ^n)SSfc*(d//;/;x) ^ ; / ; x) (n (n €G ZZ+; x €G [a, [a, 6]), 6]), £/„(/;x;A) ++; x n fc=0 fc=0

where Sfc(d/z;/;x) (k G Z+), as usual, are the partial sums of Fourier series of /(x). Since AJ>n) = 1 and A ^ = 0, then n

n) ;AA[-) = f>A< (neZ+). (n e Z+). X>A["> = I1l (n€Z+). fc=0 fc=0

Fourier Series o ) and C Series iinn L ££((l l < rr <
188 188

Hence,

n

/(x)

^ ; [n)n)5[S*(d,x; ) - /f(x)] U x; A) - / ( * ) = x) (x)] [S*(d,x;/;x) == ££f>A AAA< f c ( ^ ; / /;;xx)-/(x)] C/n(f; n (/;x;A)-/(x) = n)

*jfc=0 =0

((ne (ne n e ZZ+; Z+; +f ; xe [a,6]). Applying Holder's inequality, we derive n

1/a ^ Q ||tf lt7/ „nn( /(/;x;A)-/(x)|<(£^ A [ n ) r ) 1 /1/a ( £ |]5| 5f cf c((^d;/ x/ ;; x/ ;)x-)/-(/ x( x) )rr' '))V1 /aa ,, /;;ax; ;;AA))--/ /((xx) )| |<< |( £ | A ;iAA[»>r) =0 fc=0 fc=0fc=o fc=o fc=0 fc=0

(E

i(Ei-

where31/a 1/a + 1/a'= 1/a' = 1. So, by Theorem 3.3 (see (3.7)), one can can ooiain obtain >.f;;, one 1

1/Q 1/0

1 ||D'«(/;x;A)-/(*)| I / n ( / ; x ; A ) - / ( x ) | ==< o(l){(n + l)(|^(/;«;A)-/(*)|=o(l){(n l ) ( « -)/-(f^|AAWr) ) / - ( ^ | >Ai»>r) A A { ! - > r ) 1 / a}}, }>, fc=0 fc=o )

D*4

and our assertion follows from the last estimate and (4.9). Remark. It is easy to see that (C, 7)-means 11

A' AIZ\

*n-fc =

fc i

n

n

1 ) ^ c : ^(n ^7 1} 1 ^(n-fc)"^1 lAA^r^Cn"] n*] n ± |AA<»>r < C n ± <*C,' » - £ iAAi»>r < ch-> ± -ff ~
'£

Jfc=0 fc=0

ib=0

if we take a > 1/(1 — 7). We shall also apply another approach to the problem of A-summability almost everywhere. For A-means (see (2.J (2.55)) rb

!/„(/; x; (f;x; f{t)Knn(dn;t,x;A)dp(t) {dn t, t,x; d/i(i) fa (n €e Z+; x 6€e [a,6]) [a, 6]) (ne Un(f; x; A) A) = / f(t)K /(*)#„(<*/*; x; A)
((ne n € Z+; xxe€£ [a,b]). [a, 6]). The following assertion gives the estimate of the growth of A-means. T h e o r e m 4 . 5 . Suppose that the conditions (2.22) and (2.35) are satisfied. Then "PI for allxe[c [c-+-T),C n,d-n] (0
1

2 1/r 1/r ^(/;x;A)
JE

(4.10) (4.10)

Linear Methods of Summability Linear

189

of Fourier Series Series

holds. Here 2m/n(A) is defined by (2.37) and MM(p) is the Hardy-Littlewood max imal function for g (see (Ch.I, $4)). The proof of Theorem 4.5 is similar to the proof of Theorem 2.8, and we shall omit some details. As above,

-r

rX /(*) ' ( * ) == C \f(t)\\K\m\\Kn(dii\t^A)\dii(t) n(dvt,x;\)\d;;A)|dp lx(t)

Jaa Ja

Ja can be divided vided i:in the following w i i i g way way can be divided in the following way rC

+

rx—l/m

i

!(*) - //rC♦+ / rX—l/m /(*) = Ja «/a «/c I(x) = / + / Ja

/«X rx-l/n 1/n ++ / r>x— + // ' n

« ■ /

r>X — 1 / n

Jx—l/m Jx—l/m

+/

Jc

Jx—l/m

rX

Jx-l/n Jx—l/n

+/

■= /h(x) i(x)

+ Ja(x) h(x) + /I83(*) (x) +14(«). + h(x). = / i ( x ) + Ja(x) + h(x) + h(x).

Jx—1/n

Using Schwartz-Bunyakowskii's inequality and (2.22) and taking into account the orthonormality of the system {pn} (n e Z+), we have n

f'

n) Cl / W I - ^ £ II(X)
*TTt] * ~ * fc=0

•'«

Ja lI

1/2 l2 A n )n) WJE E w W E n

*l/2

/:

c=0 V' fk=0 fc=0

Consequently, by the definition of T£„(A) one finds 11/2 /2

U1/2 , ., h(x) < h(x) < C^,„(A){ C„T^ C,2£,nB{A){ (A){ j£ j E |/(«)| |/(*)| Jx(x) |/(*)|22dd»{t)} dM*)} M(t)}

W-

2

(4.11) (4.11)

where the constant C„ > 0 is independent of / and m,n £ Z+. Introducing the characteristic functions Xm,n(M;A) and xmjTl(t,x;A) and carrying out the arguments as in the proof of Theorem 2.8, we establish \ l/r

px—l/m n /r 1/r mL^^)) duldn(t)\ . |/ < C iAAi->r) ( ,/a(x)| (,),,c(tjAAi»>|f (f-1/m (Ei 2 n

. A 1 1/r /',

fc=0

(jP

(x-ty

One can choose positive integer N\ = iV*i (m; x) such that x_ x

^ .< w -'--iii1lil ,, x_ x TO m m TO m TO TO 2"i

^')

Fourier Series in L £ ( l < r < oo) and C C

190 then then

* x2i- ^1,m -/m ^,rx-2V » | | rrr f* r ' \m\ dMt) :/

fx—l/m «, x- -l 1/ m /m

// (( tf )) || rrr * ||l/(<)| /wrdM<) ,,

ye

(^r

(a;-*)'

"

Nl

| // (( tt )) |

I

-gX-, j = l •Jx-V,/m

JTW

/a

Nl

1 / ] )M[j *^E* E S* S5FF^ I^. - ^M.x]){ J.^/mx]) ,i/wr (*)} E 2yZi)7A»((*-2 ^/m ,a;]) * { ^- £fi 2J/m «*>} M /wr **>} 2 i / m i/(t)rd ' {n([x-2i/m,x})L j/il | / i ( [ x -- 22 ^ V7mm, ,xx]l)) y . _ ,

>a x])

a 2i/m I; Jx-2i/m

J

A^n{x)-n(x-V{\/m)) ( x( a) ;- )M- /(fx( a- 2: -^2 ( (l l/ /m m)) ) M^

A /i(»)-/i(s-2>(l/m)) —m ^E <E^ ^£ i x--((a(a; J2 i( (ll// m a; m0) " *~( ) )) ~ £i -xr--—2 2^(1/™)) x

t

x

i1

y*

r

X ^ /i([a;-2J/m,a;]) x - 23 /*( [a; Z J,x._ .^ 2i /m JJx-2i/n \M< [yn(\x-23/m,x\) K[ ~ 23/m/TO, , x\)x\)Jx-V/m X_2J

x-2J/m

d

J

_ 1i 22(»-iM ( r1- 1lW )i 2<»-

2('- W

r i/(t)r^(t)l. \m\ dn{t) J JJ

Now, by (2.35) one can derive (in the definition of function M it is convenient to consider, that the integrals over the sets don't contain in [c, d] are equal to zero) rx—l/m xx-l/m -l/m ,m

//

oo °°

\f(f\\r

\f{t)\Tryd^KCmr-'M^mix)]l \f(t)\ ^2-(r-l)j M»[\f\r}(**)E (x (x -- ty ty 1 1 3=1

g ^ d r t t )

< Cm- JMM[|/n(x)532-(- M.

So p :X

- E^

\h{x)\ |AAin)r}1/r, | / 2 ( x ) | < r}"', C(M

(4.12)

fc=0

where the constant C > 0 doesn't depend on / , x and n, m. To order the integral h(x) we choosei the positive int integer AT as in (2.43) and obtain N N /•x—1/n rx-2j-1/n B | / ( t ) | d ( t )] /" M |/(t)|dM*) . n / -
JMiM)< X X-t a; — - tt

E

^ Jx-v/n j^232-> 7 xJx-2J/n -2J/n i=l' ^ Jx-2* In

< ln^JM cc fh(l i ++ ln^W[/](a:). [/](x). i- ) » MMWWM hln^jAf M[/](a:). ^

m / Consequently, there exists a constant C > 0 independent of / , x and ra, n, such that n n B) \h{x)\
(Ei fc=0

Finally, under the hypotheses statement )otneses of or our our staten n AA n) m ix; t)pj (djjL\ x) dp{t) dfi(t) l|AAi i n)l| [' T ii/(*)i|5 |/(t)|| \ i2Pj(dKt)Pi(dKx)\ ^Pi(d«<)Pi(dM;«)| dM«) <Ei^ ^Ei^ ^\f / " jJ== o0 k=0 Jx-l/n

I^4(X)| <£ |/|744(x)| (x)| < ^

fc=0

Jx-ll „n

< C M , [[/](^)X2|AA£">I.

oEi A:=0 fc=0 A:=0 fc=0

J=0 j=

°

(4.14) (4-14)

Linear Linear Methods of Summability

191

of Fourier Series Series

Prom Holder's inequality lality follows

»V'7

v l - l // rr V r/r 1L ( I | )|rr Jlf„[/](S) ((jMt)y~ jf d M ( t ) ) 1 _ 1i/r M^[f](x)<snp^[Jjf(tW /(t dMt)y //(f)) MM*) f< SUP -M(J)V [j\f(t)\ 11/r /r <<
I 11 W(d 1 4^ we W P establish, pst.svhlish that tVint I(x) TYi^ is is *»c+ii Combining this inequality with (4.11)-(4.14), estimated by the right-hand side of (4.10) in accordance with our assertion. We introduce by oduce the de la Vallee-Poussin in means means D nnn

11

1r——- V ] ]S n-k{dfJL\f\x) V ,n-fc =V t(dfi;f;x) V ==V VVn - f c ( ^ ; // j; x) x) = == Si(d/Ji; / ;f\x) x) nn,n-fc nn,,tn-k(dfi; Vn,n-k —— V ] Si(dn; nn n-k+1j (k (A; = = 0, 0, ll ,, .. .. .. ,, nn ;; (A; = 0, l , . . . , n ; where Sk{d^f\x)';x) (k 6 Z+) are the (III. 3.1). It is easy to see that

(4.15) Z=fc n-k+ifk (4.15) n —fc+ 1 f—' nn €€ Z+; Z+; xx 66 [a, [a, 6]), 6]), n € Z+; x 6 [a, 6]), partial sums of Fourier polynomial series

>b

Vntn -k(dfJL\f\x) nynk(dfjL\f;x) f',x]

i:

== /

f{t)V f(t)Vnin ^-kk(dfl]t,x)dfjL(t) (dni t, x) dfj,(t) (xe (x € [a,b]), [a, &]), 6]), nin

where the de la Vallee-Poussin kernels are defined by (2.45). Taking into account of (2.46), we find from the last Theorem Corollary 4.6. Under the conditions of Theorem ^.5 for all x e [c + 77, d — rj\ (0 < rj < (d — c)/2) and r £ (1,2], the followingesnmi estimate I\v Vn,n-k{dn n . nfc(d - fW c ^/; /x)\ ; ! )<< ! ^cj |V„,„_ /;;;*)l x)| c^ j

71+1

\

1/2 ^i1'V2 1l ++ lIninn nn " ^+j M {[ JjTE \f{t)\ |/(t)|22 d»(t)] d/i(t)] mfdrt)] 1 J {[

U

^riTT i

1/r 1/r (M^[i/n(x))1/r] }

++ (M„[i/n(*)) (M M o/n(x)) }}

<

1/2

J

(4.16)

holds.

One can apply Theorem 4.5 to the estimate of L£-norm of A-means. As a preliminary, we establish the following auxiliary assertion. Lemma 4.7. Let f e Ls^[c, d\ (1 < s < 2). Then there exists the constant r > 1 (arbitrarily closed to vruy CLUSCU vu 1), LJ, such that rr,( lJfeL^d]; / € L ^/xl J[c,dl; 2) d d r r /r#/r s /r /Td^
^ '

where the constant C > 0 is independent of f. Proof. Take an arbitrary number q e (1,5) such that 1< - <2 9 Q

in L£(l < r < oo) and C Fourier Series in

192

and put

_ sS 5

r =_ "

Q

Applying Holder'ss inequality with exponent q, one obtains (as usual l/q+l/q' inequa

r-

r 1,q i s f(x)\ drtx) dfi(x) == = f£\f(xw 1/9

rr

M

1/9 1/9 l/

= 1)

1

l,q \ 1/9' d^x))

1/9 d/i(x)) dfi(x)J

as stated above in the section 1). Using the estimate (I. 4.15) q qq jJ {M^g){x)} {M„[g](x)} {Mli[g}(x)} dti{x) d»(x) < < Cj\g(x)\"dn(x) C J r\g(x)\'dn(x) \g(x)\* Ydd^(x)
dn( ) == f /ii(= * dM*) jT {M^mwy {M [|/r](*)}' i: d

M

1 s/r } /r d

X

(q (q 1), (9 1). ( 9>> > 11), ),

M^f^ix)}" r'«] ) i ' d d»(x) {MJ/l'^K*)}'Mx)

1). Then there exist the constants r (1 < r < 2) and Cv (independent of f), sv such that off), fd-t)

r

2 S/2 r) + £jf i(x)
Jjf

n

r

2

l

ri

/2

2

E

Jc+ri

where

,/2

H

s \f(x)\'d(i(x)), |/(x)| dM*)),

TO;*; ; A A) )

(r)) t/. tui / ir)r(f;x;A)= ((/;x;A) / ; x ; A ) == sup ^ ' *

(x€[c,d|). (x € M ) . T£2(A) A )) TT ^^((A Proof. In fact, by Theorem 4.55 for arbitrary ar r > 1 and for all x € [c 4- TJ, d — rj\ (0 < T) < (d — c)/2) the following inequality is valid V U l g 11 vana m,n€Z. m,n€Z m,n€Z++

-.1/2

r)

1/r

1/2 1/r 1/2 t/irr))(f; (/;x;A)
J

It remains to use the inequality (4.17). Lemma 4.8 is proved. Corollary 4.9. Under the hypotheses yotheses ofofLemma Lemn 4.8^ for /or operator V* (/) == K(rf/x;/;^) V* (dfi; / ; x)== v,(/)

1 sup ——-sup , 1 + ln((n + l)/(n +++1)) 1)) J fc=o,i,...,n; n€Z ln((rc++ l)/(n l)/(n- --*kA; fc=o,i,..,n; n€Z 11++ ln((rc 1)) fc=0,l,-,n; n€Z + ++ x \V -ik(rfM;/;a:)| ((z1 6€ [c,d\) [c,d]) |Vn,,n-fc(d/j;/;a;)| n-k(dfi;f]x)\

193 193

Linear Methods of Summability of Fourier Series the following is true iiuwiuy estimate cswn

4/s 1 Mx) 1/2+ /s 1/j (S(•>!>• I/(X)|2 l/(x)r dd«*)}'") {jTi/wr ">^^

K

, ^l1/4 aa [V,(a>;/;x)] [V,(d ;/;x)] d/z(x)} a>(x)} *)]•« M \y.{d f;x)]'dn(*)} K

Jc+rj

,l2,

x-ll/2

Here £/ie constant C > 0 is independent of f. In fact, this assertion follows immediately from Lemma 4.8 and the estimate (4.16). By the aid of Corollary 4.9 one can get the estimate of the norm of the operator U.:f->U.(f); U. : f -> U.{f); || tU^ ( /;;mx ; A ))A)l l| C/*(/) (7*(/;x;A)= r\ rA— V . ( / ) S=t fU.(fix;A)= . ( / ; * ; A ) = sup — U.(f) y—*%''* ,, B„(A) n€Z+ n€Z+ #n(AJ ' where B n (A) (n € Z+) are defined by (2.48). Theorem 4.10. Let P = {pn} (^ £ Z+) 6e an orthonormal polynomial system with respect to the measure /i, satisfying the conditions (2.22) and (2.35). Then the tie following jouowni statements are valid: i)

Ydriz)}r UU{pv[^(/;x;A)]^M^)j r<*-f *-V

f

Vm{fix;A)]rd,

1^ 1/r

(4.18) l 1 / 1r l (4.18) d l/(*)l[j'lfixWdnix)] '*}, <*M c f2{x)Mx) 1/2+ p<M(*)J 1/rj> E cfinite,,here the constant an£ C C> >0 0is is independent ie isisfinite, whenever the right-hand side whenever the right-hand side is finite, here the constant C > 0 is independent off; off; 2) if f e Ll(E) fl L^[c,d\ (1 < r < 2) and A-matrix (2.1) is T'-regular ma trix, for which (2.48) holds, then the limiting relation (2.3) is true almost everywhere in (c, d). V

[U.{f\x;A)]rdp(x)} -.1/2J 1/2 > EfJ2(X)Mx)} f{x)drix r>{[J ++ Hx)\

Jc+T} c+77

r

C +T

- "{[J

i/(*)i w*)] }.

] [J

Proof. The first part of our assertion follows from Theorem 4.9, using the argues of the proof of Theorem 2.10. For a given e > 0 there exists aa polynomial polynomi; II(x) G irm such that For a given e > 0 there exists a polynomial II(x) G irm such that r 1 r (4.19) -n(x)r^( {j'\f(x)-U{x)\ d^x)}l/r<e.t(x)j < 1/(*)-1 1 { £ |/(x) - U(x)\r o>(x)} V r < e. (4.19) By (II. 2.4) m By (II. 2.4) m) pj(x), n(x) = ] )4m)(•»)»

{/

r«M*n

n(x) = ^4 m

Pi (x),

) m)

n(*) = 5>< a (*), j=0

194

T Fourier FourierSeries Series in in LL^(l ^{1 << rr << 00) oo) and andCC

consequently

m m m

.J">;

t/ Unn(II;x;A) (U; x; A) = ££a l f>\™ A
(n >> m). (n(n>m). m).

0

If 71,72' are positive integers with n > m, n > mf, then by T-regularity of the matrix A the relation 17n(II;x; t/„(n; x; A) - I7 £/n//(II,x, (n, x, A) = ooxx(l) (l)

(n -^ -> oo)

(x 6e [c, [c,d\) rf])

holds. Hence

|t/„(/;x;A)-[A < |C/„(/;x; \Un(f;x; A)-C/ A) -t/„(II;x; |\U t /n„(f;x;\)-U ( / ; x ; An),(f;x;\)\ -[M /;*;A)| < n ,(/;x;A)| n (II;x; A)| + |tf„(II;x; t/n,(U;x; -(II;x; A)| + |C/„<(n |C/„-(II;x; £/„,(/; |C7„(n; x; A) A) -- U 0„,(II;x; |t/„-(II;x; U x;A)| A)| ;x; A) - l/„-(/;x; n.(f;x; <2tf„(/-II;x;A) < 2 t / » ( / - I I ; x ; A ) ++ooxx((l). l). Put

f

n - for m U{t)btt€E f /(«) for t W n(«) *6£;

//Wfl(t)= gf ~ t)=l 9(t) = 11

and

ft(t) h(t) Mf) ==

00

€ £

otherwise otherwise

'f/w-nwforteM /(*) - nU(t) for * € [c, d\ (/(*)w for* e M 1 0 otherwise.

{ 1

0

otherwise.

Then by (2.55) one obtains

U (f-U;x;A) II; x; A) == /f g(t)K d^t) + UUnn(h; £/„(/ - n; x; A) d/i(t) x; A). )Knn(t; nn(f n(t;x;A)diAQt) n{h;x;A). JE 7E

Since

9(t) (^), 9W €€ ^L;Ll(E), as above (see proof of Theorem IV. 3.2), using the T-regularity of the matrix A) we derive

h

/ 9(t)K g(t)K (t; x; A) dfi(t) d//(*) -> -► 0 (n l(t)Knn(t; (t;x;A)dv(t)^0 x; (n -> oo)

rfj).rfj). ((xx €€ [c, [c,rf]).

7E JE On the other hand, by (4.18) and (4.19)

fd-r)

{^

{J

IC+T)

r

(4.20)

-,1/r1/r d»(x)}d»(x)}
| l / r/r

:0

r l/r [U.(h;x;A] dn(x)y < \u*{h^x',K] d»{x)} KC^j '[U.ihwAYdpWy^KC^J
r

r 1/r

1/r

v

where Cv > 0 (0 < r? < (d-c)/2) independent of / . It follows from the Chebyshev inequality (Theorem I. 2.12), that 1

1/r ||{x, | x , xx €eG (c+7?,d-r/) (c+r),d-V) : C/,(/i;x; tf*(/i;x; A) > > e£,1/r } | < - f [U.(h;x; A)] rrd//(x) < ( V "r - 1 .. (c+iy,d-iy) U*(h;x;A) rf/x(x) [£/»(/i;x;A)] dM(a;)
Linear Linear Methods of Summability of Fourier Series

195 195

Consequently, the following relation n > 00) tf (h)x-,A) -)->■ 00 ((n->• (n—->• oo) Unn(/i;x;A)->0 (h)X',A) 00)

holds almost everywhere in (c + 77, d — 77) for an arbitrary 77, 0 < 77 < (d — c)/2. The second part of Theorem 4.10 is a consequence of (4.20) and the last relation. Theorem 4.10 is completely proved.

Chapter 5

Fourier polynomial series in LL iX. Analogs of Fatou Theorems 5.1

On an almost everywhere divergence of or thogonal expansions

Let $ = {tpn(dfj,;x)} (n € Z+; a; € [a,b]) be an orthonormal system with respect to the measure /i on the interval aterval [a ,b], and f(t)(fn(dfi; x) d/x(x) c«(/) Cn(f) = == /I f(t)

(n (fl € € Z+) Z+)

are the Fourier coefficients of the L^-integrable function / , provided, of course, that these integrals exist. We demonstrate that the requirement cn(f) = o(l) (n -> oo), imposed for every / € L^[a, b] which is, indeed, necessary for the construction of a convergence theory of the Fourier series of all the L*-integrable functions, is fulfilled only then when the orthogonal functions {<£n(d/x;x)} are almost everywhere uniformly bounded. To prove this, we start from the following assertion Lemma 1.1. If the integral / f(x)g(x) f(x)g(x)dn(x) dfi(x) Ja Ja Ja

exists for every L^-integrable function f(x), then 00, yioo,p < oo, oo,

< \\9\\oo,p < < ll9lloo,p

197

Fourier Polynomial Series in L*

198

where \\g\\ooltjL is defined by (I. 2.2). Proof. For in the opposite case \g(x)\ > n would hold in a set En of positive //-measure: n(En). We put ( sgnp(x) sgngjx)

( SS ^ ^ ■{■ HiEn) M^n) /»(*) fn(x)=l= fi{En) ((

00

for iorxeE x 6 £n far.€4. otherwise. otherwise.

The L^-norm of fn is 1 11/HI.M = / |sgnp(x)|d/x(x) = 1. ll/lll.M == 77F-T KEn) M(Al) E M(Ai) 7JE n nn

)L

Furthermore, if the functional ipn(f) is defined as V>n(/) = = / <M/) Ja

/n(x)p(x)rf/x(x), / n ( x ) p ( xx))rdf//x ( x ) ,

then one obtains 1 |lM/„)| hM/n)| == ■-7FT /

M^n) i)

•/£„

g(x)sgng(x)dfi(x) n, ^)s^^W ^ ( x ) s ^ ^ W dM*) >> *»

consequently lim

CO. | ^ n ( / n ) | === 00.

,!*,(/.)! n—»oo n—»oo

It follows from (Theorem I. 3.6), that there exists of an / e L* with d/j,(x) == 0oo, f(x)g(x) 0, // « f{x)g{x) = Ja ./a Ja

contrary to hypothesis. This proves the Lemma 1.1. Theorem 1.2. Fourier coefficients c n (/) (n E Z+) of every L^-integrable function satisfy the condition 0(1) —¥ oo) (1.1) Cn(/) = = ■ o ( l ) (n ( n -> o(l) (n->oo) c„(/) i/ and on/y i/ #&e following relation ||^niloo,„<
(n e z+) (neZ+) (neZ+)

is fulfilled, where C denotes the constant independent 0/ n E Z+. Proof. In order to prove the necessity, we put jcess ^n(f) == cnn(f) ^n(/) =C == V-n(/) (/) =

!>

// f(x)(p f(x)(pnn(dfi; (dfji] x) x) d/x(x) d/x(x) (n (n €EE Z+). Z+). (dfji]

/./a Ja ./a

(1.2)

199

Almost Everywhere Diverging Orthogonal Expansions

If (1.2) is not satisfied, there exists a set sequence {En} from [a, b] and a sequence {Mn} of positive numbers with Mn -> oo such that { Mn for x € En, where each En has a positive /x-measure /x(2£ n)- If fji^&n). ii we we put pux

{ff

/»(*) =

sgn Unnip^d^x) sgrup (dix\x) sgnyy (ri/z;:c) for x e En KEn) T^-7 tor M^n) T^-7 tor xx ee E Enn 0
then the L*-norm of / is ||/||i, M = 1 and we have for \ipn(fn)\ the lower estimate l/lll.M 1

<£„n (*& <£„„n(d/X; (d/x;x )x)drf/x(x) > AM„, l^n(/n)| = n{fn)\ == -Tirr - T i T -T /// > f ,n, n(d/x; llM/»)l " V>\E H(En)n) J5En n

n

n

and consequently lim IV>n(/n)| |V>n(/n)| \tpn(fn)\ = 00. 00-

n—too n—foo n—foo

Therefore we obtain from (Theorem I. 3.6) a nondenumerable infinite number of / € L^[a, 6], forming a set of the second category, with the Fourier coefficients c n (/) for which lim ssup|c u p | c n (/)| ( / ) | = oo 00 n—>oo n->oo

holds contrary to (1.1). This proves the first part. The condition (1.2) is sufficient. For, let / denote an arbitrary function be longing to V^a^b]. Let us put

fl

/'(*) /fc(a;)= /*(*) = 1

if |/(x)| ^ 1/(^)1<
0 otherwise. otherwise.

Let Fk signify the set of those points x at which |/(x)| > k is valid. Then one obtains c n ( / ) == / fk{x)
JF JFk

Ja

If we choose an arbitrary e > 0, then for sufficiently great k \f(x)\dn(x)< /l J^I/WI-W'Xjfe /ftl/(*)l«W*><5Jj? 2M JF„

and on account of the L^-integrability the limiting relation (III. 3.9) implies that for all n > rife 6 \I Z* fb fk(x)ip (dn;x)dn(x)\ II ee n // fk(x)
Fourier Polynomial Series in L*

200 200

Using the estimate (1.2), we thus havei established that for n > nk esiat e \cn(f)\
k J Fk

is true, in accordance with (1.1). Our statement is proved. We may now prove the following assertion T h e o r e m 1.3. Let {ipn{dix;x)} (n £ Z+; x € [a, b]) be an orthonormal with = oo 00 lim ssup||^„||i ffM M =

system (1.3)

n—>oo n-»oo

and iplidfi,;x)dn(x) >>00 ^>£(d/i; x) dn(x) lim inf i // JE JE -*°° for any set E C (a, b) of positive p-measure. Then there exists L^-integrable tion f(x) whose expansion nn -*°° n—»oo

(1.4) func

oo OO

:n{dn;x) (f)^n(dfi;x) /(x)~£ /f(x)~J2c ( * ) - n(f)

(1.5) (1.5)

71=0 n=0

diverges almost everywhere. Proof. If the expansion (1.5) converges in a set F of positive //-measure, then this convergence is uniform in a set E C F with fi(E) > 0 (Theorem I. 2.4) and consequently c£(/)
x X x X x) dfi(x) o(l) ->• —► oo) oo) vl^W ) )M 00) /I (pn(dw
JE JE

follows. According to (1.4) this implies (1.1). However, because of (1.3) the exis tence of L*-integrable functions with Cn(f) ^ o(l) follows from Theorem 1.2 which proves our statement. It is possible to deduce from the Theorem 1.3 a simple proof for the existence of L-integrable function with almost everywhere divergent orthogonal polynomial expansions. Corollary 1.4. There exists L1-integrable function with almost everywhere diver gent Legendre polynomial expansion. Proof. Let {p^ \x)} (x e [-1,1], n e Z+) be an orthonormal Legendre polyno mials (Ch. II, §3). For this polynomials the following estimate is true (see the end of §3 of Chapter II) /2n+l Hallooo = Halloo = — ^] f21^ ,, i.e. the condition (1.3) is fulfilled.

y-r->

at the Lebesgue Points Points

Linear Methods of Summability

201 201

Furthermore, by asymptotic (see (II. (II. 3.7 3.75) for a = 0 = 0) mla (see isynipioiic formula loriiiu

+.l\ + *<-•> --&i/=s»-[(" \)*-1] **—> [(-*v 1} +o(-1]""+o<3,2"">3,,)

IT // 2 3 2 -3/2) 2n n+ + ll cos + 0{rr ') ).V n7rsin0 = \ — r COS V rnrsmv f o r O < £ < 0 < 7 r — £ also holds good. If we map the interval [—1,1] by 0 = arccosx on the interval [0,7r], then the set E with \E\ > 0 turns to a set £0 with \E$\ > 0. Choosing therefore 0 < e < 1/2125*1, there exists a part E% of E$ which lies in [e, ir — e] and has a positive sitive measure. If E* is the inverse image of Ejj, then

cos0) = ,>»>(cos«)

[\pW(x)]Ux> r)] 2 d- > f/ P< 0 ) (*:

JE JE

JE* JE* JE* JE'

0) 0) dx ))222dx [pl (x)l [pi\p£\x)} (^)]

H>-

7T ^ 2 2n 2 n++ ll // 2 3/2 > > JH±lf f cos c o s22 2 I[ f( nn + 0---T jl l^d»-0 ' -( n00{rT Jff (-n3-~3/ 3/22/ 2))).. > n7r /E: c n + ii V 717T X " 4j d0 niz JE. Since the integral on the right-hand side converges to 1/2|25* | > 0, we obtain

> *L±I / cos f( + £V - ?"U - °(^ )'

•■h

2

lim 0 i]inf lim inf jf >i 0)\p<£\x)] » ] dxdx>0, > 0, (z JJJEE

n—KX> n-+oo n-+oo

2

i.e. (1.4) is also satisfied. Corollary 1.4 follows directly from the Theorem 1.3.

5.2

On linear methods of summability at the Lebes gue points of the orthogonal polynomial series

As above (see (IV. 2.1)), we consider the triangular matrix of real numbers x(n) x(n)n) n n) ) . == aw. A={Al A = {Ai ,A; ,fc = 0 , l , 22,, ......,,nn,,)nn ++1l l; ;; A £ °° = 1, A ^ ^ = 00,, n A<> n = 00,1,2,...}. ,1,2,...}.

(2.1) (2.1)

For every function / G Zy[a,6] one form A-means n

n)

[/„(/) U [ nL)^ (*(/)?*(*), ') = £/„(/;*; c/ fc)(/)p £/•„(/;*; ^ fc*(x), ) , n(f;x; A) = $^> A fc=o fc=0 k=0 <*(/) ')== f/

f(t)p d^t) /(*)Pfc(«) **(*) f"f(t)Pk( lp k(t) fc(t)dM(t)

(fc = 0,1,2,...) 0,l,2,...) ((k

; n= 0 ,l,2,...) (( -- 11 << Xx <<1 1; 0,1,2,...)

Ja Ja

(2.2) (2.2) and consider the following problem: to investigate the almost everywhere A-summability of the Fourier series (III. 3.1), that is, (f]X]A) lim Un{f;x;A) (f;x;A)

n->oo n-*ooO

holds almost everywhere.

= f(x) = f(x) /(x)

(2.3) (2.3)

202

Fourier Polynomial

Series in L*

We need in some auxiliary assertions. First of all, we obtain the represen tations of the classical Fejer and de la Vallee-Poussin kernels, which we will use below for the approximation of A-kernels. The next statement plays an important role in the treatment of the summability of Fourier series in general orthogonal polynomials. Lemma 2.1. For the de la Vallee-Poussin kernel in a general O.N.P.S. {pn} (n e Z+) (cf. (IV 2.45)) n

nn 1 x) = —— = ^2 Dt(dfi;t,x) (dfi;t,x) V ^ d / x j ^ x ) = —— 5 3 £>/(d/x;£, D t K + l

E

V (dfjL;t,x) Vnik ntk(d^t,x)

(2.4) (2-4)

l=n-k /=n-fc l—n-k l=n-k

2 , . . . ; tt,x,E t,x,G [0,6]) (fc = 000,1,2,..., == 0,1,2,...; 0,1,2,...; 0,1,2, (* (k = ,, ll ,, 22 ,, .. ...., .. ,, nnn;;; nn = , x , e [0,6]) the following representation is valid for all t, x € [—1,1] and n e Z+; (fc * - xx))222VVVnn,jfc(d/x;*;a t;x) (k + l){t-x) ,k(diJL;t;x) (k + + l ) (l)(t-x) n,k(diJL;t;x) 2 2 = 2a _ p . (rf/x;x)-2a^ = _ p (d/x;t)p _ (d/x;x) 2a pnn+i +i(d/x;t> (d/x; t)Pn+i (d|x; x) -k(< fcA fcfc n +i(d/x;x) nn = 2al_l kfcfcpn-nnk,-k(dfJL;t)p (dwt)p --- 2a£p 2alp n-k(dwx) n+1(dti;t)pn+i(dfjL;x) + a a +i[p„( o i [p„(d/x; o + (d/x; z) + p +2(dw t)p (dfi; x)] (a7x;x)+p + n n n(dii\t)pt)p n+2(d n + n 2 n n Pn+2(rfM;^)Pn(rfM;a:)] 2 n 2 + a n a n +i [p„(d/x; t)pn+2(d/x; x) + pn+2(d/x; t)pn(dfi; x)] - ann__fcf _ia c _ i na_jk[p„_A:-i(rf/i;t)p ppn_fc+i(d/x;£)p n_ (d/x; x ) + (^M; ^Pn-fc+1 (< n-fc[Pn-fc-l(rf/i n_fc +i(d/i;x) n_jfc+i(d/x;x)] !^;^)Pn-fc+i(fl +i(rf/x;x)] - a _ _ia _jk[p„_A:-i(rf/i;t)p _fcfc+i i(d/i;x) ++ Pn-fc+i _fc+i(d/x;£)p _jfc+i(d/x;x)] n fc

22

n n

n

+

n

n

*)Pm(d/x;x) m " °m-l)Pm(d/i; m -i)Pm(rfM;t)p - l ) dK m-l)Pm( t)Pm(dn 5E33 ■(( ™~ ™~ t)p mm(dfl\ x)

+ + ■ + m=n—fc+1

+ +

n

fl aa

+1 m=n—fc+1 m=n—fc+1 m=n—fc+1 n n a 5 31 a fl m(u +i Tn{u +i m mm m=n—fc m=in—k

E5 1>™(

aa

(2.5)

(2.5)

*)p +1 (d/x; x) x) +1 (d/x; (d/x; t)p t)p (d/x; x)], - umm)\p)[Pm(^;*)Prr t)Pm+i(rfM; ^) ++ ppPm+l(dM t)pmm^(d/xjx)], (di x)], m(d/x; ;t)p m+1 mm+1(d/x; m m(d/x;

w/iere a n , u n (n e Z+) are £/ie recurrence coefficients (see (II. 2.17)). Proof. It follows from Christoffel-Darboux's formula that n n n

2 (A: l)(t-x) , fc{dfji]t;x)= (d/x;t;x) =■ £ (k ++ l)(t - x)2 K Vlntk 5 3[[o>itpi+i(dfji,t)p/(d/x,x) ^2 [a/tpi+i(d/x,t)p/(d/x,x) «[o>itpi+i(dfji,*)p/(d/x,x)

/=n-fc

- tpi(dfi,t)pi+i(dfi,x)

, -p/+i(rf/x,t)xp/(d/x,x) + +pj(a +pj(a7x,*)xp/+i(d/x,x)]. /x,*)xp/+i(d/x,x)]. -pi+\(d^t)xpi(d^x) pi(dii,t)xpi+i(d^x)}.

Hence, in consequence of this relation, on account of the recurrence relation (II. 2.17), one has 2 2V (dn;t;x) (k + i)(t-a?) l)(t-x) ntk (fc + (fc v n t *(^;*;»)

n n

= E{< i+iPi+2(dfJ,,t)pi(dp,,x) + a/u/+ipi+i(o7i,t)pz(d/x,x) = 5^ 3 {a/iaa/+iP/+2(c?M,t)pi(dp,,x) a/tx/+ipi+i(a7*,t)pi(d^,x)

/=n-fc /=n-fc 2 +i(dn,t) + afpi(dn, t)p x) -2-paafo+i(d/x, pj+i(d/x, i(d/i, a/ix/p/(d/x, t)pt(dn, (dn,x) ^)p t)pi/+/++xi(rf/i, {d^x)x) --- a/ix/p/(rf/x, a/*x/p/(d/x, aiuipi t)p t)pi+i(dfi, x)x) + a,?pi(dn, a 2 p/(d/x,t)pi(c x/p/(d/x,t)p/ (d/x,x) lafo+i(d/i, i [L,t)pi+i(dii,x) t(dn,x)-a /+1 + 1(d/x, 2

lH,t)pi+i+l+i(d/j,,x)-a + a/_ia/p/_i(d/x + a/_ia/p/_i(d/x,t)p ai-iaipi-i(dix,t)pi^i{d^x) a/_ia/p/_i(d/x,t)p i(rf/x,x) i(d/x,x) - ptfpi+i(dp,t)pi+i(diJL al.2p/+i(d/x,*)p/+i(d/i,x) a2pi+i{d^t)pi3x)+i(d^x) ,x) -

aiutpi+i(dii,tt)pl(dp,,x)-ai-ia aiuipi+i(dn,t)pt(dp,,x) - lpuai-iaipi^.i(dfi,t)pi-i(dfi,x) HPi+i(d^t)pi-i(d^x) x) 2 41011rf/x.t)pi+ ^ +22(dfi, (dn,x)x) + a/tx/ aluu.i-iVi(du.t)viA.i(du.x) + a?a;-J_ aiai+ipi(dfi, t)pi (dfi,x) a/tx/+ipj(d/x, i^(d/x, t)pi^i(dfjL, *)/>/+i(d/i, xx) x)) -1++ afpi(dfjL, aafp^dfi, pi(dp,,t)pi(dfi, t)p t)p xx)}. )}. l+n ++i^(rf/x, t(dfi, t(dfi,x)}.

Linear Methods of Summability

at the Lebesgue Points Points

203 203

We regroup the terms of the last relation 22 {k V Vnik (dKt;x) (k + 1H« {dr,t;x) n,k(dti;t;x) (* + l) l)(t-x) -l)(t-x) •

n

= 2

5Z X ] [a?P*(dMi*)-0|Wi(^i*)Pi+i(dM»»)]

=

n n

a + 5n1 a>i(ui+i

- 5] <

-ui)\pi+i{dfi1t)pi(dnJx)-\-pi(dfi1t)pi+i(dfi,x)]

l=n-k n n ai a a + ^2 i i+iPi+2{dfrt)pi{dii,x) n /=n-fc + ^nn aiai+ipi+2{d^t)pi(d^x) /=n-fca + 5n^ a>i<*>i+m{dii,t)pi+2(dii,x) J=n-fc l=n-k l=n-k

5Z

- 5Z <

n n a ^n a>l-iG,iPi+i(dfj,,t)pt-i(diJ,,x) l=n-k l=n-k ^nn a/_iafp/+i(d/z,*)p/_i(d/i,x) l=n-k - n ^2 o/-ia/pi-i(d/i > t)pi + i(d/x,x). l=n-k /=n-ik

5Z '-

E«

By the aid of Abel's summation by parts (I. 2.30), one can obtain n n

rf/i, x) ^2 (d^t)pi(d^x)^2 aiai aiai++ipi ipi+2 +2(d^t)pi(d^x)-

5Z ^

l=n—k

n n n

^ ^

l=n—k

a/_ia/pz+i(d//,t)pj_i(d/i,x) ai-iaipi+i{d^t)pi-i(d^x)

— anan+ipn+2(dii,t)pn(dii x)-— an-k-ia >n(rf/x,x) 1 n-kPn-k+i(dii,t)pn-k-i(d/j,yx) n — anan+ipn+2(dii,t)pn(dii x) - a n _fc_ia an-k-ia 1 n-kPn-k+i(dii,t)pn-k-i(d/j,yx) and

n n

n

2 2Pi+i(dfi,t)pi i{dn,x) ] nT a2pi(d^t)pi(dfi,x)^2 + n a>aPi+ Z=n-fc /=n-ik l=n-k l=n-k ] T afpi(d^t)pi(dfi,x)^2 a2Pi+i(dfi,t)pi+i{dn,x) o = al_kpn-k(d^t)p - ' GnPn4 alpn+1(dfi,t)pn+i(d^x). [dii,t)p . Z=n-fc /=n-Jk n-kn(dfi,x) 2 = al_ p (dii, t)p (dn,x) a p k n k n k n n+i(dn, t)pn+i(dfr x). This shows the validity of Lemma 2.1. This shows the validity of Lemma 2.1. 2 Dividing both sides of (2.5) on (t - x)2 and tending to limit as t -> x, we both sides of (2.5) on (t - x) and tending to limit as t -> x, we obtainDividing by the PHopital rule obtain by the PHopital rule n I/ n xx 2a kP k ; x 5Y,1 Hrf( ^2PI(X) x) == 2^2a^n-kPn-k(d^x)p 2alp^ ^- ^ )Pn-k(dfi; x)pn+i(d/*; x) n-k(dfi;x) x) -- 2a^ +1{dfi)x)p n+i(d/*;x) + 1 (d/x;

E

=0 l=n-k ii=0 /=n-fc

+ aannaan+n i+ i [Pn(rf/i; x)pn+2(dM; x) + Pn+2(rfM; x)Pn(d»\ x)] x)] + +

n n fl Z m " " aa m )Pm ((rfrfM5 M5 ^)Pm ^)Pm (( ^ ^ 55 xx )) 55 Z (( fl m m -- 11 )Pm m=n—fc+1

2

2 2

E («

- an_fc_iann_jfe[Pn-fc-i{d\x\ _fc[Pn-fc-i(rfM; x)p x)Pn-k+i{dp, x) n_jb+i(d//; x) rf + Pn-k+l(dK x)] + Pn-fc+l( M; x)p nn-fc-i(d/Z; -k-i(dlX\ X)] +

n n n a m ( U m + l - ^m)[Pm(^M; x)p 51 x ) pmm++i(rf/Z; i(fi^; x) m=n—k m=n—k m=n—fcrf

-E-

m(dti;x)}}. + Pm+i( /*; x)p z)Pm (<*/*; x)]}.

Fourier Polynomial Series in iA Fourier

204 204

r, using the initialI conditions In particular, conditions a_i a_; = 0, p_i = 0 , one can get, putting k = n n 1 l n+l[Pn(^M;^)Pn+2 ^ ( n - /+l)p?( / + l)pf(x) ^{annaann+ i\p^(du; l)p?(x) = -{a +i\pZ(dii; x)p>n+2(d/*;x) n + 2 (d/z; x) 1=0 J=o r' dfi;x)p + Pn+2(dM X)p X) -- o„2 X) 2a£pJ( t (d/x;x)-2a^p; n(n(dfi; nmP n+1+(d/z; l ( ^ ^X)p Kn+1 + t{dfJL' l+(i <( d ^1 ; xxx)))

J2(n-l n

/ 2

«E( + 2f>

+

22

■I-

2

a H ((°m m ~ x)) - am-i)Pm(<^; a m - l ) iPj ^m' ( ^ ; x)p x)Pm{du\ m(du; x a:)Pm(^;a:) m=2 m=2

nm=2 n

m +■E« $^23 aaa™( (uw u™+i ~ lm)\p'^ umm)\pn(d/JL\x)pmm+i(dfjL;x) mn m+ + l1 --^m)[Pm(d/x; !/x;x)p i(d/z;x)

+m=2 E' m=2

m+

dx)pn, m=2 Pm+l( rfda: a; f(d/x; x)v lfi;x)]}. m (d/*;x)]}. + Wx)Pm(dfi;x)]}. + + i Pm+i( C + i ( « M; )Pm(rfM;a:)]}.

Corollary 2.2. For the Fejer kernel nn 1 F (dfjL;t,x) {dii\t,x) = = — — ^2,D n n m{d^t,x) m{dii\t,x)*,*) F n (d//;*,x) = — — ^2,DD m{d^t,x) n + l m=0 m=0

(n [[aa,,66 ]] )) (71 (n = = 00 ,, 1ll ,,,222 ,, .. .. .. ;; t*t, ,x, ix6€€[<*,&!)

of general O.N.P.S. {pn} (n e Z+) the following representation

is valid

dfi; i, £, x) x ) == £fn(*,x), t, (n + l)(t l ) ( t- - :x):)22F Fnn(dfi; (d/x; n (t, x), 2

(2.6)

w/iere £^n(*, = anann+i\pn(dfj,; fn(*, n(t, x) =

+ 2 (d/i; t)pn(du) (dw t)pn+2 *)p (d/z; x) + pnn+2(du; +2(^M; {dp, x)] x)) n+2(rf/x;

-- 2alp 2alp (du\ t)p *)p +i (d/z; x) x) 2a pnn+i(dfi; +i(du\ t)pn+ x) ni(dfi; n+i n+i(du; n% 2

pmm(dfl\ + 22H ^D a i -"- aom-l)Pm(rf/i; ^ _ 1 ) p m ( ^ ; t)p t)p (dM;«) 2( (°m (°m flm-l)Pm(^; t)p (df. x ) m m(dfi]x) + 22E(«

(2.7)

m=0 =0 m=0 n n a ;mt)p t)pmm(du; {dfji; x)] + ^ |aa m (um+i w m + i -- uUm)\Pm{dfi;t)p +i(du;x) m( i(dfi;x) ++ ppmm+\{dn\ i m( +i(dp,]t)p d | i ; t)p x)] flm(Wm+l U(du;t)p m)\pm m(du;x)] m)\pm(dfi;t)p m. i(rf/x; x) + m+m+ +p m=0 m=0 m=0

+^E E '-

is va/id for all t, x e [a, b] and n e Z+.

Corollary 2.2 follows from Lemma 2.1 because Corollary 2.2 follows from Lemma 2.1 because

F Fnn(dfi]t,x) (d/x;*,x) == Vn K ^tn(d[A;t,x) d / / ; *,x) and and a_i p_i(x) a _ i == 00,, p _ i ( x )= 0

F n (d/x;*,x) = VntTl(dfi; t,x) and a_i = 0, p_i(x) = 0

(x ( x G [[-1,1]). -l,l]). (x G [-1,1]).

In particular, putting t = x in (2.7), we obtain an important formula In particular, putting t = x in (2.7), we obtain an important formula

n n f Yl K a m -- aa^_!)p^(rfMi m-l)Pm(lr{dwx) M; 2?) ++ Om(«m+l ~- Uumm)p)p x)prn+mi{du\ x)] am (d/z;x)p +i(d/*;x)] am(«m+i m(dfi; m(dfjL;: X ) K°m " m - l ) P m ( ^ ; * ) + a m ( w m + i - Um)pm(dfJL; x)pm+i(dfJL\ x)] m=0 =0 m=0 a

d

dfi;x) x« - u - pl Mx) i(rf/x;x) + x)]. = ln \pl( |Pn(^M; X) - — —- (0* ((x Wnu+i)Pn(<^; ++i)j l )nP(dii;x)p n ( ^ ; x)p +i(d/Z; x ))+Pn+i(dM;*)]P*+i(d/i; *)]• n+i(dii;x) +1(dfi; nnn)p nn+ n

(2.8) (2.8)

Linear

Methods

of Summability

at the Lebesgue

205 205

Points Points

In fact, one can infer from (2.7) n

nn

E<

-E<

rf x umm)p)p x)pmm+ i(d^ xx)) Yl >( m ° m- «" - ma -ml -)l P ) Pmm(( < M flm(«m+l +i( f o; * )) ++ J ^ | a m(dfi; m= =0 0 m==00 m m

a rf = /*;*) a+n++iPn{dii; llPj > nn ( d( /^i ; x )x)p p n +n+ (d/z; / i ; x x) ) = a nPn+i( n P n + l ( r f /M *; *) " " a> a n ann 2 (2d

and one remains to use the relation 0>n+lPn+2(dH] x - U n + l ) P n + l ( d / x ; x)) ~ 0>nPn(d^ a n + l P n + 2 ( ^ ; xX)) = ((X

x),

which follows from (II. 2.17). Corollary 2.3. For the de la Vallee-Poussin kernel Vnik(dfi\ t, x) for all t,x e [a, b] and neZ+ the following representation 2 (k ++ l)(t x) == vvnn,,fc(*, (A; 1)(* -- x) x)2V V^ *»*) *) ++ vvnn,*(x, ,fc(x,t) *) n,k{dii; fc(*, x) ffc(dM;t,

(2.9)

holds, where ,k = = aann-ib{(an-fc -ib{(an-fc -- an_jk+i)p vn,fc an-k+i)Pn-k(du,t)p n _fc(d/x,t)p n _ib(d/z,x) n-k(du,x) + an_jb+ipn_fc(rf|X,t)[pn_fc(d/X,x)

-pn_ik+i(d/x,x)]

+ fl„ _ ffc+ U_LIT) + aa„_ ~Pn-fc-i(rfM,a:)]} + c +iPn-ik+i(^*)[Pn-ib(rfM^) i P n - i a.. k +il ^/f//. ( ^ * ) [^^\n P n - i b LTrfix. ( r f M ^rr^ ) — - Pon - f tc_- il (Trf/i. d M i ^xW\ )]}

+ a n { ( a n + i - an )pn (d/x, t)p t)pnn++22(, a n P n - f l(<*/*, ( ^ , 0[Pn(rfM» t)\p X) )] + OnPn+ilfllJL, ppnn OnPn+1 ) X)~-- -P l (l d( /^X,, z«X)\ )z)] n+ n(dfJL, n(dfJL,X) n++l(d/X,

n

(°m E ((a^

+ +

]SC

(°m -

a

((dp, d / i , t)p x) ) Palt^pmidn^pmid^x) m (dfi, ^m_-i 1) P m t)pmm (rf/i, (d|ii »)

m=n—fc+1 m=n—fc+l n a am(u i +l --W U^lprnidfrtfprn+iidfrx) 5Z m ( ^m+m + p m ( d / i , tPrnidlXitfprn+iid^x)]. )pm+i(rf/X,x)]. + m ) [ p m ( ( f / i , t ) p m + i ( r f / X , x )+ —k m=n—k m=n m = n—fc m=n—fc (2.10) (2.10)

+ E<

In particular, for the Fejer kernel we obtain Drain

(n + l)(t l ) ( t-- xx)) 222FF Fnnn(d/r,*,x) ( t, X) t), (dw x) == |ln(*,a:)+ln(a:^), t , X) a?) + |6i(*. n{dM n ((t, n ( x , *),

(2.11) (2-11)

where WUC1C (d/x,x) = On{(a* a n { ( a nn++ i -" aaTn )p (d/z,t)p>n+2 (d/x,x) ^nn(t,x) (t,x)=. Q>n)Pn(dll,t)pn+ n (d/z,t)p nn+2 + 2(cf/x,x) + OnPn OnPnidp, +n 2+{dli, -- Pn+l(<*M,*)] pn+n+l(dfl, x)] i(ll,\ P P nn++li((^^^)[Pn(^M,a:)^^^))[[PPnn((^^, ,aa::)) -Pn+i(rfM,a:)]} - J * n + 1 (<*/*>*)]} nn n

E

d i,t)p m (d^,x) a ^a ™"" - °m-l)j*n( M)Pm (<*/*»*) ►^ (Z O(m fll-l)Pm(*,<)Pm(*^) +++ ^SE« *m-l)Pm(dp,t)

m—0 =0 m = 0 m—0 nre n

+

JZ

a

" Um)\pm(dfi,t)pm+l(d^x) H (d/z,x) + +

m(um+l

^ Om(WmH +mE U i ~ W )[Pm(« =0 m+

m=0

m

P m + i ( ^ ,p*m+)i(d/J,,t)p P m ( ^ma(d^x)]. :)](2.12) (2.12)

206 206

Fourier Polynomial Series in L*

In fact, the representations (2.9)-(2.12) follow from (2.7) by straightforward cal culation. Lemma 2.4. Let the O.N.P.S. {pn} (n € Z+) is uniformly bounded on [c,d] C [a, b] (see (IV. 2.22)) and there exists a positive function <po(x) on the set E: E=[a,b]\[c,d], (2.13) E=[a,b]\[c,d\,c,d\, such that for some r (1 < r < oo; 1/r 4- 1/r' = 1)

u (/.^♦"P" r \ i/r'

' i E i f e ^ rr f i xd//(x) l ]

holds. Furthermore, let

|/(x)|d/z(x) < OO
and

r x rr r [(f I \f( \f(x)\ )\
l"

(2.14)

< 00

(z) '

(2.15) (2.15)

\l/r

T r 1/r yjifi ( ^ j> ^S(x)d < <
Then at every Lebesgue point x G (c, d) for the partial sums Sn(f\x) series (HI. 3.1) we have the following estimate Sn(f, x) = ox(l) ln(n + 1) ^n(/,

(2.16) of Fourier

(n -> -►oo). oo).

(2.17)

Proof. It is easy to see that t;t,t,x) x) dji(t) d/x(£) dfi(t) 5Sni (f; (/; *) /(*) [/(*) n(dn;t Sn{fi ( / ; x) * ) -" - / f{x) ( *0 ) == = //[* [[f(t) /'(t) ( * -) -f(x)]D /f(x)]Dn(dn; «/a Ja «/a ) (t)-/(x)]Z> (« /(x)]£> (d/j; t, x) dn(t) rf/x(t) = " f(x)]Dnnn(dia t, x) dn{t) t, x) dp(t) n(dfi;t,x)dfi(t)= f/ [/(*) (dn dfi{t) + j f [f(t) -- f(x)}D f(x)]Dnnn(d/x; (dii;

JE

= Ji(x) Ji(x) + JMx) 2 (x) J\{x) +

(2.18) (2 18) '

(xz eG 6G[c,d]; (x [c,d];nGZ+). nnneG GZ+). Z+ [c,d]; Z+).

The ChristofFel-Darboux summation formula gives together with (II. 2.22) and (IV. 2.22): |D (d//;*,x)| + ll \Ds8s(dfi;t,x)\ <ss + {dti\t,x)\<8<

((t,x€[c,d), € Z+) (*,xG {t,x€[c,> 00,, ssG seZ+) Z+)

(2.19)

and „ 1 |Ds5(dn; (d^;t,x)| (t, [c,d], d\,\t\t---xx\x| Z + ), \D t,x)\< xx €G [c, | > >>0,0, s5G €6 Z+), {dii\t,x)\< C-r- r (t, (t,xe[c,d\, Z+), ) l < |\t~x\ -t -xx|| \t-x\

(2.20) (2.20)

207 207

Linear Methods of Summability at the Lebesgue Points

where the constant C > 0 independent of s € Z+ and t,x e [c,d\. It follows from (2.19) and (2.20)

jfl/W

t)dn(t)\ t)-/(x)]2> |Ji(z)| Wi(*)\ " \f(t)-f(x)]D f(x)\D n (
+C

+ l) + 1) i ) /f

| / ( t )\f(t)-f(x)\dn(t) - / ( x ) | d\di*(t) /i(t) l/(«)-/(*)l
J|t-x|l/(n+l),t,i€M] lf-a;l y|t-x|>l/(n+l),t,a;6[c,d] J|t-x|>l/(n+l),t,xe[c,d)

I*l t ~- a ;x\l

Using the argument in the proof of Lemma III. 4.6 closely (see also the proof of Lemma 2.6 below), we get at every Lebesgue point x e (c, d) Ji(x) 11) ) (n -> Ji(x) = o x (l) ln(n + 4-1) -»■ -► — > oo).

(2.21)

Now application of Christoffel-Darboux formula (II. 2.32), the Holder inequality and (IV. 2.22), (2.14), (2.16) gives

I-M*)| [[f(t)-f(x)}an(d») \Mx)\ == \ JE/[/(*)-/(*)M«W I JE «/£; JE

xx

pn+1(dfji;t)pn(dfJL;x) - pn(d^i;t)pn+i(d^x) (dn; £-j—— 1/ — O/

x)

\ d/*(*) dfi(t)\

= O \f(t) f(x)$\p \\Pn+l (d^t)$dfi(t) \pn+1 i{diJ,;t)\)dfi(t) | / ( t ) ---/ (f(x)$\p x ) | ( | pnnn(dfi-t)\ ( di/u;t)| p;t)| + (dM;OI)rfM*) n+ = 0,(1) Oxx(l) (l) jj/ \f(t) (dfi-t)\ + |p (d^t)$dfi(t) Pn+l r

fdp(t) 1/rj| U IIbn(<^;*)l PPP n ^^))))| rrr 'd»(t)\ I/WIV5W<*/*(*)]171/r = oo,(i)[jf ss (i)[jT |/(«)|>5(t)^(t)] ^ )(*) «*"('>J] [7E "^5 1/T/ 1/T/Uo, i). [ / ^ y 5 J ++ (( r _ 1 /-'

[JE L^£

fV5o(t)(*)

JJ JJ

Thus J^•M*) ( l ) ln(n l n ( n++1l ) z ) == oOx(l) 2((a;) x (l)

(n-t (n-^oo). (n -> oo).

(2.22)

The estimate (2.17) follows from (2.18), (2.21), (2.22). Lemma 2.5. Suppose that the condition (IV. 2.22) is satisfied for the O.N.P.S. {Pn} (rc £ Z+). Then for all t,x e [c,rf] and n e Z+ the following estimates are

Fourier Polynomial Series in L*

208 208 valid

=I 1n - f11c +- 1

|K = f n _fc(d|i;t,x)| \V = ntn\Vntn -kk(dir,t (dir,t99x)\ x)\ ="

II

II

_ _

II

^2,Dm{d^t,x)\ D Drn{dp,t,x)\ rn{dp,t,x)\

m—k

C(n (n + k A: ++ 1) 1) C("

f ((

< < jJ

ff c c + +

nn U n

53 1 1 5Z m—k m=fc m—k

C(n + C(n + 1k k+ + l) l) c ^JTZ~1 i l*-*l TT-T

C

c

I I

for / o r\t|*-- x\ > 0 for\t-x\ > for\t-x\ >0 0 for \t-x\>Q(t,xe r f° ' o r \t~^\>0(t,xe I* - ^1 > 0

(t,x e [c,cf])(2.23) (t,x€ (t,x€ [c,d])(2.23) [c,d])(2.23) [c,rf])(2.24) (<, x 6 [c, [c, d])(2.24) d])(2.24)

e ■ /or for \t-x\>0(t,xe x| t x € [c[c, d]),(2(2.25) l[\ <^nr^r^A"-i«-*i^°(*.* (n — k (n-k + l)(t-x) ? (^TTTH^F "" " ° { M),(2-26) ' ' ^ -25) Knje

2

J

w/iere n

n

^E

« » fc, *=s l1 ++ X ! |
1m=fc m—k

and the constants C > 0 in (2.23)-(2.25) are independent of k,n e Z+ and tjX G [c,rf]. Proof. It follows from (2.5) (as well (2.9)-(2.12)), that for Vn,n-k(dn,t,x) the following is valid loiiuwiiig representation represeii n --kA; + + l)(t 1)(* t-x) a 2 Ki,fc(^M;*;a:)(nV -- 2 x) n,h{dii\ t\ x)(n , ;rf/x; x) t)pkkk(dfjL]x) (dfjL]x)-2a = 2alp 2a|pfc(a (dfjL' /x;*)p (dfjL]x) -- 2a\\pn+i2alp 2alp (dw +1(dwt)p (d^]t)p t)pn+i nn+ +i(dwx) (rf/x;: i(d^x) k(dfjL;t)p 1t)p nn+1

+ anaannna+\ + ann++ ii [ pon(dfjL;t)p ( \pn(dfjL;t)p (rf/i;x) +a (dfi]x) 2n+2 n+2(df. + + + +

)pkk+i(dn;x) \pk-\(dn; t)p +i(dn; Pn+2(dfj,] + k-ia k\pk-i(dn;t)p P n + 2 ( ^ \t)Pn(dfJL\x)] ^ )t)p j n(dfjL;x)]] --- i ak-iaka\p k-i(dn; x(dn', x) pk+i (rf/x;\t)p t)pk-i(dwx)] -i(dw x)] p fc+ i(rf/x;t)j n n

+ 2 E< £ (al

(2.27)

2 2

a a 5 (dfJi;x) m(dfji;x) >m 53 3 ( a m-- f l<4-l)Pm< m( -ml ) )Pm(dfi]t)p Pm-l)Pm(d^t)p m(rfM;0 Pmm (^^) m=k+l m=k+l n uw n afl + 5 33 ™(Uu«m+l ™+i ~ m)\pm(dKt)p +i(d/j,;x) +p i(rf/x;t)p + 5 ™+l ~- W m)[Pm(rfM;^)Pm+l(^;^) + Ppm l ( ^ ;itt)pm(dfi)x)] ) p mm( (rf^;x)], r f ^ ; x1 ) ] , m ))\pm(dfi; bm m+i(d/j,; x))-^Pm+i(dfi' mt)p m ++ a u m=fc m=fc

+ +

E

+ 5 3 ™( ™

, xt,x,€[a,b)). , e [ fa, a , 661). ]). (fc m=fc = 0 , l , 2 ,,. — . . ;: *t.x.G \a.b

(fc (II. = 02.22), , l , 2 , . (IV. . Using 2.22) and definition (2.26), one can infer from formula (2.27) the estimate (2.25). Using (II. 2.22), (] On the other hand, the estimates (2.23), (2.24) follow from inequalities (2.19), (2.20) and the definition of the kernel V^>n_fc(d/r,£,x). We have thus proved our assertion. ssume ]that the o.Ar.p.i L e m m a 2.6. Assume O.N.P.S. {pn} (n € Z+) satisfies (IV. 2.22) and n

n

£|A(a£)| £|A( (ngZ € Z+), « ffcc)|(«2)l +++ 53|A(u fc=0 fc=0

fc=0

(2.28) (2.28)

Linear Methods of Summability

209

at the Lebesgue Points Points

where aniun (n € Z+) are recurrence coefficients for pn(x) (cf. (II. 2.17)). If x: eE (c,d) (2.15) isisvavalid, then jc, a; is w the me Lebesgue point of f, for which{2.J5)

fc[m~i

T ]<«

n+l . 11 T A - /(*)] /(z)] V„, _fc(dM; (d« t,*>x)*)(fo(t) oi,(l) ( 1 ) 11 1+-In IInIn n - "n+; t *l i- (n B fc / [/(*) /(*)]V , (dM;*>*)Mt) =
Proof. Let x £ (c, d) be thej Lebesgue Lebesgue point of / :

x lim0iJ ±— xj f^\m-n*)\Mt)=o. * |/(t) \f(t)-f(- )\dfi(t) / ( * ) | d (t) = 0. &

fe£>

X

h->o 22//ii

M

(2.30)

In order to estimate the left-hand side of (2.29) we put A "^n.fc ^n,k ==

_

11

11

£n,fc == —, ,I +,. n1l'>i'' ^n,fc n +i fc n - f cTTT* + l' n +fc+ 1 n —fc+ 1

(2.31) (2.31)

One can assume that £n,k -> 0 (n —► oo; A; = 0 , 1 , . . . , n). If otherwise this relation is not valid, then we can use the estimate (2.17). We obtain by (2.23), (2.24) and (2.25) the following relations In%h f(x)]Vnfcn^(d; ^[dti;t,x)dfi(t)\ (dfi;tix)d (x)5== 1111 [f(t) [/(*)) - - /(x)]V„,„_ f(x)]V x)dffi(t)\ i(t)\ n%h(x) kk(dfi;t
+c

+ c

\f(t) - f(x)\ dn(t) dp(t) \f(t)-f(x)\d>

J0<\t-x\<8 Jo<\t-x\<8 J0<\t-x\<8nnt,k kn,k

d
i/w -

L<„-„<...nfeF " "'\f(t)-f(x)\ ,<|t-*|<e„,fc

+
I* "

.

+ l)(t-x)* nkfc(n-k r|<£n, J\t-x\
dn(t) ^

"O^+^iw+^te (2.32)

Obviously, by the defining relations (2.30) and (2.31) (fc = 00,1,2...,n; n-)-oo). O * ) = °*Q) , l , 2 . . . , n ; nn->oo). 4Ifc(«) fl-U) ^(* -J- oo). c (l) 4!i(^) fl-(i) The integrals rx+tn.k rx+£n,k

rx—6n,k rx—6 n,k

/

and

Jx-e„, Jx-en,kk

/ Jx+S n,k Jx+&n,k

can be estimated exactly in a similar way, so we estimate the integral only

^2(«)■=C ^

1 ■'i/w-zwi^ 0I 1

^ ** ., -it^r*'

rx+en,k

Jx+6n,k

iM(t).

(2.33)

Fourier Polynomial Series in L* Fourier

210 210

We define the positive integer iVi = Ni(n, k) by NV l 11 *-- 66J„, <2N*5 ,(5„,jfe. 2NJ >~ < Nln2^«„, k:f<e n,k£„,* k. nn,k,■,*<£»,fc<2 c < fc.

Then

m rx+2 rx+2mS6n,nk,k

*3<»)

,f

1

x+2"" 61n6,kn,k Jx+2™-

/(*)-/(*)!, |t-*| *> m mm x+2 S„

Ni Ni

(2.34)

5ntk ■+2 rrx+2S„, klk > 11 f^ \dfl(t) T , m^ \f(t)-Hx)\dfi(t) \f(t)))-f(x)\dfi(t) l m 6„,k , +\-2"2 —is m = li 22 5„, ^ dn,fefc yy xx+2 n k

and by (2.30) = = *S<«) ** $$ (( ** )) I = =
Using the definition of iV*i (cf.;. (2.34)), we obtain n+ +fc + 11 t l- " g ^ oxx(l)ln^^ (l)ln ' ** + ^nl^) O ) == oM l ,) In ln^ ^' | n-k + 1

. (A; = 0,1,2...,n; 00 , 1l ,,22 .. .. .. ,,nn;; n -+ -»• (* -* oo).

O*) =

Thus N gf c W===c^0,(1) 1) ( l11+h+illn lnnn nn ^+= |1 1 j n ! ( l *%(*) O ) = ^( )( n !!t+i) n-h+ll >(' /

n; n -— (k .....,,nn; oo). (* ^►00 oo). (* == 0°., 1l1,,.22.---. ; "-+ )-

It still remains to estimate the last term in (2.32). We estimate the following integral only

^ - d b j L 1e-*)^ * *

1 !!/(*)-/(*)l / ( « ) - / (2* ) ! ddn(t), ,. y Jx+e n- H A; l4-1 :. ntk

jaw= because the integral

li

r

X-£n,fc I

„n-k i. +, li

/ - ' " *■ !* /! (/ W «( * )) -- // ((W **) )!I!_ , rfll,rt dn(t) Jc

can be estimated exactly in the same manner. •uuw a pusiuvt As above, we introduce a positive integer iV*2 = iV^x; k\ n) such that N N1 1 xx + 2N2*2 *-e e, ,k
Then m

N2 /.x+2 en,fc r.+a-. 1 n> ""n-fc n - f c + 1l m =1lJx+2^-^€ Jx+2m-l , £n kn,k l "

JSM

1

c

<
"

1IIi

JJ5^ ' ^2

t)-/(a l

(t -

X)2

-d»(t)

m

j1 1

L-/ ZOm.2 Omc-2 + X £ n-k+1 nyk n - A *: + l m = i 71

i|y// (

/»x+2 />x+2mmmeennn,,,fffccc

/

m m Jx+2Jx+2 -ienik -*enik

f{x)\dn{t). I/O |/(*)-/(x)|d/i(*)-

(2.35)

Linear Methods of Summability at the Lebesgue Points

211 211

It is easy to see that from (2.30), (2.31) one can derive

«0*) = <

1

1

E CX)

1 = (fc== 0.1.2 nn^— >•oo) oo) _ffc T 0 ^x(1) )L)==°* °*W M( 1 l) ) (* (fc 00,,11,,22•....■..,,•,»; nn;; n 2^°«(^ •-ffitfr) O * ) = nnn---fc c +++lxi1^ JT E ^ '^2^° -► oo) ' m=l

= C
m=l ' m=l

and consequently

O*)

J

O * ) = 0,(1) , 1 , 2 . . . , n; nn-► ->■ oo). n!fc(^) ox(l) (fc (A: ==00,1,2...,n;

(2.36) (2.36)

The estimate (2.29) follows from (2.32), (2.33), (2.35), (2.36). This shows the validity of Lemma 2.6. Corollary 2.7. Under the assumptions of Lemma 2.6 at every Lebesgue's point x € (c, d) (and, consequently, almost everywhere in (c, d)) the following estimate f(x))Fn(dii; (dn; x) dfi(t) d/i(t) = J/ [/(*)) -" f(x)]F /(x)]F (d/i; t, t, x) d»(t) = 0,(1) o x (l) (n (n -> -►oo) oo) /'[/<«•

(2.37) (2.37)

/io/ds, where Fn(dn\t,x) is Fejer's kernel. Lemma 2.8. Let the O.N.P.S. and the function f satisfy (IV. 2.22), (2.14), (2.16), (2.28). Then at every point x € (c,d) the following relation f{t)-f(x)}F = -" f(x)]F (d -oOO (n /I [fit) [/(') f(x)]Fnnn(dwt,x)d»(t) (dfifi]]t,x)dn(t) t,x)dn(t) = {\)-±— (n e €e Z+) Z+) [) xx{\)^— x(i)Z+i nn+l +l holds, where the set E is defined by (2.13). Proof. First of all,observe, that for a; x €€ (c, it for (c d) and t e E the relation JE JE JE

1

(2.38)

_

=0o-(i) ((rW-o-W « vhpt --x* )) 2» ' -w =

is valid. So, by (2.6), (2.7) (or (2.11), (2.12)) and (IV. 2.22) we obtain

|| jJ {f(t) \f(t) - f(x)]F f(x)}FBn(d/i;*,x)<^i(t)| (d t,x) dfi(t)\ ;t)-/(x)]F fi;ti;t,x)dp(t)\ n(d = = O il ) - L - { // \f(t) - f(x)\(l f(x)\(1 (dn,t)\ \pn+1 (d^t)\ (d»,t)\++|P«+2(<*M)I \Pn+2(dfl,t)\ \pn+2(dn,t)\ /(x)|(l + \\p | PPnn(dfi,t)\ (dM)l + ++ \p |Pn+l(dM.OI xx((l)-L-{ n+1 =O o.(i) n

-E

n+l(J 1 n + l EI JR

+ -l •• IM<*M)I \Pm{d^,t)\ E |A(u \&(u )\[\Pm(d»,t)\ + E l a --i -- aa-l ++E+£\Mn (d^t)\ \Pm(d^t)\ m m)\[\ Pm m)|[|p m(dM)| +E EI"-"! m=0 m=0 m=0 m=(

m=0

+ + |p,„\p + i(d/i,t)|])d/i(t)}. m+1(dii,t)\])M*)}+ |Pm By the aid ia of oi the tne Holder inequality and (2.14), (2.16) one can derive ( \f(t)-f(x)\\ Pm(d^t)\dti Pm(d/*;<)MM >-/(*)llft. /i/(*)/"l/W-/(*)IM
JE JE JE

<{jm)

{/E^r«»}'"'

f(*)|V5(*)d/*}1/r'

FourierPolynomial Polynomial Series in L* Fourier Series in L*

212

The estimate (2.38) follows from the last two relations. Lemma 2.8 is proved. Let us define the A-kernel (see also (IV. 2.5)) nn

== Y,xlk)Pk(dti /?„(d/i,t,z;A) nn(dp,t,x;A) tt)pk(dfJL9x) K (dfjL7t,x;A) ^Yl^^id^Pkid^x) bn)Pfc(dM)Pfc(d/x,aO

e ZZ+). Z+). ( t , « €€ [-1,1]; nn € (t,x +).

(2.39)

fc=0 fc=0

Corollary 2.9. Under the hypotheses of Lemma 2.8 at every point x e (c,d) the following relation is valid n

[fit)>)-f(x))K - f(x)]K f(x)]Knn(dp,t,x;A)dti(t) /( [/(*) " (dfjL;t x;A)dn(t) lX1;A)dn(t) n(dfjL;t JE •^

E

O . ( l ) £ ; I||A = O.(l) ^ 222A< AW ^n)|| A A o«(i)

(n € Z+), Z+), (n Z+),

(2.40) (2.40)

fc=0 Ar=0

2

} n A(AA< n) ) == A £ ° - 2AJ£)1 + AAJ£_ where, as IMU<4; A £ ° == A(AA^ A2 AA£° fc+22 ((n = 0 , 11 , . . . , n ; ne Z+). In fact, by Abel summation by parts

nn

Kn(dn;t,x;A) K„(dfi;t,x;A)

E'

22 2 n ) ) = £ (l(k* + 1)A A£ . \
Hence tence f if(t) i-- / ( af(x)]K„(dfi;t,x;A)dn(t) ) ] * ; ^ * , * ; A) <*/*(*)

JE JE

n

A22A< l)A A[ nn)) //" [/(*) [/(*) - f{x)]F /(x)]Fnn{dn (d/i; i, t, x; A) cfo(i) d^t) = f£<■ > : + 1)A A£ = / .^ fc=0 ik=0

(x € (c, d); d); n € Z+) (x

fc=0

and it still remains to use the estimate (2.38). Corollary 2.9 proves. The main statement of the present paragraph is T h e o r e m 2.10. Let the O.N.P.S. {pn} (n G Z+) satisfy (IV 2.22), (2.14), (2.28) and the entries:s oj of T-regular cond (see also (IV. 2.48)) l -regular A-matrix i\-mai ~ix (2.1) the condition

(fc + + 1) 1+ln 11+ ln A2A gZgZ 2 42 (fc+ +1) A n) (A).t (A)^ T ? ? [ [ ^Tir1' ^fe1' ^ ^^(" '^(" ^(^ - ) "f 5n ++

n + 11 1. 2 ) (fc + l ) ( n - f c + l ) r |A A^|| < C ( n € Z + ) (2.42) + l n n - f c; + 7l l 1 n+1 fc=0 fc=o L J n-k+iy J fc=o L holds. If for the function f the conditions (2.15), (2.16) be fulfilled for some r (1 < r < oo), then at every Lebesgue's point x e (c,d) (and, consequently, almost everywhere) limit relation (2.3) holds. Proof. It follows from (IV. 2.55) and (IV. that )0) ana j i v . 4.4) 4 S„(A) =

= f[f(t)-f

Un(f\ (f;x;A)-f(x) x; A) - f(x) = { [f(t) - •/(aj)]lf„(dAi;t,s;A)
[/(*) [*)-/(x)]Jif " (d W t,x;A)d/i(«) t, - f(x)}K f{x)]Knn(d»; (dr, *, x; A) d/i(«) dfi(t)

/(x)]A"„(d^; t, x; d//(*), + // [/(<) [/(*) f(x)]Knidp\ ', + [/(<)I -"- /(*)]#„(<*« *; A) A) d|i(t), dfi(t), JE

Linear Methods

of Summability

213 213

at the Lebesgue Points Points

where the set E is defined by (2.13). Therefore, in view of representation (IV. 2.50) Lemmas 2.4, 2.6 and vCorollary 2.9 one has ^ u i u i i a i y ^..^ u i i e ii
222 n) n) n) n 1 AAA< : CT„(/;x, Un(f;x,A) (f;x,A)A )-- /f{x) /(*) (*) = ==oOx(l){X> l)|A + (t/+l)|AA(, (i/ (i/ + 1)|AA^>| A[ || + )| x(l){ £ ( * ++ l)|A ) l A<

fc=0 fc=0

n n -- 11

n ) + (n-„)|AA< („-„)|AAW|+ g _fc)(l ++ llnn ^±i)| A82Ai:>|}+O (n)|. A (»)|} + 0 xx( (l)^|A 11)£|A ) ^| A 22 8 A + + £ A<»>|. + (n-v)|AAW (n-*)(l+ln^±l)|A AW|. 5Z (n (n-*)(l + 1 ||+ UDE

fc=i/+i fc=0 fc=i/+l

fc=0

By estimates (IV. 2.52) and (IV. 2.53) one finds n

2

Unn{f; U (f;x,\) x, A) - f{x) = = «,(l) o0,(1) +O Osxx(l) (l) £ £ :IA \A |A22A<")|. X^\. A^>|. tf»(/;x,A)-/(x) x (l) + Jfe=0 fc=0

It still remains to prove thatfromcondition (2.42) follows n 2 < |A 2 Ai n) | = o(l) £:=£|A A(">|=o(l)

n

(n —> oo). (»->OO).

(2.43)

fc=0 fc=0 fc=0

In fact, for every integral r > 1 one can choose n 0 = no(r) such that £™ =m 1/i > r holds for all m < r and n > no. no. Foi For n > rao(r) we have ve r-l

g £ £53 < Ei' 53; << n

fc=0

+

1/-1

-

n-r

;Eo

n) fc2 aA2A |A |A22Al")| A( *+++l^)^| All 2^A^^^ l| +++ig(n-*)|A )lAl*£°l|in)l |A Ai ;;^£5( »> ""*)|A

! + ;fc=r£ «

fc=0 n-1

fc=r

fc=i/

! -WA<">| E E I E- »z*W i n

fc=r \ »t =t=n—fc n-fc

fc=n-r+l fc=n-r+l fc=n-r+l

and by (2.42)

<

/

■1 \ t = n - f c r-l

£^

a 2 ££<£|A < 2 | Aa A<,">| W•}|| +++ £C'£ |AAA<"> (n >nno(r)). (n>no(r)). (n> o(r)). r n fc=0 fc=0 n

For a given e > 0 we choose, at first, such that C7/r < e/2, and then by T-regularity of A one can select n so larger: that the estimate that tJ e

r-l 2 n) |A A< A<">| i
fc=0 fc=0

is valid, and (2.43) is proved. This shows the validity of our statement. Corollary 2.11. Let the O.N.P.S. {pn} (n e Z+) be uniformly bounded in [c,d\ C (-1,1) (see (IV. 2.22)) and the condition (2.28) fulfills. Furthermore, let for the entries of T-regular matrix (2.1) the condition (2.42) is valid. Then for

214

Fourier Polynomial Series in L*

/ G Lfajd) C) L^(E) at every Lebesgue's point x G (c,d) (and, consequently, almost everywhere) the Fourier series (HI. 3.1) A-summability to f{x). In fact, if we put ipo(x) = 1 and r = 2 in Theorem 2.10, then Corollary 2.11 follows. Putting r = 1 in Theorem 2.10, we have the following assertion Corollary 2.12. Let the O.N.P.S. {pn} (n G Z+) satisfy to (IV. 2.22), (2.28) and (2.44) {dli\x)\
(2.45) (2.45)

JE JE

then at every Lebesgue 's point x G (c, d) (and, consequently, almost everywhere) the Fourier series (III. 3.1) A-summable to f(x), whenever for the entries ofTregular matrix (2.1) the condition (2.42) is valid. Remark. 1°. Jacobi's orthonormal polynomial system {/>n(«>/?;#)} (^ € Z+; — 1 < x < 1; a , / ? > —1) satisfy the conditions of Corollary 2.11 and Corollary 2.12 on all compact sets in ( - 1 , 1 ) (cf. (II. 3.22)-(II. 3.25), (II. 3.76) and Corol lary II. 3.15); 2°. (C,a) (a > 0) - means satisfy the condition (2.42) (see Ch.IV, §2). We apply received results to the following problem of //-moments: given func tion / G L]j[—1,1] by finite //-moments momei n

■A

Tn= T = n

du i f(t)tndfi f f(t)t Ja Ja

( nnG Z+ ) , €Z

Ja

to restore the function / . For the solving of this problem we construct the polynomial system {pn} (n G Z+) orthonormal on thee interval [a, b] with respect to the measure d//(x): inter n n

n ) ff cc Pn(x) = x Pnfr) = £^ / !l /<">**

(x G [a, 6]; n G Z+). Z+).

fc=0 Ar=0

Putting An) An) l(n) Sn-l t-1

An) ,(n) i(n)

aa

l

;/,((n»n+++1ll)) _v l

*** n—1 \ 2l _ jnn;——rr (/(n n»G r» Z+), U^ == 7sr-^+i) - = Tc^ny "' i(n) i(n) "~T(nTTT v( ^z + ), t*n n n+l * +i

n ~= (n+l) ' " liAnTi)'

n

ll

nn + n+l + ll

*n

**nnn ++ll

we obtain that the polynomial pn(x) satisfy to threerterm recurrence relation (II. 2.17), moreover, one can easy to calculate the Fourier coefficients of / : C n(/) Cn(/)

n)

) = =£ £ 44 niWr. T T 33

*s=0 =0

(n («€Z ( « €€ ZZ+). ). + ). +

Poisson-Abel's

215

Summability Summability

Therefore, if the conditions on system {pn} (n £ Z+) and the entries of matrix A (see (2.1)) of Corollary 2.12 are satisfied, then almost everywhere in (c,d) the relation n 1

:i T.)p (x) = =f{x)Jn-»oo ™°E> E ^(E^)*^*) ^ fc)

lim ^

fc

=

/ ( * ) ■

3=0 3=0

fc=0 fc=0

Furthermore, in view of Corollary IV.. 2.12 L.VL ifit the tne condition condit] (IV, 2.22) is valid and const, max |J|/(x)| < const, l v /(x)| < a<x
|I>l n) (Eev,) ;4fc)rPfe (z)|
s=0

holds uniformly on all compact subsets of (c, d).

5.3

Poisson-Abel's summability of the orthogonal polynomial series

We will say that the O.N.P.S. {pn} (n € Z+) (with respect to the weight w(x) in the interval [a, b]) belongs to the class Qo = Qo([c, d\;
and Dl»( Dl»( X) X) 3 J|Pfc(*)l Mnnn === max sup ssupi y— 7-7- == o(n) 0
—»• 00); (n -»■ —>oo); 00);

(3.1) (3.1)

c) the condition (2.28) holds. For every function / G Ljja, b) let us denote u(r,x) of Poisson-Abel's means of the orthogonal Fourier series of / (III. 3.1) for O.N.P.S. {pn} (n € Z+), that is cOO 00 00

-E'

k ]rktf)Pk(x) ck(f)Pk(x) u{r,x) u(r, ] rhc u(r,x)x) = ][Vcfc(/)/> Y, fc (x) fc=0 A: *=o fc=0

[ f(t) f(t)Pk(t)w(t)dt cCk(f)= k(f)= Pk(t)w(t)dt

(0 < r < [a, 6]); < 1; x eG [o,6]); [a,6]); (3.2)

(fcGZ (fc€Z+), ( * e Z ++)),,

216

Fourier Polynomial Series in L*

where the system {pn} (ra G Z+) belongs to the class QQ. Observe that in this case the series (3.2) converges absolutely and uniformly for 0 < r < 1 - rj (0 < rj < 1), x G [c, d], if the condition

h

\f(x)\tpi(x)w(x)dx / \f(x)\ip\(x)w(x)dx \f(x)\(p\(x)w(x)dx

< 00 oo

(3.3)

JE JE

holds. We will say that u(r, x) is w-harmonic extension of function f(x) to the region (3.4) (3.4) D = {(r,x), 0 < r < 1; c < x < d}. d}. We will call the track r = {(r,x), 0 < r < 1; a < x < b} nontangential at the point Mo(l,xo) (a < xo < 6), if T = {(r, {(r,x), x), 0 < rr < < 1; a < xx < < 6; |x - x 0 | < 7(1 - r)},

(3.5) (3.5)

where the constant 7 > 0 is independent of r, x. The Fourier series in orthonormal polynomials {pn} (n G Z+) will be A*-summable at the point xo € (—1,1) to the value /?, if u(r, x) ix(r, -+ 0, /?, 0 --► * '■0,

when the point M(r,x) tends to the point Mo(l,xo) f° r a n v nontangential track T (see (3.5)). The following statements on boundary values of w-harmonic functions are valid. Theorem 3.1. Let the O.N.P.S. {pn} (n G Z+) belongs to the class Qo([c, d\; ipi\Mn). Then for every f G £jja, b], for which the estimate (3.3) holds at any point of continuity XQ G (c,rf), the following relation is valid u(r,x)->-* /(x u(r,x) f(x00))

(3.6) (3.6)

however the point M(r,x) G D (cf (3.4)) tends to the point Mo(l,xo) remaining inside D. Theorem 3.2. Under the hypotheses of Theorem 2.1 at every Lebesguefs point xo G (c,rf) the series (HI. 3.1) is A*-summable to the value for f{xo). In order to prove Theorems 3.1 and 3.2, we need some auxiliary assertions. Lemma 3.3. Let the O.N.P.S. {pn} (n G Z+) belongs to the class Q0. Then for the Poisson kernel 00 OO

k (t,x)x) = = ^2r ^rkpkpk•(t)p Pr(t,x) P (t)pk(x) k(x) r(t, kPk(t)p k(x)

(a,6); [c,rf]) (0 < r < 1; t €G (a, 6); a;x €G [c, rf])

(3.7) (3.7)

Jt=0 fc=0

the following representation ion is is valid vai; 2 ( 1 - r )^-^f2r%(t,x) >*&(*, x) ( l - xx) 2 {l OO

Pr(«,x) PAt,x)= =

fc=0 > fc=0

(0 ( 0 < rr < 1; \t |\t-x\ *-- xx\| > > 00;; t*G te€ (o,fe); (a,6); xxe xG€ [c,d]),

(3.8)

Poisson-Abei Summability Poisson-Abel's 's Summability

217

where &(*,£) is defined by (2.7). In fact, if the O.N.P.S. {pn} (n € Z+) satisfy the conditions (IV. 2.22), (IV. 2.35), and (3.1),then the series (3.2) is converges absolutely and uniformly for all t, x € [c, d\ and the series

wp W £< XSi^w sw !/si ' CX)

fc=0

is absolutely and uniformly converges for a l H 6 E and x 6 [c, d\, too. So, we can use the formulas (2.6), (2.7) and n

oo

fc

2 ( l - r 2 )fy(fc ))£';r*(fc + + l)a (n € Z+), fVur fcti == (l-r lK; ^ =-" -n^^+1 yY1 EEE^M ,' (»6Z+), (»6 Z+), + lK; ^<7n fc ; OO

fc

=

fc

fc=0 ife=0

fc=0 0/=0 fc=OJ=0 fc=0i=0

Jb=0 fc-0

where {ti n } (n € Z+) is an arbitrary numerical sequence. L e m m a 3.4. Suppose that {8n} (n € Z+) is an arbitrary sequence of nonnegative numbers, for which o(n) > oo). Sn = o(n) o(n) (n —► oo). Then for r -> 1 22 2 fc OO

' £'

(l-r) fy«S rf = ((l-r) 1 - r ) 2f> r*Jfcfc=o(l) = o(l) o(l)

(3.9) (3.9)

fc=0 fc=0

and

3 [k+l)rkSk fc = = o(l) o(l) ((l-r) 1 - r ) 3f>+l)r*<S OO

3

Ec

(3.10)

fc=0 fc=0

are valid. Proof. We should show the relation (3.9), formula (3.10) can be derived exactly in the same way. Let us denote by == o(l) 0(1) ui/nn = == 6^*» -± = o(l) n

(n ( n -— >>ooo). o).

If N = [1/(1 - r)], where [a] is the integral part of a, u , then tUCIl oo

oo 2N f c 2 % = r ^ + (( ll -- rr )) 222 £ 6k = = 5 j S+1+S r kS 5 22. = ((ll -- rr))i22 ^]E(( 1l -- rr ))2 ; f ^ r •% k=0 fc=0 fc=N+l fc=0) Jfe=AT+l ifc=o

> £ '

Since i/n = o(l) (as n -^ oo), then

Si< 5 1

_11

^

^ fc=0 fc=0 fc=0

N

i/ fc A;r fcl

._1_ ' N2

«3

N

E"fc■k = N]fc=0 Jfc=0

iVk < • k=0 *k=0 =0

o(l) (TV -> oo)

218

Fourier Polynomial Series in L*

2

and

f; *** oo

oo

kv < (12(sup» - r) 2 k(sup ) i/i*) krfck kukkrk<(l-r) £ ) fcr fc) fY^

5S < ((1l 2--r r) ) 22 f£; ft22<(l-r)

k N h>N k>N

> ^

fc=JV+l k=N+l e=JV+l fc= fc=JV+l

fc=i/+i fc=I/+l k^rl k=i>+l fc=i/+l

222

SUp j/I/fc ( l ) (n ( n -► ) Cjr» r^ sup i/fcfc == oo(l) -> -►oo). oo). < (1 - rr)r) C 7 ; -r=n k>N (.1 k>N (1 -- r) ry It follows from the estimates of S\ and 52 that the relation (3.9) is valid. This completes the proof of Lemma 3.4. Lemma 3.5. Let the O.N.P.S. {pn} (n € Z+) and the function f(x) satisfy to the hypotheses of the Theorem 3.1. Then the following assertions are valid:

1) if 8 > 0 is fixed such that (XQ — 6, xo + S) C (c, d), and \x — XQ\ < 6/2, then for allr > 1 — 6/2 the isnmaie estimate rxo+8 J»Xo+<$

//I

Jxo—S Jxo—S

(3.11)

(t,x)\w(t |Prr(*,x)|u;(t)
is true, where the constant C > 0 doesn't depend on r, x; 2) the relation (*)-/(z)]P l(t,a;)u;(<)dt -- f(x)]P x)w(t) /j [f(t) [/(*) f(x)]PP(t, (t, x)w(t) dt dt === o(l) o(l) jTi/w rr r

(r ((rr-> --► > 1) l )1)

(3.12) (3.12)

holds, when the point M(r,x) tends to Mo(l,#o) (#o € (c,d)) for any track from region D (see (3.4))Proof. We divide the integral rXo+5

L(r,x,xo) L(r,x,x00)= L(r,x,x )= = ///

JJ xo—6 XQ — 6 J xo—6

(t,x)\w(t)dt |P\P r(£,:r)|u;(t)cfe r(t,x)\w(t)dt I\P

in the following way (it is not difficult icult to see, that [x - (1 - r),x + (1 - r)] C (x0 -6,x0 + 5)): /•xo-(l-r)

L(r,x,x L(r, x, XQ) L(r,x,x == 00))=

/

/ Jxo—S JXQ—S JXQ—S fXo+6 rxo+6 /•Xo+5

+ // + /

- X~o0Xo ++(l-r) (+(l-r) i_r) /•x

(t,x)\w{t)dt+ (t,x)\w{t)dt + /I ]\P i r(t,x)\w(t)dt

Jxx00+(l-r) ^ +(l-r) Jx0+(l-r)

Jxo — Jxo ^ x o— - ((l—r) l(l—r) -r)

\P x)\w(t) dt |P (t,x)|t/;(t)^ r (t, |Pr(*v

| Pr(t,x)\w(t)dt r ( * , *\P) r|(t,x)\w(t)dt=Ii+I w ( * )=* h =/+ l +^2 / 2 ++2+h. / /33. I\P

In view of hypotheses (IV. 2.22), (IV. 2.35) and formulas (3.7), (3.8), one can derive the estimates for all t,.,*,x €^. ^[c,, U,J. d\: alU, ( 1 | C 7 3 -: C:; C—— 11 —r —r ||Pr(*,*)|<4 Pr(t,x)|< \Pr{t,x)\<{ 1 i-^S/ r_ / r ll CCC ( ^ ) ~(«T- ^z ))22» V

for > 00 for I* - x\ x\ > > foi for \t\tt -—x|

(3.13) (3.13)

>0. for flfouo rl |r|tt-x\ l t*| t- -^xl |>>00..

(3.14) (3.14)

Poisson-Abel's

219

Summability Summability

In consequence of hypothesis (IV. 2.35) from (3.13) and the relations follow ma (3.14) (0.14; tnc /h +hl-r

w(t) dt "®*< ^ < l-r I(* ~ X)2 J\t-x\>l-r

and

■><>£

w h 0 such that \t — xo\ > 6, because XQ e (c,d) and t € E. Since the point M(r,x) tends to Mo(l,zo), then one can choose the variable x such that \x — #o| < 5/2. Prom the last two remarks,one oneBhas \t-x\>66-.-.> \t~x\>\ \t~x\>

S

(3.15) (3.15)

- 2'

By Lemma 3.4 the relation will be established, if we will show the relation i;(*)d£ o(l) [f(t) - /(*o)]F„(*, /(x 0 )]F x)w{t) dt = = o(l) )]Fnn(t, x)iu / [/(*) x)u;(t)

oo), -+ (n — -»•>>oo),

(3.16)

JE

where Fn(t,x) is the Fejer kernel for the O.N.P.S. {pn} (n G Z+). Consider

7"

^n(x) [f(t) - f(xa)]t {t,x)w(t)dt In(x) == / \f(t)-f(XQ)]t f(xo)]Z (t,x)w(t)dt nn n(t,x)w(t)dt /»(*)

((nn €€ Z x £eG [c,d]). [c,d]). Z+; [c,J\). + ; ix

JE JE

Applying conditions (2.28), (IV. 2.22), (3.1), (3.3) and estimates (II. 2.32), (3.15), and the representation (2.7), we obtain

;ion {4.1), we ODiain

. f/• / ( * [f(t)-f(xo)]vi(t)i ) - / ( *nxo)) o ) WV1 <(t:) f / „Oa„Oa +ii \\p \ 2(t)\] |Pn+2ft)|1 f/"f [—7[/(«)-/(*o)]pi(*)f »l(«) [jPnWI [/(*),,,, Mi , - <<, [/(«)-/(*o)]Pi(«)L |\\Pn(t)\ W ) | ,. IP |P»+»(*)ll n(t)\ —— T-T|/„(X)| + \I < / ^ —77 ToT—, a nan n+i —777H Pn+777— ■fnW , x
2 „2a„2 I.IPmWl IPm j ^•2U2 ,\Pm(t)\ 2sbn+l(*)| |P»+l(*)l ., o„■>\ V^ pVL + g 2^bn-uft)! a ^ l a m - a m - l 11| - — ¥>l(t) vi(*) m=0 ^ v ' n

) ||"bm(t)|

^ lPm+l(t)|1

++"-EE |Um+1 umi\ [ J ^ te»|i § i ++JS^ll \b*m] U)W)(t)d*t m|« m+1 ~- Tn\ Um 2 J a™l "i+l m=0 V l W VlW m l m=0 v>i(t) JJ J $£> L[ ^ W c m /{xo)l < CC i jT lfit) |/(t) - / ( x o J I ^ W - ^ M n n 1 a

ufl

U

T

&L ~

[[

a

Mn

_ a

m

a m 1 W f L1 + m=0 m l ++ m=0 5E3 I""* l^m-ttm+ll E 1^-1 I°™-1 ~~ - "ml "ml + l \w(t) ( ) dt ^ 1 + m=0 E + m=0 E K - "Um+il m=0 ( Jn -> 00), = o(l) 0(] f m=0 \f(t) f(x m=0 J 0)[ = o(l) JE / \f{t) - f{xro)\ipi(t)w(t)dt = o(l) o(l) {n (n -» 00), 0)\ oo),

Fourier Polynomial Series in L*

220 220

from which the relation (3.16) follows. Lemma 3.5 completely proves. Remark. As in the proof of Lemma 3.5, one can obtain the following relation "" W ) " ^ v,»**A w^rvu.*** l

1 M

v

)]P r (t,x)w(t)dt J B[/(*) [f{t) - f(x x)w(t) dt = o(l) /J f(*o)]Pr(t, o(l) \f(t)-f(xo)]i 0)]Pr(t,

1) ( r --» + l1) ) (r -)•

(3.17)

for any function / € L*,(E) if the polynomial orthonormal system {pn} (n€ Z+) belongs to the class QoProof of Theorem 3.1. Since the series (3.2) converges absolutely and uniformly, one has

r >6

u{r,x) u(r,x) u(r, x) = j/ f(t)Pr(t,x)w(t)dt (t,x)w(t)dt and

(0 << r << < 1; 1; xxG6e [c,rf]) [c,rf]) t (0 ( 0<

J:

/ Pr{t,x)w(t)dt (t, x)w(t) x)w(t) dt = = l1 ((00 < rr < < 11;; x €€ [c,<*]). [c, d]). d]).

Therefore

(3.18) (3.18)

(3.19)

tx(r, ( 0<< r < [c,d\). w(r, f(*)-/(xo)]Pr(t,x)u;(t)ift /(xo)]P r (t, x)w(t) rf]). (3.20) tx(r, x) - /(x f(x00) ) = / [/(*) - /(xo)]Pr(*, * M i ) dt: (0 < 1; x €6 [c, [c, d\). Ja chc For given e > 0 we choose 6 > 0 such that (xo — £, xo + £) C (c, d) holds, and for |t — x| < 6 one has (3.21) |/(t) - /(x 0 )| < e. (3.21) Let us |x - x0\ < 6/2 and r > 1 - £/2. The formula (3.20) gives u(r, x) - /(x /F((*o) /(x ) = = // / 00 )

)]Prrr(t, (t,x)w(t)dt [f{t) [/(*) - /(x /(x00)]P )]P (t, x)w{t) dt (/(*)-/(xo)]J

J (c,xo-8)U(x (c,x0-8)U(x +8,d) J(c,xo-*)U(x 0x+«,d) 00+5,d) fXo-\-8

1/

/(t)-/(xo)]P + / !/(*) - f{Xo)]Pr(t, - f(x (/(<) / ( l 0 ) ] PP(*,x)ti;(*) r ( * , X)w(t) X)w(t) dt+ dt + / \f(t) [f(t) )-f(zo)]P f{x0)]Pr(t, )]Pr{t, r(t,x)w{t)dtx)w(t) JE Jxo-8 JE Jxo-S Jxo—8

+[ =

dt

h+h-

In consequence of the estimates (3.11) and (3.21) it follows that L-lIIlclieS ^O pxo+8 pXo+8 /•Xo+5

/h<2 < /

/

XQ—8 JXQ—8 s JXQ—8 rxo+8 +

) | | Pr (t,x)K*)
:::rxo+8 s

< < //

||\P P r(t,x)\w(t)dt (t,x)Ht)dt p |P (t,x)H*)cfe Jxn—8 Jxo—6 JXQ—8 Jxo—8 \x-x |x - Qx\ 0 | < <5/2i 6/2 andI \t then |t -—■xxx0o0|\| > >> 6, d,6,1 the] \t-x\>

/

If (3.14) gives

-}

h == /[ /l

r (c,x0-6)u(xo+6,d) J{c,x 0 -$)u(x 0 +<*,d) -djU^xc J(c,xo—i

^^i

<
[f(t) "- f(x /(x )]Pr (t, X)w(t) x)w(t) dt dt [/(t)-/(x [/(t) 0 )]Pr(t,x)ti;(t)cft 00)]Pr(t,

< \f(t)-f(x0)\w(t)dt. < C1^ I-rj T \Ht)-f(x 0)\w(t)dt.

p

221 221

Poisson-Abel's Summability Summ&bility

So h = o(l) h=o(l)

(3.23)

as the point M(r,x) tends to Mo(l,£o)Now, the statement of Theorem 3.1 follows from the relations (3.12), (3.22), and (3.23). Proof of Theorem 3.2. We assign an arbitrary positive number S > 0 and we choose a positive integer N\ from the condition Nl+1 - r) < S < 2 Afl+1 2 NN,l7(1 7(l 7 ( l -r)<6< 7 ( l - r),

where the number 7 > 0 we introduced in the defining relation of nontangential track T (see (3.5)). Then following estimate itimate is is obviously obviously xueii the m e iuu |u(r,x)-/(a \u(r,x) \u(r,x) - f(x 0)\ < /I/ /(*o)| J\t-x \<27(l-r) 0\<2f(l-r) !-xo|<27(l-r)

\f(t) - f(x f(xo)\\P 0)\\Pr(t,x)\w(t)dt i-/(x r(t,x)\w(t)dt i/w0 )||Pr(*,«)|w(t)A

Nl AT, Nl rf + £ / fc + k^[ £ /2 /./2 f c7(l-r)<(t-a 7(l-r)<(t-:ro)<2*+i7(l-r) -xo)<2 f c + 1 7(l-r)

r

+ + // J\t-x J\t-xo\>6; \>6; te(c,d) 0*o|><&; t€(c,d) «/|t-x 0 |>

++ I /

l/ 1

J\t-xo\>6; t€E J\t-x0\>5; t£E 1 -x 0 |>£; J\t-x \ *€#

(t)-/(:ro)||P \f(t)-f(xo)\\Pr(!t,x)\w{t)dt r (t,x)Kt)
%x)\w(t)dt \f(t)-f(x \f(t)-f{xo)\\P 0)\\Pr(t,x)\w(t)dt r(t,z)\w(t)dt 1/\f(t)-f(xo)\\Pr(t,'<

(3.24)

dt\ f(X0)}Pr(t, X)w(t) dt\ [/(') dt\ I/O W - /-( ^f(*0)]Pr(t, 0 ) ] P r ( t , xx)w(t) )X)w(t) l ■'

= Si SX + S ^2 5 44. 2 0+ S3 + S = 5i + 5?22 + H- S3 + &l.

^FVom the defining relation of Lebesgue's point >oint and the estimate (3.13), (3.13) one can obtain the estimate *Si = = //

t - x0\<2f(l-r) |<27(l-r) ^|t-x 0 |<27( \<2
\f(t)-f(x \f(t) |/(t) ~ - /(x f(x )\\P 0)\\P r0(t,x)\w(t)dt 0 )||P r (t,x)k(t)dt r(t,x)\w(t)dt

C c L < ( rr~; iI. f m - fram^iwmift. ^ z fI| t - x | < 2 7 ( l - r\nt)-f(x \m-f(x 0)\w(t)dt. ) 1

~

r

^ | t - x00 | < 2 7 ( l - r )

Hence Si = 0(1), o(l), ft

(3.25)

as the point M(r,x) tends to Mo(l,a;o). Since the point M(r,x) tends to Mo(l,xo) f° r a nontangential track T (see (3.5)), then from the condition \t — XQ\ > 2fc7(l — r) one can deduce fc

ni - r ) . |t - x | > (((222f*c -_ 1)7(1 1)7(1

Fourier Polynomial Series in L*

222

In view of the definition of Lebesgue's point and the estimate (3.14), we derive Ni

\P |P\P(t,x)\ ( (t,x)\ max 5 2sE„. <x |t-x |>2 r) 7(l-r) S2
fc=l'

0

rr r

fc

r

x /

ffcc + 1 7(l-r) * 7/ 22fcf c 7(l-r)<|t-x 0 |<2*+i-r(l-r) 7 (l-r)<|t-xo|<2 + 0 |<2

\f(t)-f(xo)\w(t)dt \f(t)~f-/(a?o)|u;(*)*

k

J2 *yi

r

^ (2* - 1) 2 (1 - r)

)) = 0 ( 1 ) ^ 3 ( 2 * - 1 ) 2 -

So S22 = o(l), as the point M(r,x)

(3.26)

tends to Mo(l,£o) along track T (cf. (3.5)).

We use the inequality (3.15) and the estimate (3.14) once again, then for a nontangential track T one obtains the following estimate: 533= == j/I S

J\t-x00\>6; <5; t€(c,d) te(c,d) 7|t-xo|>5;

= ==

\w(t)dtdt (t)-/(x )||P (i,x)Kt )\\Pr(t,x)\w(t)dt |/(t) --f(x /(xo)||P X)\w(t) 0\f(t)-f(x 0)\\P r(t,X Pr0(*, l/(t) )\w(t) dt (t,x)|jT |/(*)-/(xo)h ) \ f \\f(t) m - -f f(x0xo)\w(t)dt

m; max max I.l-Pr(i,i |P (t, x)| \PrP(t, x)\ /f IX v " I *\t-x0\>6; |>(5;; tt6(c,d)

max

1 \f(t)\ + ||/(*o)IM«) )\)w(t)dt. jj -- ^i -—^^2j /f (|/(t)| / ( s\f(x « ) *■ dt. (\f(t)\ 0 ) I0M Jc

* )(t( t-- zxy) l*-xo|>|; l«-io|>|; te(< te(c,d)

Consequently, 5 3 = 0(1) o(l) $3

(( rr-->Hl ) .

(3.27)

The estimate of the last sum 54 have been established in (3.12). ^From here and from the relations (3.24)-(3.27) the assertion of Theorem 3.2 follows. In view of (3.17), the following statement may be proved exactly in the same way. T h e o r e m 3.6. Assume that the O.N.P.S. {pn} (n G Z+) satisfy the assumptions (2.28), (IV 2.22), (IV. 2.35). Then at every Lebesgue's point x0 G (c,d) for any function f € £j,[—1,1] and f € L^E) the Fourier series (III. 3.1) is A*-summable to f(x0). Remark. It is easy to see that the Jacobi polynomials {p n (a> /?, x)} (n € Z+; x G [—1,1]; a,/3> - 1 ) satisfy the hypotheses of the Theorem 3.6.

5.4. Analog of Fatou's Theorem for Differentiated

5.4 5.4

Fourier Expansions Expansions

223 223

Analog of Fatou's Theorem for differentiated Fourier expansions

Differentiating t h e Fourier expansion (III. 3.1) of f(x) in orthonormal polynomials {Pn} (n £ Z + ) term by term w e obtain t h e series oo

f>(/K(*), E< fc=o

f > (cfc(/K(^), /yfc(s),

-A

ck(f)== /f f(t)p f(t)Pk(t)dp(t) <*(/)= (t)dp(t) Ck(f) k(t)d»(t) Ja ./a

fc=0

(* (fceZ € Z+++)) (fceZ

(4.1) (4.1)

which is, in general, not an orthogonal expansion (we observe that t h e first term in t h e s u m (4.1) is equal zero). We consider t h e following problem: t o treat A*-summability of (4.1). We start from t h e following auxiliary assertions. L e m m a 4 . 1 . Let the O.N.P.S.{pn} (n € Z+) belongs to the class QQ, then for all tt,, xx GG [c, [c,a*] d\ and and t £ [a, b] (\t — x\ > 0) the following representation is valid

Vx) «

-

^ ax oxX

oo f(I1 -_ rr)2 )2 ff ~~ 2 2 rk 2 2p2 {x) r) + [ 2a2 ( a ?2 --- aaaa^aoPop'iW] + = ( T ^ ) 2 i -22 ^(t,x) + —{-2a p (x) + 22(a 2(af ^), afloPop'iW] oPoPiW] = (t-x)*\ ( T ^ ) 2 I " E»"*&(*'*) +— ~ o) r~ 1— r 0 °0 °

( - f y &(*,*)+fc=0 K

V

OO oo

fc

2 3 ky £a2l))| .1[(A + 2 f y fcX^a/_i(A £k o,-12(A p _i i2_ali(M t)p {(t)p' ( x )-l-(x)p (Pl(t)p'l_(x)) az2f^r _2a?_ )[pj. 2()\p l-1(x) 1(x)}1 W Pli(t)pU :(*)rf-i(*)]

k=0 fc=0 oo oo oo

1=2 Z=2 1=2

fc 2 5A + ^ A aa 22 :: + + A + ii [[ p p^ ^ k+2(x) +i\pk+i(t)p' ]a>kfc+ :: }a + ff ccit + ^ rr f c [[ 55;5Aa2 + Aag A aa 2+1 + + > k=o ib=0 fc=0

+ E^

-pk+2(t)p'k+1(x)} )-Pfc+2(*)Pfc+l

oo

fe a 4o a ^ )rrrfcafc | + 2 - 2-"a |2a [ gp)f|cpH fc+2 2)afc)[Pfc+l(^A:+2(^) fz+l(i)Pk+2( c + i ( t ) p t + 2x()x ) ~ -Pfe+2(*)Pfc+i(aO] )) Z H ^* a+*j f+ci+i (i^( 4f+ac|+l+ i1L ++a°*+2 -Pfc+2(tK+iW] ( l - r ) ]k=0 k=o fc=0 fc=o

+ 0- -

+

r

•)E oo oo uo

2

~ ( ~

fc c a f c + i a f c + 2 P f c rJ/ C^ >k+3(t)p' 2rka ^^ kak+ia r))) ^ rafcf afcafc+iafc+2[p* a k+2\pk(t)p'k+3(x) [2[Pfc(*)Pfc+ W ^ + 3 ( X3() a?~ Pk+3(t)p'k(x)] )-P* k(x)]

rr

- ( l - r ) ] fc=0 k=0 fc=0 oo oo oo

A

x + T V (u uk+i)\pk(t)p' (x) -pk+2(t)p'k(x)} [uk u+ki)[pfcWpik +i)\pk(t)p' + ]]TV(u ) - Pk+2(t)p\ k+2>k+2( k+: +2W-Pfc+2(^KW] ffcc -- wfc

k=0 fc=0 oo oo

fc ik

fc^ a ( A t * i2)\pi+i(t)p 2 + E 7 *f *c Y^ai{&ut) | p z + i ( t(t)p{(*)-pi(t)pj )p{(x) -Pi(t)p' il+l(x)\ z + ■l(*)l fc=0 z= z=o 1=0 fc=0 z=o oo oo fck r fc^ a zaa o/ _/ i (i AA2 2uu/ _ i )i [ p ( i i)bi-i(*)pi+i(*)-P/+I(*)PI-I(^ z=o fc=0 Z=0 1=0 z=

+E^ + E ^: E«'

E«<' - ( +E^ ++]C E ^r*]C

w/iere & ( t , x ) is defined by

'- )fr'- (^^ (2.7).

(4.2) (4.2)

Fourier Fourier Polynomial Series in L*

224 224

Proof. In view of the three-term recurrence relation (II. 2.17) it is not difficult to see that akPk+iW ;) (a_i ((aa_- i = = 0, p_i p_i = == 0, 0,fc Z+). xpite) xp' UfcPfc(x) offcc-iPfc_i(x)-pib(x) k GG Z+). x J ++U UkPkW fuc P f c C x J+ + Oak-iPk-xW-pkix) --iPfc-i(s)-Pfc(s) lPfc., k(x) = afcPfc +1((x) afcPfe+i(z) (4.3) (4.3) Taking into consideration (II. 2.17), (2.7), (4.3) and Abel's summation by parts, one obtains

2 + 2(o? 2 ( a 2 - a>o)a>oPo(t)p'i(x) (t - *z )) ^^^ ^^ =« & &(*,*) 2, a » ) + aaftc (t)p'i(x) < * *) , . )-"- 22agpg(*) (* &(*. 2(a? ag)oo 0)a0P0 Po(*)pi(x)

fc-1 fe-i fe-1

2 2 + + 22£a,( £ai(A af_ (a:) (a - pj+i(*)p|(x)]\ 1 )|pi(t)p{ 1)|pi(t)p{ +1+1 >p{+i(*)-Pi+i(*)PiO"0] +2>£° ^2 ^>i(A ( *af_ V-i)iw(*)pi+i(*) - pi+i wrfc*)]

£J=i =1

+i[pfc+i(*)p' 3a2ojk i(t)pfcfc+2 - Pfc+2(*)pJb+i(«)] H3a fc afc ii[pik++i(*)pik (s)-Pfc+2(*)p'fc+i(z)] 20r)-j +2+ + 3a^a i[pfc (x) Pfc+2(*)p'fc+i(z)] ++ fc+ + -

2a afcfc(4a£fc+H+a^«fe+ii(t)Pfc(*)] -i)[Pfc(0Pfc+i(z)-Pfc+i(*)j (t)pi(x)] l(')Pfe(*)] fc-i)[Pfc( +1 - 2a£_ 1)[pfc(t)pJk+1(x) - pfc+ fc+1 x x afe-iflfea jk W-Pfc+2Wpik-iW] ak-iakak+i\pk-i(t)j/ feGfe+1 [Pfe-1 (*)Pfc+2 + (x) ( ) -Pk+2(t)p -i(x)] ak-ia ak+i\pk-i(t)p' ( ) ~Pfc+2(*)plb-i(«)] +2 *)] k 2 k k afeajk [Pfe(t)p' pfe+3(*)p'fc(x)] aafeafc+iafc+ (*)p'fc+ " P*+3(*)p'fc(z)] kak+ia +iajb+2 k^3(x) 2[pjk 3(aO (z) kfe fc iafe+2[Pfe(*)p'fc+3 3(»)-Pik+3(t)Pfc(»)] fe 2 +^ ^0a|/((iwi £( + i - ui)2 |pi+i(t)pj(x) -Pi(t)p{ ]Ta/(uj+i -pi(t)p{ +1 (x)] + i(x)] 0] (li£ + l - u i ) [ p i hl(*)p{(*)-Pl(t)p{ i(t)p|i l(*)]

1=0 j=o j=o

+

+

x + aafcfcaafcfc+i(u -- u pfc+2(t)p^(x)j H-afcajk +2(x) + i( fc + +i(u ufc+i )i)[pfc(t)pi i)[pfc(t)pi (x) -- ~pk+2(t)Pk( )} . i ( ufcf c - t i f cfc+ ( t ) p k +2 Pk+2{t)p'k(x) + [ p f cfc(*)Pfc+2(*) fe-1 fe-i 22ui)\pi(t)pi (x) + ^a/ai ^2aiai^i(A -pi +2 (t)p{(x)]. +2 +2 (x) )]. x 2 + i(A u/)[p((*)pJ +2 )Pl+2( aj+i(A u/)[pj(*)p{ (; ) - Pl+2(t)p'l(x)].

£=0

+2

225 225

5.4. Analog of Fatou's Theorem for Differentiated Fourier Expansions Expansions

Therefore

9J z)fy (<-*)fy^=fy^.*) <±<*3&(*,*) dx £ oo

e

fc=0 fe=0 fc=0 fc=0

fc=0

1 + j-^[-2ogpg(t) + 2(of - 2aDpop'^x)} ag)p0p'i(a;)] »] ^7[-2aSpg( .— r \pl(t) + 2{a -a20)p0p'l(x) oo

fc

k2 2 + ^al.1-(A al )\p -1(t)p' + 2 E rfcf c ^ a J _2f^r i ((*)-pi(t)p{-i(*)] x l)(x)-p - p l (l*(t)p' ) p ;l_-1i(x)] W] 1 ( A 2a ? 2 ) h - 2i ( * l) p + 2X

E«;ai-i(A o?_ )|pi-i(t)ii{(x; E' E«
2

1=2 1=2 1=2

Ar=0

oo OO

r [5Aa£ + [V[5Aa£ +Z / **i $3^*[5AaJ J]ajH-i|pik+i(*)l4 2(*) -Pik+2(*)l4+l(»)] -Pfc+2(t)i4 (x)] WPfc+2^) +E fc4-+ (AaJ (Aafc+i)]flik+i|Pik+i(*); =o fc=o +

+

+1

fc=0 fc=0

oo

k i(ty fc+2 (x) --ppf cfc+2 y afc a2 |+2 a£ - 2a^)[p 2a2 )\pkfc+ +l(t)p' (*yf fc+1 (x)] + (1 (11-- r) r) £5^ + 2 (ty c + 1 (x)] k+2(x) rrffcca+a1f (c 4+ 1a(£4+a1^ + 1 + a k+2 +2-2( k

)E' k=o fc=0 fc=0 oo

-r)fy

r a00fc 0iafc+2 x r 10 2 p " *+ +^iaik ^ ^fc+3( ^ ^b+sW-Pfc+sWpJbW] ^X)) -Pfc+aWP/tO*)] r^** ajkafc 3(i " ((X1 --- r) 0) ^^^ ]CZrfca [P*W "Pfc+3W^fc(X)] +2[pik(*)pj

fc=0 k=0 fc=0

oo OO oo

k

Pfc ^ r1**•*(** ( u(uf ck-u --- ku -- p + 5>2r uk+i)\pk(t)p' - +2(*)Pfc(aO] p(*)Pfc(x)] k+i)\p k(t)p'k+2(x) k+2(x) fc+2(*yfc(x)] M pfc+2 +5> fc+i)[pjk(tyfc+2(a:) k. +E fc=0 =o fc=0 k=0 oo oo

fc

rr frf ee Au Au 22 2 £>(4 + + E EE> E°'( E°'( ') ') bb\pi+i+Mi(*)pi(x) i(*)pi(x) -P/WP{ I(X)] Pi(*)pi+i(z)] ++i(x)i H{^ui) {t)pfl{x)-'Pi( -«(*w E° +E

+E

fc=0 fc=n fc=0 oo oo oo oo

1=0 fc fc k

+ J^r* ^ r * 5 ^53oio|_i(A 0 | 0 |i|0|.i(A . 1 ( A 2 t i22|ti|_i)^_i(t)l>{ -ti|-i)[pi_i(t)p| i ) [ p i - i ( t ) | i { + 'i+iW-Pi+i(*)Pi-i(«)]( :f l p ) - - pP/+i(t)Pi_i(a;)]i+i(t)|i{.1(«)]. 1+ + i(*)

E« -5> fc=0 «=o +E . n fc=0

li = A 0

fc=0

The assertion of Lemma 4.1 follows differentiating term by term the expansion (3.8), which converges absolutely and uniformly for t G [a, b] and x e [c,d\: ,dPr(t,x) ( * - « ') - dx

> = (1-r) 2 I

I"

2

( * - * )' |\ v

2 1*6k(*,x) + ■aE' OO

A:=0

fc=0 fc=0

oo

( t - xc)

r )E fc=0

kdikit,x)\

k=0

fc=o

5x

' ) ■

)

Lemma 4.1 is proved. Lemma 4.2. Let the O.N.P.S.{pn} (n e Z+) satisfy (IV. 2.22), (IV. 2.35), (2.28) and for recursion coefficients an,un (II- 2.17) the inequality n

2

2

|A22t»«| «,| ++ |Aui| |A«,|2 ]2]]
EC 1=0

J=0

( n€€ Z (ne Z+) (n +)

(4.4)

226

Fourier Polynomial

Series in iA

holds. Then the following estimates are valid (c \t-x\ C _|£-£L r Cfort e [c, d], x e jj vv := :=[c + 77,t],dr/] \22: f° te[c,d\,xe [c + d - rj] (l_ I ^ i( 1 - r ) ' I 8P (t r) I ap r (t,g)L (0 < < i?t] < < (d (d -- c)/2), c)/2), \t |t -- x\ x| > > 0(4.5) 0(4.5) a);x-) ( * -■*x) Ut QX J\ -< <1 (0 —9aT" , 1-r 4 6 [c,d\,x xe6ii)» j',),I*|t-- x\x\>>°>( 0, (4.6) c c, ~\J /or f°r l<e€Ml' -) f

£

°(^^

2

w/&ere the where £/&e constants C > 0 in m (^.5,) (4-5) and (^.^ (4-6) are independent o/r,t, ofr,t,x. x. Proof. By Bernstein's inequality (see Theorem I. 2.11) and (IV. 2.22) one has tfnB(x)\ < C(n + 1) (x € e jU |l/ (x)| < v , n e Z+).

(4.7) (4.7)

Since the system {pn} (n 6 € Z+) Z + ) satisfy (IV. 2.22), then the series oo OO

fc

VPk(t)Pk(x) pfc(i)p'fe(i) Y,rk

((00 < < rr <
fc=0

converges absolutely and uniformly. It can be inferred from (4.7) for all t e [c,d], a; £ jrj jrj that that a; G

"1 =

r (t,g)| L_ x),gP ap^£)| J ^2 , rkp(iy Pfc fc(k(x)\ + l)rfc==_ cC -cM. ( * - * ) ■ '

9a:

I

OO

I

OO

fc=0

fc=0

( ^l - r ) ' '

A fc=0 fc=o

in accordance with (4.5). On the other hand, in consequence of (2.28) and (4.4) it follows, that (n + l)|A(« n ) + (n + l)|Aa 2 1 < C (n = 1,2,...) (n + l)|A(tin) + (n + l ) | A a J | < C (n = l,2,...) and, therefore, by Lemma 4.1 and the last estimate one can derive

+ C |A»i|] +£i |<«-'>^|* ^{£^ £*'-«' £H "I(

<

-

!

)

,dPr(t,x)\ ox

■

-

1

c

oo

kk

fc=0 fc=0

/ ==11

OO

fe i+ r {E-*l[l + oo

_ f l - r ) 22 r < S ci ( * - * ) 2 i ;fc

fc=0

C

1=0 =o

oo OO

2 l)|A2«?l a | +£(* + |Aa fc+1 ++ TT7 + £ r *r*B B^* ++ !)l^ + B ;(fc * ++ l)r l)^ +[|Aa^| [l^| ++ \Aal 1 + +l|]\] J

2 2

L

1- r

fc+2

fc=0 fc=o

oo

oo

k

fc=0

1=0

-B' OO

OO

+B

2

2

fc 2 + £rV (fc lAttfc+il + |Awfc+2|) + + £ r * £ (y+i)\A Z + l)|Au,| (fc + l)(|Au fc | + |A«fc+i| Ul\

fe=0 fe=0

fc=0

:U-

„|A«,|} i J / ++ l)|A u,| - ; c^£< + f^2B S( cfif^{ E )2\i I ; °(T^\i oo

fc

fc=0 oo

2

4*

oo

fc e+1 [|A^| f£r* + f> > + l)r ffe+1 [|A^| + |Aa^ |Aa2+1 |] ++ f >y(fc+i)[iA (ft + l)[|Au |AUfc+1 1 |] Ufefc| | + iA« fc+1|]}, fc=0 =0 fc=0

n fc=0

5.4. Analog of Fatou's Theorem for Differentiated Fourier Expansions

227

whence we finally obtain the estimate (4.6). We have completed of our statement. Let OO

oo

Wr(f;x) = ]rkck{f)p'k{x) Wr(f',x) = ^ r * c f c ( / y f c ( x ) fc=0 fc=0

(0 < r < 1; a < x < b) (0 < r < 1; a < x < b)

(4.8)

(4.8)

be the Poisson-Abel means for the series (4.1). Our main aim is to prove the following analog of Fatou's theorem about A*-summability of (4.1). Theorem 4.3. Let the O.N.P.S. belongs to the class Qo and for the recurrence coefficients (II. 2.27) the estimate (4-4) is valid. Then for every function f G Ljy[a,6], for which (3.3) holds, at every point XQ G j v , where the functions ff,uj' are exist and finite, the series (4-1) is A* -summability to f'{xo), that is (f,x)->f'(xol Wr(f, X)->f'(x0), when the point M(r,x) tends to Mo(l,xo) for any nontangential track To = {(r,z), 0 < r < 1; x G ji), \x - x§\ < 7 ( l - r ) } . Proof. First, to observe, that under the hypotheses of Theorem 4.3 the series (4.8) converges absolutely and uniformly for all x G [c, d\, 0 < r < 1 — <5, 0 < 5 < 1. In fact, by (IV. 2.22), (IV. 2.35), (3.1), (3.3) one has M/)| < < ||j M/)l

-r

\u

"I.

f(t)pk(t)w(t)dt\

+ |j +

f(t)f(t)p Pk(t)w(t)dt\ k(t)w(t)dt\

< C{k C(k + 1) (fc (k G 6 Z+).

Therefore, by the aid of (4.7), one can obtain oo

2 k k n± C^ Cp'kf(x)\
(l-r)3

fc=0

k=0

<(0 < r << 1 <°

6).

Hence, in accordance with this remark it follows from (4.8), that ,^dP (t,x) ^ w(t)-w{t)dtdt m■A w^r"* dPr r x)

WT(f, x) = J" f(t) Wr (/,*)

((00 < r << 1; 1; xz G € jjnv))..

Ja

Taking into the consideration that

A

/ P r (t, x)w(t) dt = l

one can deduce

Ja Ja

(0 < r < 1; x G jv)

b

dPr{t,x) ftb*2i9Pr ^t,x)w(t) —w(t)

Ja Ja

dx °X

dt = 0 (0 < r < 1; x 6Gj jv v)). .

(4.9)

228

Fourier Polynomial Series in L*

We can assume without loss of generality that /(xo) = 0. In fact, on the other hand, we consider the function h(x) = f(x) — /(xo). It is easy to see, that = W ( / , x ) -WrP(f,x)

Wr(h,x)

Wr(f(x (f{x0),x),

and in consequence of (4.9) the last term on the right-hand side vanishes. Thus /(*o) = 0. Further on we suppose temporarily, that /'(xo) = 0. Then for an arbitrary e > 0 one can find S > 0 such that (xo — 6, xo + S) C jrj and < e\t -- x x 00|\ <

\f{t)\w(t) \f(t)\w(t)

6. for \t - x0\ < 6.

(4.10)

Obviously \Wr{f,x)\6 1 /|t-x 0 |><5

I

(4.11) (4.11) J1(x) + ^ (Ja2(x). = Ji(a:) O-

We estimate the Lebesgue function

>^H*^U

f I (t x) I . dP,dPr(t,x)\ ^■\w(t)dt L(r; x; XQ) = (0 < r < 1; a; € j„). ( t -- xx0 ))—l^-L-L\ L(r;x;x 0 ) = /|t—x |<<5<5 I\(t dt (0
I

"I-

,dPr(t,x) 7 Jt_-zo[ |((£,_ Ji^ol (|t-„|> - x 0I )o ) ^^|< C (|t - *ol > 00)) — /

r

dx

"(l-r)»

holds. On the other hand, the point M(r,x) gential track I V Hence

I*** ~- X* °^ o | - .<\t-x\ < \t - x\ 2

(4.12) (4.12)

tends to Mo(l,xo) for any nontan-

> 227(1 for |\tt - xx00|| > ). 7 ( l --r r).

(4.13)

Therefore, by Lemma 4.2

—I "

1- r ,dP 1/1r(t,x) X) ry* «*/ | < < ( t - x dPr \(t-^o) c ? ( / _ " ^2 r 0 )^ 0 ( t -x0) #X

o

f o r | tfor|t-a:o|>27(l-r) - x 0 | > 2 7 ( l - r l) ,

(4.14)

where the constant C > 0 is independent of r,t,x (0 < r < 1; t e [c,rf], x G j v ) . It follows from (IV. 2.35), (4.12) and (4.14), that 11 c L(r, X,XQ) x , x 0 ) <"CCT( l - r )r^ L(r, 2 / 1 r »

|£ —\t-x xo\dt 0\dt

i* ') 7o<|t <J0<\t-xo\<2'y(l-r) |t-x |<2 (l-r) dt + CC(l-r)f + ( l - r) 7 - ^ 2) V - )

L

0

7

2 7 ( l - r ) < | t --xs0 o | l(C J2y(l-r)<\t-xo\ ' - * -x^ 0)) '

5.4. Analog of Fatou's Theorem for Differentiated Fourier Expansions

229

whence one can obtain, that the function L(r, x, XQ) are uniformly bounded for all r (0 < r < 1) and x £ j v . In consequence of (4.10) we have sL(r, ix;x JJi(x) , x00)) < eC, x (x) < eL(r,

(4.15)

where the constant C > 0 is independent of r and x. We avail ourselves of the decomposition

Hl«-

1 mI

L

.dPr(t,x)\ w Mx)== f/ Ut G L - ((t_« * - x 0■J-gr^*) 0) )£^if>Lt)dt Ja(*) t-x0\>5, , t6[c,d) t€[c,d) A I|| **"- ~z ox)0| | 11 J\t-xo\>S, t€[c,d] OX | I ft °ll ,dPr{t,x) OA d -^ JjW(x ^ ^ +)+J^^x). +/ | / «X1 L(*-*) _ ,X)) « * MV^L ( M jf)(x). —dx—\ J\t-i t-x0|><5, ^|t-x |>5, tt£E €ESI|| *t ~~ x || || ^x I

fcH <*>*

I'

"h

I /(*) I 51*

0

^From (4.12) one can deduce

•r<

1}

^ J I ^< 0 depends only <5 > 0, and consequently one obtains 4 1 ) ( x ) = o(l)

(r-H).

(4.16)

For the estimate of J2 (x) one can use the representation (4.2). Observe, that by (3.1) and (3.3), it can be derived / \f(t)\\ \f(t)\\Pk(t)\w(t)dt < CMk Pk(t)\w(t)dt

JE

(k (fc € Z+),

where the constant C > 0 is independent of / and fceZ+. So, by (4.2), OO -I

1

OO

*2>

2) f y Mfc + - i - +1 f V r*M A f cfc £ ( ;
00

OO oo

fc fc

f c + l)rkfcM Mkfc + ^ r frcfcM M* ( / + l)|A«,| 1)|AW||82 + (1 ( l -- rr)) ]^ (l(k f e ^^(/ fc=0

fc=0 fc=0

ii==00

OO

2 + f V rMfc M Z; ( / +Il)|A ) | A 2 «u,_ }) 1 _ 11||}, fc£ fe(£( fc=0

1=1 /=i

where the constant C > 0 independent on r. In view of (4.4), we obtain 00 OO

OO 00

) Mffcc + (1 ( l - rr) ) ^£ (^(fc+l)r 422\x) C(l * + l)rfcfcM Mfcfc}. }. (x)< C ( l --rr) )22■{. .{ l ++ $>]rf efcM r)£(' .n fc=o fc=0

1

fc=0

230

Fourier Polynomial Series in £*

By the aid of Lemma 3.4 and under the assumption (3.1) one has 2) J< (z) = 42\x) = o(l) ((rr-->>ll))..

(4.17)

The statement of the Theorem 4.3 are consequence of (4.11), (4.15) - (4.17), if /'(*o) = 0. In a general case, we put

[/(

f(x) = /(*) = /(*) /

(

*

)

_1_

-

f l -ffjf—f'{x (xo)pi(x), -ff'(xo)pi(x)\ (x0)pl(x)\ ) + +r,■— y*< 0)Pl(x), " /L ■

where p\{x) = a\X + b\. Differentiating of the function under square brackets, we see that it vanishes at the point xo and for the function l/aif'(xo)rpi(x) and, consequently, the means Wr(f;x) in this case are equal to r/'(xo). We have thus proved our assertion. Corollary 4.4.

WW\f,x): «;<«•»(/, x):==

OO

rkct>0\f)p'k{a,frx), fVc^/KC",/?;*),

fc=0 k=0

a,/3) 4c^if) (/) = =/

/.l i

f(t)p {a,0; t)w f(t)Pkk(a,p;t)w € Z+) a>p(t) dt < oo (k e a,0(t)dt
is a differentiated Fourier-Jacobi expansion for the function f, satisfying 0 df dt < oo and and I 'l/(*)l(l-*)° \f(t)\(l-t)a(l (l + t) tfdt

*) " -V*dt
/ , / 3 a1 / 4

1

Then W^(f,x) ^ f'(x0), Wl"'V(f,x)^f(xo), when the point M(r,x) tends to Mo(l,#o) ( - 1 < XQ < I) for any nontangential track T (see (3.5)), whenever f'(xo) is finite. In fact, it is not difficult to see that the orthonormal Jacobi polynomials satisfy conditions of Theorem 3.3 with (px(t) = (1 - ^ - ( " / a + i / ^ ! + t)-(/3/2+i/4) by (II. 3.22)-(II. 3.24), (II. 3.76) and Corollary II. 3.15.

Chapter 6

The representations of the trilinear kernels. Generalized Translation Operator in orthogonal polynomials 6.1

The representations of the trilinear kernel

Let /^(x), v(x), T(X) be measures in the interval [—1,1]. Associated with /x(x), v(x), r(x) are the sequences of orthonormal polynomials P = {pn}> Q = {<7n}j H = {hn} (n e Z+), respectively, that are, polynomials pnn(x) (x) == kfcnxnxn n 44--......,, qqnn(x) (x) == llnnxxnn ++ ......,, hhnn(x) (x) == m mnnxxn n ++ ...... have degree n with the leading coefficients kn = kn(dfi) > 0, ln = ln{dv) > 0,ran = mn(dr) > 0, and

Lpm /

1

/

// >il

Pm(x)p Pm{x)pn(x)

dfi{x) dfJL{x) =

Smnn I

qm(x)qnn(x)di/(x) qm{x)q (x) du(x) = SJmn m n1 >

(m,n {m,n eG Z+),

I °

hm(x)hn(x)

dr(x) = Smn nI

)

where Smn is Kronecker's symbol. The orthonormal polynomials satisfy the three-term recurrence relation

231

232

The Representations of the Trilinear Kernels

(see (II. 2.17)): xpnn(x) xp (x) = annppn+i(x) n + i ( x ) + unpn(x)

+ a n _ i p n _ i ( x ) , p - i ( x ) = 0, a_i = 0 1

xqn(x) = b6nnq^n+n i(x) _ig nn_i(x), 6_i = 0 x?n(x) (x) + 6 n _i^f _i(x), g_i(x) = 0, fe_i + i(x) + vnnqgnn(x) x/inn(x) (x) = c n /i nn+i(x) xh + i(x) + wnhn(x)

\

((1.1) i.i)

+ Cn_i/i Cn_i/inn_i(a:), _i(x), /i_i(x) = 0, c_i = :0,. J)

where, for n E Z+, the recurrence coefficients satisfy &n

_

tn

= 1,, Cc n n = »,. b On **nn++ll «n+l

ann === 1T and unn= = j

■JC

wn= = Wn

,

/ :

xp xp22nn{x)dp.(x), {x)dp.(x),

mn — = -■

m ^ lnn+i +1

*=L

vvnn==

((n e Z+)

xqxql{x)du{x) xql(x)du(x), 1

i

x/i n (x)dr(x), ((nn €€ Z+).

Note, that sup { a n , 6 n , c n , | i i n | , | v n | , | i i ; n | } < 1.

(1.2)

n€Z+

We consider orthonormal polynomial systems {p n } 5 {?n}> {hn} belong to Nevai's «M-class, i.e. the following relations 1 lim an = lim bn = lim c n = - , ~ 22' n—»oo n—»oo n-»oo n—►<» n-*oo

(n € Z+), which

lim tt unn = lim vnn = lim wnn = 0

n—>oo n—foo

n-*oo

(1.3)

n—>oo

hold. Introduce the trilinear kernels n

Vnn(x,y,z) =Vn(x,y,z',{p D (x, y, z) =£> {/>*}, {/ifc}) = k},{hto}, k}) n (x, 2/, *;k},{q

^Pk(x)qk{y)hk(z) Ip/bC^ft^JftfcW

fc=0 fc=o (n 6 Z+; - 1 < x , y , 2 < 1)

which possess the following reproduction property: for every polynomial n

7r = ^dkPkW ^fcPfc(x) *n{x) n(x) =

n ((n € Z+; - 1 < x < 1)

fc=0 fc=0

the relation

dii{x) == y2d^dkqkqk(y)h / *n(x)V 7rn(x)Vn(x,y,z) (xJyJz)dfji(x) € Z+;; - 1 < »-\
(1.4) v1*4)

6.1. The Representations of the Trilinear Kernel

233

First, we consider the simplest case. Let {Tn} (n € Z+) and {Un(x)} (n e Z+) be the Chebyshev polynomials of the first and the second kind, respectively (see Ch. II, §3). These systems belong to Nevai's .M-class. Putting c = cos a, y = cos/?, £+ = cos(a — /?), C- = cos(a + /?) we obtain n n i n _ 1±Y,T J2Tk(x)T yk(x)Tk(y)U (y)U (z) (z) = = rT (C-)U (c_)^ (z) (z)+ + \ k k k fe k k fc ~ 2 =i fc=i fc=i fc=i k=i

I>(a

Y,T k{^)Uk{z\ :(C+)^W,

and by the analog of the Christoffel-Darboux formula (II. 2.32) the sums of the right-hand side become a sum of two fractions with denominators z—£_ and z—£+. So,if the recurrence relations (1.1) belong to Nevai's class, one expects that the trilinear kernel has two peaks near z = £ - and z = £+• The next statement plays a fundamental role throughout this Chapter.

0.7V.P.5. P = {p },Q = {?»},# = {hn} L e m m a 1.1. For the general O.N.P.S. {pnn},Q {qn},H {hn} G Z+) /or for a// all x,y,z x,y,z € £ [-1,1] [—1,1] and n £ Z+ the following representation is (n € imliti valid £ nn(x,y,z), {z - C-XC+ - z)Vn(x, y, z) = A i4nn{x, (x, y, z) + £Bnn((x, x , y, z) + E {x, y, z), («-C-)(C+-2)©n(*,»,«)

(1.5) (1.5)

tu/iere C-= = xxyy- -v V / ((l i---x^2x2)2)()((l il- --yy2j/2)2),),, C C++===x^xy+y/(l-x*){l-v>), y+ +\ /V( (l l- -ax: 22 )) (( ll -- yy 22 )) ., , ((((
/I * (1.6) (1.6)

A n ( x , yy,, zz)) = Ann(x,y,z,P,Q,H) pn+i{x)q +i(z) ^n(x, ( i , j/, z; P, Q, H) = 2anbncn2a i(y)hn+i(z) n+i(y)h nn+ nbnCnp n+i(x)q

}

2 + [1 + 2 a n _ i 6 n _ i c n _ i - ((a^.x a 2 _ x + aa*) )

(£-1 + £ ) -" (4-i (<£-! + 4)]|>»(*)fr(v)M*) -" (#-i 4)fo»(*)*.(v)M*)

I

)f

+ 2a nn& Cn_ipnn++i(x)g _i(z) + 2a 2ann66nn_ic _icnnppnn+i(x)g +i(x)gnn_i(y)/i _i(y)/inn+i(z) +i( 6 nnCn_ip i(x)9 nn++i(y)/i i(y)/i nn_i(2) |

+ 2ann-ib -ibnnccnpnpn-i{x)q anaan+ ipip+n2+(x)q (z n-i{x)q n+i(y)h n+i(z) n(y)h + 2a -n+i(y)h n+i(z) nan+n 2(x)q n(y)hnn(z) -- ■bbOnbn+lJ b +ip {y)h cnnCn+ip .ipnn(x)q (x)g (y)/i„(; --- cncnc+ip +2(Z) n+2 n{z) nC nn{x)q 2{z). n+2 n+lPn(X)qn(y)h {y)h {x)q (y)hnn(y)h +2{z).nn+ nnbnn+ipn(x)q n+2 n{z)

J (1.7) (1.7)

234

The Representations

of the Trilinear Kernels

Bnn(x, B (x, y, z) z) = = J9 J5n(x, y, z; z; P, Q, H) H)

\

1 '. .j ? i bk » f c (c ( c -i)-k -)-ft*(fc. = g { 26fcCfc kccfe(af(a I) - h(h i-\ I)- (-i«*-y(4- ! - I)-(4 -*-yj (4-M- J)| ( fe - i)+ + * " 2 J fcfc(< 2 / -° \ -2)~ n-l

(

{[«

k

+ i

k

fc

k Uk

fc=o IL fc=0 I

J

D +h - 1

1\ / 26fc_iCfc_ifafc_ -l(^C/b_l ); -- 66/k-i fafc-i1 ■i(bk-: » 2 "" 2) 2bk-iCk-i [ak-i - - J + &fc-i ^c/b_i - - J - 6fc-i (&fc-i - 2 ) IN

+ +

i) - (4-1 - -\)~

- («2_i, - \ ) n-2 n—2

.-i^l -"(ct \ " '-I* ~ 4/ (4-1 - 7) J)

-I)

1

.,-iV

r ^ ) ^k(y)h ) ^k2(z)) I}Pk(x)q

rT i l l I a Cfc+1 Pk 2 qkk(z) 2 hk \p +2(x)qk+2(y)h (cfc+i pafc+if^ k -I\Q>k+i ; n ( f c f c + i 2 + ^ c J 2 o f e + i ^ + i - 2 ) + ( *+i ~ 2 ) ~ ( " 2) \ ^^ ^^ ^ fc=o fc=o LL J

cfc

+E 4 „n _- o 2

-\h

r

Tr : 2 c f cH+i i ((aafc+i + 11 + ^2 frfc 2Cfc 2c -~ 2 )) fc k+i n—2

DJ^

i)-(

>-D]

j 1x1 C fe+1 c+1 ~ 2 x h 2 (z) ) P * +\pk+2(x)q - V (bfk+i Cfc+1 ( ~ 2 ) ~ ~ 2 / ( ) 0 f c f ok)(y)h * +k+2 (*) - 5 ) <+ (<

-D-

+

I

[

fc=o fc=o LL J I T 1 1 I 11\ i1\ 1\1 6 fc x 2 /lfc + 2 2; PfcW^+2 (y)h )\Pk k (2{z) + ^ a J 2ifc&2&fc+i f c + i (Lcf ffccCfc+1 ++i1 "-" 22 )) +f ((frfc+1 *+i ~~ 2 ) " (flfc+i ( a j k +h 1 l -~" i2 ) K ( )^+ (2/) + ) 2 ^ fc=o L J I n-2 n-2

n-l n—1 f

-IJ-

■\h X<(-"i)-j)

f

-i

-i

-i "I

I

a 26>fcfc(Ck ((c*: ~- 2 ) + ( 6 * ~ 2 ) " ( o a>k+i + 5 1 1 afc + x1 \2bk *ffcc"2)|^ ~- 2 ) PH2Wgfc(y)/ifcW \P^2^k(y)hk(z) fc=0 fc=o I L J

-i)

-j)-

-a]

x 2ckk\a>k + &fc+l f(a c( k \pk(x)qk+2(y)hk(z) bk+i L 2c -- 2 )+(■ ) +f (cfc ( c * ~ 2 ) -~( f\tbk* - ~ 2 )) P*( )^+2(2/)M*)

I w „ . _A\

*>}

a + 2akfc(b ((ftfe jfc_ b -~ 2Pl) Pfc(a;)9fc(y)/»fc+2(2) + Cfc+l c fc+ i1 22a ~ ( ((«cCfc ^W^^)^2^) \ k - 2 ) + ( * ~ 2 ), 2)

2; ^ r*

2;

-\)\

(1.8)

and and nn £ „n ( x , y, y, z) 2) = = E £ nn(x, ( x , y, y, z; z\ P, P, Q, Q,H) = ^ )eefkc(x, H) = y, z) (x,y,*) fc=o fc=0 (n€Z+; - l < a f , y , z < l )

(1.9) (!- 9 )

6.1. The Representations 6.1. Representations of the Trihnear Trihnear Kernel Kernel

235

while ekkfb(x, (x,y,z) ( r,y,z) x , y, y , zz) (x y, ek(x,y,z;P,Q,H z)) == eefcekk(x, (x, y, z\ z; P, P, Q, Q, H) H) c(x, = 2w 2ukfckpkpPik(x)[6A pk(x)[b gfc. == 2u bbk-iq ^u {x)[b ( {x)[b qk+1 +(y) vkkfcqgkfc(y) n ( y ) ++ Vfc9fc(y) +4- &fc-igjfc-i(y)] k{3 kkkqq kk;+ 1(y) k-i(y)] k-iq k-i(y)\ +i(z) + u [c (z) 4w 4Cfc_ihfc_i(2)] i(z) h (z) cjb_ihfc_i(2)] c fkch^k+1 i ( 2 ; ) h (z) + c xc [cfch [Cfcfcfc+l k+fc+ )kkkhkkk(z) I +-r fc cck-ih -iiik-i(z)} -i\t)\ k

k

i(x) + 2[a ++ aa>k-iPk-i{x)]{2[b + &fc_iflfe_i(y)]ii;fc/ifc(*) 4--2[a 2[afcPfc _ip _i(x)]{2[b 4- b6&fc-i9fc-i(y)]n;fc/ifc(z) ![flJkPfc+iW afcf c_iPfc-i(x)]{2[6 _ipfc-i(x) fc fc+ +i(a:) fcp fc+ kfc-iq k-i{y)]wkhk(z) 1 fc +i(x) fcz— fc g fc +i(y) 4-iQ kqk+i(y) 4- w wkkhhkk(x)]{2[6 {z) 44- fcCfc_i/i cgfcfc+1 _i/ifc_i(z)]} + ^fc^(y)[cfc/ifc+i(2) (yffcc _i(2)]} )]} (z) 4{z) Cfc_i/i _i(2)]} w h (z) + c -ih . - aifc(^fc4i a (u +i (ufc+i 4)p +i(x)q (y)h (z) u p (x)qkk(z) (y)h (^fc+i + u u p (x)q (y)h k k k k k k +v>k)Pk+i{x)q (y)< fc k k?ife+i(aOflk(y)M*) k k k k(z) k k k kkQk{y)h k{z) " t*fcPfc(x)g*(y).

(1.10)

2

(tzfc_i + 4-1 / fe)Pfc-lW9fc(l/)^(2) (z) k(z) - u kpk uku)pk)p -i(x)q - - aafik-iiuk-x a_i(ufc_i c_ k)Qk(y)h k( kk(y)h fc k-i(x)q ki-i(uk }

(x)q - b bkk(v+i {z) h(vk+i 4- vk)pk(x)q (y)h ke(y)h 'vfk)pk(x)q cW +k+i(y)} i kf(z) ) ^ ^k(z) ) >k(v k+i + k(v k+1^ k+i - k-i(v -b b)kk-i(v -i(v k kk-i-i

44-

k(y)h (x)q (y)h (z) vvf.p Vk(x)q k(z) kk{x)q kk{y)h k[z) kv\p k{y)h

+i(y)h (z) k(z) - vlpk\ (x)q :)Pk{x)q (z) vkv)pk)p kkk-i(y)h kk-i{y)h k(x)q k-i(i

\qk-i{y)h - c*k(wk+i 4- wk)pWk)Pk(x)q Pk(x)q +i(z) (z) + k{y)h k+1 k(wk+i k(x)qk(y)h kk(y)h h k(z) - ck-i(wk-i

)qk(y)h wlp wkkp(x)q k(y k{z) k(x)q k(y)h k(z)

wkk)pkkk(y)h (x)qk-i{z), 4- >k)Pk(x)q -i(z), kk(y)hkk-i{z),

are £/ie recurrence coefficients here { a n } , { 6 n } , {en},. lu {unn\.}> lv„\. { v n } , lw { wm\n } (n (rc € Z_i_^ (nG Z+) are for the polynomials pn(x),qn(x),hn(x) (cf. (1.1)). Proof. By (II. 2.19)

x2pnn{x)

= anran+ipn+2{x) X) 4= + (a2n_x 4- ao?2 n 44- Un)Pn(z) tin)Pn(^) + an(u (tX TX n n + u n+i)p n+i(x) n +l)Pn+l(^) + u„)p u n ; n _i(x) _ 22 aannn__ip 4i ( >u nnn__i: 4Un)Pn-i(z) _iippnnn__2(x) _22^( xx;) + aann__i(iz L(x) 4- arni _

(n €e Z+;; p_i(x) - 22((x) x ) = 0); (ne pp-- i W == 0, p_ 0); 222 = b xX qgtfnn{x) (x) b +iq (x) (v 4vn+i)q 4(6*_i + &n &2 +4- v2)g n = b +ig (x) b b +iq (x) 46 ( v + V +l)gn+i(x) ^ ) ^ nn((x) x) 0 + fc (v + vV + (&n-l n n+2 n n nn+i)q n n n n n+27 nn ^n n + n +l(a?) + n +l)g n +l(a?) + (&n-l + * + 6 _i(v _i )g 4- &-2b -iqn-2(x) 4- 6 nn_i(vv nnn_i -ign-2(x) -iq (x) + _bn2n_-2b fe - i 4+ vvnnn)q ) n-i(x) n _i(x) 26 n-2(x) nfr nn-l9n-2(^) (n e€ Z+;; <7_i(x) g_i(x) g_ 0); -2(x) 2 (x) = o); gq-- i ( z ) == 0, a_ 2

2 2 2 T 22h Ih I n.n_(x) fi^ = = cC /•_/•_ . 1 i/>._ . «fi?^ f in»n_ 4-I- W ID_nn+l)/l .+ 1 Mi._ . i *) f i * ^+-4-((Cn-l+<2+*tt ^/)._ xX x) 4--UC c/»nn(w it; i)/i (x) +2(x) (w 4(r - 1 .+A-Cnr + A- ^ii) M * fir^ ) cTnC c n+ih + i(x) +i/n nn+l(x)

+ -i nn__i 4+ w n )/i wn)h 4- ccnn-i{w i (nw n__i(w n-i( ) 4- C n _ 2 C n _ i / l n _ 2 ( x ) n _i(x) _ 2 C n _ i # l n _ 2 (x) = 0, /i_ 2 (x) =C n0).

h (n € Z+; ' h-i(x) (nr* C.I..

(n € Z+; ft_i(ar) = 0, /i_ 2 (x) = 0). So that So that 2 lk(y)h - xa22 - y2 - z2z2)p )Pk{x)q pkp(x)q (1 4- 2xyz — )p k{z)k{z) k(x)q k(y)h k{z) k(z):(z) = k{x)q k(y) k{y)h k{ k(y)h 2 2 2 (1 42xy;z x y z )p (x)q (y)h {z) = p {x)q {y)h k k k k k k{z) 4- 2a 2a b cb vc +i(x)q +i(y)h +\(z) y)h 4- 2a', b c -ip2a+i(x)q b c -iv .{x)q + (x)q i{y)h +i(y)h -i(z) + p k+i(x)qk+i(y)h ^ i +i(z) + k kk kk kk k

k

k k

k

k kk k

k k k k

k k+ 1

kk 1

k

4i(z) 4- 2afc6fcCfc-iPfc+i(x)gfc+i(y)/iA:_i(2) +i(x)9fc +i(y)h .\{y)hfc+ 4- 2ofc6jbCfcPfc 2afc6fc_iCfcPfc kVk+\{x)q k+l{z)+i(z) +i(a;)flffc_i(i/)/ifc k-\{y)l 42abfc 6- fcc-ic p fe+ i(x)g i(z) 4fc+i(x)q fc\_i(y)/i (y)hkfc+ -l(z) [y)hk+1+(z) -i(x)q i(2:) ++ 22afc_i6fcC 2ak-ibk:cpfckkppfc-i(x)gfc 4- 2a 2afc6fe_iCfc. k-iPk+i(x)q -i(i k-kl-i(z) k+i(y) +i(y)/ifc kk k 1 k-ip k k-1k(y)h 42ofcfefc-iCfc-iPfc+i(x)gfc-i(y)fcfc-i(z) 42a fc_i6fcCfcpfc-i(x)gfc+i(y)/ifc+i(2:) -i(z) _i6fcCA;_iPfc_i(x)^ i(y)/iib_i(2) + 2a 2a -ib c-ip . -i(x)q +i(i Jk+i(y)h k

k kk k k fc k + i^;f4fc-l^; ^-^ajk-ic/fcCjb_i//fc_i^;gib + + 2a-ib fc _i6fcCA;_iPfc_i(x)^ + i(y)/iib_i(2) + ickk-i(x)q pk-i(x)q 2ak-\b\c -\{x)q + 2a 2afc_i&fc_i< 44- 22ajb_i6fc_iC k-i(i \k-\(y)h k-\c k-fck_ip k-\p k- qk-i(y)hk-] k k-ickp k-i(y)h k+i(z) k+\{z) fc _i(x)gA:-i(y)^fc-i(2)

4- 2ak-ibk-ickpk-i(x)qk-i(y)hk+i(z)

4- 2ajb_i6fc_iCfc_ipfc_i(x)gA:-i(y)^fc-i(2)

236

The Representations

2

{a\_ +f a:(x)qk(y)h hk(z) 4)pk(x) x T k)pk(x)qkk(y)h k(z) ) --- \u (4^1 uk)Vk\<*')Hk\y}">k\*) k_1 (y)hk(z) hk(z) ak-k2.ak{z)qk{ -ipz)Qk(y)h -lPk-2(z -\p afc-iPfc-: k(y)hki k-2(z)q k(y)h k(z) k-2{ a k-ip k-2aka W 2 2 2 bl)v bicby^vic(x)ai bkkb bkk+ip (z) )hk{z) (b 4--b& (x)Q (z) -- (1(&2i -iPk(x)qi b+ip ipk(x)q (x kk(x)q k+2k(y)h cu.o(y)hk(z) k2(y)hk(< k-_i x -+ kk)p k(x)q k(y)h k(y)h kk(z) +i :+2(y)/ kb k+ k(x)i k)pk(x)q k( >k(z) W-(44£-i ) b{x)qk(y)h

-- u, akkO>k+lPk+', ak+ipfiPfc+2(a (x)qk(y)h (z) y)hk(z) k+2ik+2{x)q kik(y)hk( )
of the Trilinear Kernels

- bkkk-2b -ip {z)q -2)Qk-2{\ (y)h ckckk+l hk{z) cckkck \-iPk(x)q --22bbkbk-ip %-\Pk{z -2kb{z)q -ipkk-(2k(y)h ckp +ip +22{z) (z) » - CfcCife+iPfc (k(x)q (y)hk(z) kkiPk(z)qk k{z) k)Qk{y)h k(y)h k-2(z) k{z) k(y)h k+ k-2(y)h k(x)q k+2( kk-iPk
where ek(x, y, z) is definedI by by (1.10). (1.10 W e regroup similar terms of t h e last relation ' (1.10). V 2 (1 44- 2xyz + 22xyz x y z -—:xx2z22 - 2yy'2 -2 z222)p )p pklk(y)h (x)q Pk(x)q k(x)q f c (x)9i k{y)hk{z))hk(z)0 = pfc(: k(y)h k(z) k\ k(z) k(y)i

+- 2afc6fcCfePfc +i(x)g +i(y {z k++i(z) 2akkcbkkpc:kkkp+\{i pp2a 2a kb 4i(x)g i(y)/ifc i(z) +i(y)/i/b+i(z) ikbkCkPk+i(x)qk+i (z) fkkc +\\ kbkc+kp k+i(x)q f c+i{y)h k+i(y)h k+1 fc+{y)h k+i 4-2a/k_i6fc_ic + fc_lCfc_iPfc_i(x 2ak-ib -i{y)h 2akk-ic -i(x)q - i/k_i6fc_iCjfe_iPik_i(x)qfib_i(y)/ifc_i(2:) ,k2a -ib -\b -\cfkc._iPik_i(x)^_i(y)/ifc_i(2:) k-ick-ip k-i(x)q k-i{z) k-i(y)hk-i(z)1 kk-i(y) kkk--\ 22 2<4) k(y)h ~ 4- ((4-i+4)]Pfc(*)kQ>k+\Pk+2{x) + 2a -ic b -ip -ic +i(x)q -i -i(y)h {x)q -i{z) -\(y)h -i(z) [z) a a a a +i +ip a + a +ip (x)q [y)hk(z k fkkck_iCib_ip i(x)^_i(y)/iA:-i(2:) k k k k f c + {x)qk-i(y)hk kk k kk kk kk+2(x)q k k k+2 k(y)hk(z) ik(y)h k 2a kC -ip -i(x)q (x)q bfkkcb(: . )ippgk)(x)q + 22ajk_ifefcCfc ak-ib j-ib k k_ckickf-i f ck-i(x)qi _ i kP+i(y)h A :{x)qk+i(y)ht [x)q - lt)qk+i(y)h ( k^-i(z) g f+ci(y)/ijk_i(2:) + lk(k-i(z) 2-i(z) / ) ^ - l ( [z) 2 ) - b6fc6fc+iPfc(^)g)k+2(y)/ifc(^) ^kb6bbk+ fkkb cip b+k+k+ Pip ([x)qk+2( ^ k + 2k(+ y2)(y)h / i f ck((z) ^) v)hk(z] 4--2a 2ajk_ifefcCfc_iPA;-i(x)gfc ki>k-ib kek-ip kC kii} kk+1 k+ k)+i(y)h cf kc-ip 2{y)hk ll\h.L. . r» (x)au-^ (v)hi 42ojb_i6fc_] +- 2a 2ofc-i6fc-ic -ib -ic k-\c p p (x)q -\{x)q\ pfc_i(x)g -i(y)h +i(z) _i(y)/i i(z) CfcCfc ip (x)qr (2/)/i 2('Z) 2a -\b -\c [x)q s)q -i(y)h -i(y)h +i(z) +i(z) c c c ip c +ip (x)q c c (: +ip (x)q (y)h + (z) ik-\b k k kk-\c k kp k k kk fc fc+ [z) fc+k 2 k k1 fc k kk k k+ kk u+ kkfc kk k fc khk+2(z] kkck+

+-eke(x,y,z). (x,y,2). (x,y,2) f c(x.v.z). For the proof of the formula (1.5)-(1.10) we consider the following "principal" terms (the other terms are treated in a similar manner): n

n n

S i == ^2pk(x)q Pk(x)q k(y)h k(y)h k(z) k(z) E fc=0 fc=o fc=o

+ 2^2 24 aak2ib1*T,a i kckp(x)q (y)hk+i(z) kckpk+ kb k+i(x)q k+i k+i(y)h fc=0

n

n n

fc=0 Jfc=0

fc=0)

+ 2] ak-ibk-ick-ipk-i(x)q -i(z)-fc_i(2) - J2[(al-i +4) + (bl-i+bl) 2^afc-i6fc-iCfe_ip k-i(y)h Jfe_i(x) gfc_1k(2/)/i + (y)hk(z) + (4-i(4-l+4))Pk(x)q + 4))Pk(x)qkk(y)h k (z) n

E'

^2okbfck6ibCfc_iPfc ^22 = ck-ipk+i(x)qk+i(y)h S = ^>22a +i(a:)^ +i(y)/ifc-i(2:) k-i(z) fc=o fc=0 fc=0

n n

E<)

- '

-^ck-2ck-ipk(x)qk(y)hkk-2(z) -2(z) Ck-2Ck-iPk(x)qk(y)h

fc=0

and n s

n n

^ajfebfc-iCfc-iPjk+i^fc-i^/ifc.^^)i £ 33 == ^22o>kb k-iCk-ipk+i(x)qk-i(y)hk-i(z) fc=0 fc=o fc=o ik=0

^akak+ipk+2(x)q a>kQ>k+iPk+ (z). k(y)h 2(x)q k(y)hkk(z).

Using Abel's summation by parts and the initial conditions p-\(x)

= 0, q~i(x)

=

6.1. The Representations

of the Triline&r Kernel

237

0, /h-i(x) i - i ( x ) = 0, we have E +i(x)qn+i{y)h CnPn+i (•*) nbnCnpn(x)q n+i(z) Eii = = 2a„bn2a n+i (y)/»„+i 2 2 + [1 + 2 ao n _ 1i 6 n _ i c n _ i1 - (a*_i (62_! ( a * . ! + aa*) ^ + b2n) n) - (b

~- (4-1 (Cn-l + +

cl)]pn(x)qn(y)hn(z) 4)bn(a:)9n(y)ft»(«)

n - l (\ 1\ g { f 226 6fcfCfc(. i h(ck)b+ 6 f c"( co 1f) c-h ) - b6 f c ( 6\ f- c - i ) - ( 4 - i ) - ( a ? -M i ) I| c c(af kc ( a f c -+ ~- i>k(h )-(«>-2) ~2) + k **--2)-(* fc=o I.k LL fc=o

+

A-\ ?. *-yj)]J *-i)j

-I)

+£ h +

1\

+

,-I^ - {4-r .-1\ s-i)

r

26fc_iCfc_i ._i fafc_i + bk-i\Ck-i *-i(b -i c f c - i-- 22 )) - f r6jk 2&fc_icfc_i (a fc _i - \ -) J + 6fc-i (cfc_i - 2 ) " " 1 Vffcc _ 1 " - 52 ;) -~ v * " 1 ~ 4 /

-te .-^1ilk c C ^ M M * ) - \(H4f c- -1l - j ) f P f(*)9fc(y)M2) - - * ) ) '

Next, note that p-i(x) = pp_ p_i(x) - 22((x) x ) = 0 "j g_i(x) = q. g_2{x) q^(x) = =0 \ 2 (x) =

(xG [-l,l]), (X €[-1,1]),

/i_i(x) = /i_2(x) = •0) J so that we obtain n-2 n-2 r 1> E 2 = 2annb6nCn-ip Cn_ip +i(x)q i(x)g +i(y)h i(y)/i -i(z) _i(z) + + ] T c 2afc+i 2a +i(& +i -J n n n+ n n+ n n fc fc fc (fcfc+i fc=0

-M I

fc=o L \pk+2{x)qk+2(y)hk(z).

+ (ajfc+i " 5 ) " (Cjk+1 " 2 )

\Pk+2(x^k+2^hk^'

a

--j--

- ^ + (<*fc+l fc+1

)]'

■i)J'

(Cfc+i -

-2J

In a similar way way S3 = -anann+ip +iPn+2(x)qn{y)hn(z) n+2{x)qn(y)hn(z)

n-l n—1 r _I> + ^ a f e + -1 + i 26/b 26fc(c J + (& fc(cfe) + ( &fc *-2) --J fc -2;+ n fc=o

--

L

4)

l Pk \Pk+2{x)qk(y)h \pk+2{x)qk(y)hkk(z). (afc (z). (flfc - 2 )) *-2>Jj

i)\

._!>!

The formulae (1.5)-(1.10) are a consequence of the last three formulae for E i , E 2 , E3. We have proved our assertion. Corollary 1.2. For a general I O.N.P.S. O.N.P.S. P = {p„} {pn} (n e€ Z+) Z+) the following representation is valid n

(z - C-XC+ -+ -^)y z) ^2pk(x)p = {x, y, z) + B„(x, Bn(x,y,y, z) + En(x, y, y, z), lpk(x)pk(y)pk(z) = An(x, k(y)Pk{z) (z-C-)(C k=0 k=o

238

The Representations

of the Trilinear Kernels

where y)Pn+i(z) An(x, y, z)) == An(x, P) == 2a*p 2a^p An(x,]y, z;;P) * a n P nnn+i(x)p ++l(^)Pn+l(y)Pi ii(x)p ( z nn+i(y)p + i(y)p nn+i(z) + i(z) 2

+ + 22a a nin_-_l!x ~~ -- 3(aJ_! 3(aJ_! a an2)]p (x)p (y)pnn{z) {z) n)]p n(x)p n)Pn(y)Pn( ( « n - l++ +" ^0>l)]Pn(x)p + [1 [1 ++ n nn(y)p (y)p [p n+ \pi(x)p i(z)) + 2a 2ann-ial . xbn+ ^ ) P nn -_i(y)p ln(-i(y)p y ) j n+i(z) n+i(x)p n+ n+i(z) s )4++pPp„-l x(x)p„+i pnn++i1( (y)Pn+l + p + i ((x)p x )Pn+1< p nn++ ii (y)Pn-1 {?)] + pPn+] (y)Pn-1 (^) ) P n - l(2) W »)PP y(y)p ) n +l (*)] n +i nn_-i1( W 1 [Pn+2 )Pn(y)PnO -a - na Onan+l[Pn+2(^)Pn(y)Pn(2) OTn an+l[Pn+2(^)Pn(y)Pn(2) hi(z) 4O^TJ: On(x)p + Pn (^)Pn+2 (^) + Pn (y)Pn+2 {z)\, ( z ) p n ++2(y) 2 (y )Pnn(*)+Pn(: (*) ((^)p x ) p nn(y)p (y )pn.n +2 (*)],

Bn(*, y, ^j ^) == r>n\x, B„(x, y, y, z; z; i^; P) &n\x, y, n - 1L r

-

= £ K -2)V;)(24fc=o L fc=0> L

1\

ak

2o fc - 1)

_I\

2 1 -2a f c _i-l)]p i fc [x)Pk(y)Pk(z) + (afc-i - ; )(2al_ n-2

+ 2^ fe=0 fc=0

"i)(

-

l\r

a fc a fc+ i1(ofc+i - - J *[Pfc+2(aOPfc(y)Pfc+2(2) afeOfc+i( (ofc+i +2 (2) i " 2 >l[Pfc+2(x)pfe(y)pfe

+ Pfc+2(^)Pfc+2(y)Pfc(2) + pfc(x)pfc+2(y)pfc+2(z)] n n -- 1l

-

a + 22'^2^afcafcafc i f afc.-IV» - )[Pfc+2(a:)pfc(y)pfc(z) +1 fc+i(a*: 2)\Pk+2(x)Pk(y)Pk(z) fc=0 fc=0

**

2JW

+ Pfc Pit (x)Pk+2 (z)Pfc+2 (y (y)Pk )Pfc (^) (z)++Pfe Pfe{x)p (*)pfc (y)Pfc+2 )Pfc+2(^)] (*)] k (y

and

n

£ nn(x, E (x, y, z) =En{x, (x, y, z; P) = ] P ekfc(x,y,z) (x, y, z) fc=0 fc=o (ne ( n € Z+; - 1 < x , y , z < l ) while

e fc (x,y,z) == Cfc(5 e ffcc (x,y,z;P) ,y,z) = = 2w 2u^fcPfc(z)[afcP pfc(x)[a pfc(x)[a +i(y) i(y) L(y)+u + ufcfckpfc(y) ppkfc(y) (y) _ip -i(y)l[a ukfcfcppfc(z) pk(z ;y) + aafc_ip _i/;fcfc-i(y)][a afcPik + i(2) + u fcfc fcfcppfcfc+ fc fc fcpfc+i(z) fc(z) k-iPk-i{y)][akl )[afcPfc+i(y)+' i(y)][ofcPfc+i z pfcfc+ +i(j +-h2u 2uk[a afc-iPfc-i(z)]{Pfc(y)[afcPfc+i(z) + afc-iPfc-i(2 afc_ipfc_i(z)] 2u +■flfc-lpfc-l ak:-l(«)]{Pfc(y -ip(«)]{] pk+i(z) )[akpk+ )]{Pk(y)[a fc-i(z)] + 2ufc[< fc kPk+\{x) ■i(x) lW + ajtfc[a fc fc k-i(x)]{p k(y)[a k>k+l( ) .ip fcfc-i(z)]H-2w + ^fcPfcW ukpk(z)z) ++cafc_ip -i(z)] ++ 2tifc[afcPfc+ 2ufc[a p i(y) + afc_iPfc_i(y)p (z)} UkPk(z) afc_ipjfe_i(z)] 2ufc[afcPfc+i(2/) p i(y) + a -iPk-i(y)Pk(z)} afc_ipfc-i( 0 a -iPfcfcfc fc+ fc fc+ k i(y)pk( fc + k-iPk-i(y afc-] >k(z)}

x ')Pk{y)Pk(z - afc(^fc akfcb(ufc (u (uk fc +-+ Ufc+i)[p u + ■Pfc(»)pfc+i(i pfcp(x)pfc+i(y)p OPk(z) fc+i(£)pfc(y)pfc(z) fck(z) k+i)\pk+i(x)p k(y)pk(z) k(x)p k+i(y)p )p/b+i(y)p Ufc+i)[pi -fc+i)bfc+i( )PA Pfcw+pfc(^)p/fc z) f 2 + Pfc(aOPfc(y)p*+i(*)] Pk(x)P) )Pfc+iW]-3w - 3u 3u|p (x)p fc (y)p WPik(y)Pfc(^) kpfc k(x)pi fc (z) k(^)pik(y)pik+i( ) )p*W >-i(u . f i-i i. _1_ i t )\p MV». - (f\nr\. f*i\n. - I -a -i(u -- knaajfc-i(ufc_i + (z)+ ( * \i+PfcWpfcPfc(x)p 1.. +w/b) \(v)Vh( >k(z) kb_i(ujfe_i k k k)\p k-i(x)p k(y)pk[yJpfcW fc _i(y)p fc (z) i(x)Pk(y)Pk k-i(x + lu+u Ufc)[pfc_i(x)p^ + PibW

+ pb(^)Pfc(y)Pfc-i( k(x)pi Pk(x)Pk(y)Pk-i(z)]> + Pfc(^)Pfc(y)PA:-i(^)]. Here {afc}, {i/fc} (he Z+) are the recurrence coefficients for the polynomial system P = {pk} (k <E Z+).

6.1. The Representations

of the Trilinear Kernel

239

The following curious result can be immediately inferred from Corollary 1.2. Corollary 1.3. For an O.N.P.S. {pn} (n G Z+) with respect to an even weight function w(x) = d/j,/dx on [—1,1], we have the representation for all x £ [—1,1] and n € Z+: 2 ( x - l ) 22(z

+

n

1\

+

yk(*) )£< ^fc=00 fc=0

( 4n _! . ! + a£)]p3 (x) = 2a„P n + 1 (*) + [1 + 2<4_i 2a n _i - 33(a a n )]p n (z) ean-xa^p^^-,.) _ 3o a ip„ (a;)p^(a;) + 6a„_io p (a;) 3o„o ip (a;)p (x) n n+ +2 n n+1 n+ n+2 n n-l r n-i 1 1> + afe | ((22 o J -_22oaffce--l1)) (afc-

+£E

[(»(

fc=o fc=0 L

-2) * -DP

- 2at;_i - 1 ) \PI(X) (ak-i1. -- ^ 2 )(2ai_!^

+

T

n-2

1

2 ^-)Pk(x)p + + 66' ^ |afcGfc+i( afcafc+il(afc+i (a fc +i •-- 1l)pk(x)p k+2{x) k+2{x) >£<

- 2>"

k=0 fc=0 n-l

x + + 6 6<£< ^ a * a fc+ii(a (aakfck-^) ~ Pk'i)P )Pk+2(x). k(x)Pk+2(x). ( -j)Pk( f.i- —

—

fc=0 fc=0

In fact, since w(x) is even, then unn = 0

(n (neG Z+)

(cf. (II. 2.33)), and for ►rxx== t/y == zz (z - C-)(C+ - * ) = ( * - l) 2 (2x + 1)

( --1! < *x << 1).

The following problem often arises in mathematical physics and in the rep resentations theory [Dougall, 1919], [Rahman, Shah, 1985]: to determine the sums of the series OO

£ oo

)hk{z) lkPk{x)qk{y ^2lkPk(x)q k{y)hk(z)

(-Kx,y,z
< 1)

(1.11)

( - 1
(1.12)

fc=0 fc=0

and

00

oo

7 = {{7n,7n 7n,7n e R R 1\ , In 7n ±7^ 0,0, Un G€ ZZ+}, +},

(1.13)

and 0>n}, {pn}> {?n}, {<7n}> {{^n} (rc € Z+) are the orthonormal polynomial systems with M (™ x <<1 1), respect to the measures /x(x),i/(x),r(x) ( -—1 1<< X ) , respectively.

240

The Representations

of the Trilinear Kernels

Let n

>±l(h (h-- 2 Mrr

n

ii

ll

^n = l +

SI)* 2 i T r 21; /

k=0 *\

y\uk\+\vk\+\wk\)

1(n 6 Z+),

fc=0 k=0n

(1.14) (n 6 Z+) are the coefficients of recurrence formulae

where aklbk,ck,uklvklwk (1.1). The estimate

(1.15) 7V = o(n) o(n) (n (n -> oo) Nnn = holds by virtue of a limit equality (1.3). In view of Lemma 1.1 we have Lemma 1.4. Suppose that the O.N.P.S. P = {pn}, Q = {qn}, H = {hn} (n € Z+) belong to Nevai's Ai-class and satisfy the following hypotheses: there exist positive functions
(1.16) (1.16)

\qn(x)\ (x) ^(x) |fl»(x)|

(-1 ( - K< xx << 1; n n €G Z+; Z+; tf(x)€Lj(-l,l), f/,(x) € Lj(-1,1),

(1.17) (1.17)

\hn(x)\ < 0(x) |ft»(x)|

(-1 ( - K< zx << 1; n n €€ Z+; *0(x) IrJ(-l, ( x )€€ L j ( - l , l1). ).

(1.18)

Moreover suppose that for an arbitrary sequence (1-13) the relations oo OO

£ w>ik|A(7fc)l
( n(neZ € Z ++))

(1.19)

fc=0

and lim 7n7Vn = 0

(1.20)

n-*oo n-»oo

are valid. Then the following equation

(* - C-)(Cf - *)^

00

Pfc(x) Vk Pi

gfc(y) hk(z)

7' (y) 9{z) ^* * = 0;

° _ = ^ ) &{7k)[A A(7fc)[ifck(x, (x, y, y, z) z) ++ £Bfcfc(x, (x, y, y, 2) 2) ++ E2S y, z)] z)] k(x, fc(x, y, fc=o k=o OO

(1.21)

holds, where the series on the right-hand side of (1.21) is uniformly and absolutely converges for x,y,z€(—1,1), and i4 f c (x,y,2?) Ah{x y z) == =

''

B Bkk(x,y,z) {x,y,z)

1 -i4fc(x,y,z) y(g)^(y)g(z) iPfrMyW*)'4i*l f c ( a r , y , z )

1 ==■^^e^Bk{x,y,z) —B (x,y,z) = v(xMy)0(z) k

[xmyw

_ 11 E Bkfc(x,y,z) (x >y> z) = - _ — ^ ^ (—E a rk{x,y,z) .y.z)

x , y , z<< ((-1 1 << X,ViZ

1, fce Z+), *' k e Z + ) '

( L(1.22) 22)

( - K<x x,y,z , y , z <
(1.23)

(-1 ( - K<xx, ,yy,,zz < l 1, , A:G A: € Z+), Z+),

(1.24) (1.24)

"

6.2. A Generalized Product Formula

241

here Ak(x, y,z),Bk{x,y,z),Ek{x,y,z) are defined by (1.7)-(1.10), respectively. Proof. By the aid of Abel's summation by parts one can obtain n n

\lkPk{x)q ^2lkPk(x)qk(y)h k(y)h k(z) k{z) / ^ lKfK\

n n—1 -- 11 n

=7 + ^ ^A(7 =-)hPk(x,y,z) A(7fcfc)Pfe(x,y,z), )Pfe(x,y,z), fc P fc (x,y,z) + A

fc=0 fc=0

fc=0 fc=0

(1.25)

s,y,z < 1, nnG€ Z+), ((-1 - 1 < i.J/.z where the kernel Vk(x, y, z) is defined by (1.4). It follows from the formulae (1.5)(1.10) that oo

(*-<_)(C+-z)

gfe(y) M * ) :wfePfc(^) (x) v(y) *(«) V h—r-:

-r-T-

-. >

fc=0

= lim 7 n [i n (x,y,z) + £ n (x,y,z) + £ n (x,y,z)] n—►oo n—►oo

n -1 n—x

z) + BBib(x,y,z)+Bfc(x,y,z)]. H- lim (x,yy,l z) fc(x, k(x, y, z) + £ fc (x, y, z)]. im VA(7fc)[A V AA(7fc)[i ( n—>oo ' n—►oo

* fc=0 fc=0

In view of the relation (1.2), (1.16)-(1.18) and (1.20) the first term of the last sum vanishes for all x, y, z € (—1,1); the second term can be estimated by oo

cf>fcJVfc|A( |A(7*)| 7*)l C

fc=0

and from (1.19) we derive, that the series oo

£a

A(7fc)[ifc(x, (x, y, y,z) z) ++ B£kfc(x, (x, y, y,z) z) ++ E£kfc(x, (x, y, y,z)] z)] J2 ) A(7fc)[A fc=0

uniformly convergent for x, y, z € (—1,1) in accordance with our statement. Lemma 1.4 is proved.

6.2

A Generalized Product Formula

Let the polynomial systems P = {pn}, Q = {tfn}> H = {hn} (n £ Z+) be orthonormal with respect to the absolutely continuous measures / x / x _ d/x da w(x) , w(x) = = wi(x) wi(x) == —, rfx'

rfi/ . x dv w2(x) (z) == —, w 2 dx" -, v

_drdr , „ ^x f. x\ ( - lK x < 1) w w33(x) (x) = =~~—dx (-1 < x < 1)

respectively, and satisfy the conditions (1.16)-(1.18). Put (the kernel P n (x,y,z) is defined by (1.4)) 1 X> (x,y,z) := - , v w w ^Vnn((x,y,z) T>nn(x,y,z):= x,y,z) (*My)0(z)

((x,y,z<E x , y , z € (-1,1); (-1,1); n e€ ZZ+) +)

(2.1) (2.1)

242

The Representations of the Trilinear Kernels

and introduce the Lebesgue

quasifunctions

\T>n(x, y, z)\p{z) dz i((x, a : ,y eG ((--11,,11));; nn €€ZZ+), +), |P„(x,y,z)|p(z)dz

Ln{x,y) {x, y) := /

= /

>

(2.2) (2.2)

where p{z) is a given weight. In this section we get a Generalized Product Formula. First, we obtain the estimate of Ln(x, y). L e m m a 2 . 1 . Let the O.N.P.S. P == {p {Pn}, = {h {hn} € Z+) n}, Q = {qn}, H = n} (n e satisfy (1.3), (1.16)-(1.18), and let p{x) he a given weight. Then for almost every x e (—1,1) the following estimate n + l ^Lnn{x, ( x , yy)) = O Oxyx (l)N lna ^ — l / (l)iV n n In

,/zxed; n e£ Z+) - K< x, y2/ < 1, y is fixed; Z+) l((—1

lvT iv

(2.3) (2.3)

n

holds, where Oxy{l) is independent ofn€ Z + . Proof. It follows from (1.2), (1.4)-(1.10), (1.14), (1.16)-(1.18) and (2.1) that the estimates f

r

f o r a l l x , y , z e ( -(-1,1)(2.4) l,l)(2.4)

nn ++ l1 N

i Vn

ff o rall /*" " —-for a l lxx, y, y, z, 2€G ( --1l , 1l ) , - \ "|(*-C-)(C+-*)I T l(z-C-HC+ -z)\

\V n{x,y,z)\ > 0(2.5)

{

are valid, where the constant C > 0 depends neither on n € Z + nor on x, y, z e (—1,1). We divide the integral on the right-hand side of (2.2) in two parts

/ >L

|Pn(x,y,2)|p(z)dz= / \Vn(x,y,z)\p(z)dz

^

-

= / egralJ-l and estimate only the integral -1

pi

rxy

l

In(x,y)= In(x,y) = I

Jxy

r

\f>n(x,y,z)\p(z)dz+ 1

\Vn(x,y,z)\p(z)dz

\V n(x,y1z)\p(z)dz |Ai(z,2/,2)|p(2)ck

/ |^|A,(z,y,z)|p(z)
Jxy f

+ /

Jxy

\Vn(x,y,z)\p{z)dz

(x,t/G ( x , y e ((-- l1,, l1)) , nn € Z Z+ +) ,

ofthe t h integral because the estimate> of xy

/

\Vn{x,y,z)\p(z)dz |X> n (z,2/,2)|p(2)ck

( x , 2 / € ( - l , l ) , n<EZ+) neZ+)

can be obtained exactly in a similar way. At first, consider the case 2 y . C+ == xy (l l--* x2 2))(l ^/ li1, C+ xy++Vx (/f{T^T( (( li ---2yy/2 2)) )V , , i.e. i /x^y.

Put Put ), /(A i (- l*- 2l 2) () (l l- -2J // 222)), *n =" -nK ^+V. lj-Va-^Cl-y 4.--J

n+l

N

,.

n e„ ^ Z= T 1nT+T l^■(1-C+). -Cf). n +1

(2.6) (2.6)

6.2. A Generalized Product Formula

243

By the relation (1.15) <* o(l), eenn == o(l) o(l) (n (n -> -> oo). oo). Snn == o(l), We divide the integral In(x,y) in three parts

f

rC+-*n

JIn(x,y)= n (x,y) = /

Jxy Jxy

+ /[' +

+

i

\Vn{x,y,z)\p{z)dz+ |©„(x,y > 2)|p(z)cfa+ I/

I|P |Dn w(x,?/,2:)|p(z)d2: («,y,«)|p(z)(fe

C+-*n n

^ ^ ^ ^ ^ ( ^ ^ ^ =/ ^l£\x,y,z) (^^^^+ / ^I
and estimate every term separately. By the aid of (2.4) we obtain rC++en

2 |4 )(x,y)|
(2.7)

n

'C+-*n

p(z)dz Z + ; - 11 < x,y < 1). p(z) dz: (n ( n € Z+;

By Theorem I. 2.5 and (2.6) wee have h /<++«»

JCk+Sn

JC+Sn

Consequently,

W nl - * p(z)dz < CXiyN + 1 1 nn +

|/£ >(z,y)|(x,y)|

(2.8) (2.8)

where the constant C > 0 is independent of n £ Z+. Now by (2.5) one can derive the following estimate p(z) dz

|J<3>(x,y)|
;++£n,|(*-C-)(C+-*)I -A'

CNn

22 / ' -V(l-x V(i-^ )(l-!/ )(i-y h'C++«n ) ( i - y 2 )) )i ++en 252!

yo^ :

p(z) dz

z"- c+'

Z

where the constant C > 0 is independent of n 6 Z+ and x, y € (—1,1). Introduce positive integer K K 1 C+ + 22*- £„ < 1 < C+ + 2K£„

p(z) dz CNn + 3) (a; y), < |l4 ™* y [)fe4 i = l "+ 2,-U„ *"<+

7

K

CN

" >

a+^en 1

1 l

PM±

<++2 i e w

n p(z) dz. < —= 1 2 ) ( l-- 2,2) y 2 )£^f Vtn J<; - vV/ (( ll -- xX»)(l »-ie +2J£nn " fi^h C +++ ') 7-i"*

Consequently, almost everywhere in (—1,1) 1n + 1 \lW(x,y)\ < CxyNnnK n < CvyNnln \I^{x,y)\ CxyNn\n n1^ —TT—

lv7

((n n €e Z+; , y < 1) Z+; --11 <
(2.9) (2.9)

244

The Representations

of the Trilinear Kernels

€ Z+. The 1 last term holds, where the constant Cxy > 0 doc does not depend on n G In (x, y) can be estimated in the same way

tfW)'

I..2 »(Xy( y)\
JV„

( n € ZZ+; + ; --1K<s x, t, /y<<11). ).

^1

Our result is a consequence of last inequality and the estimates (2.8), (2.9). It still remains to consider the case C+ = 1, i.e. i.e. x = y, C- = 2x2 - 1. We have the following estimates

b

n + 1l for all 1 x,z G (-1,1) ,26 iVn 2 ' ( l - z ) | z - 2 x 3—Yjfor + l| ° r all x, z € (-1,1), satisfying \z - 2x2 + 1| > 0. ! c fcrall 6( 1 1)

\Vn(x,y,z)\<\

d-^V+ii

Therefore, as above,

r

Therefore, as above, -. 1 \V {x,y,z)\p{z)dz
f1 °T^A l-X* J2

Jx*

Jxx 2

where

2

» 5n = -i((1l) - x 2 ) <*« =: -^-(l-x N

^TT n+1

and, consequently,

*'* - ' '

satisfying \z - 2x2 + 1| > 0. n p(z) dz C(n + 1) , p(z) dz, +hC(n " ^T^T ^ + + 1) Ji-s t n5n p(z) dz, l-Z

Jl-8n

Kxx<
^2+1 z)\p(z) dz < C N In / \V |P (x,y, (x,2/,z)|/>(z)rf2:
nn

x

nx

n

n

where the constant Cx > 0 does not depend on n G Z+. In a similar way, one obtains £

n+ 1 \Vn(x,y, (x,y,z)\p(z)dz z)\ p(z) dz < CxNn
where the constant Cx > 0 does not depend on n G Z+. Our assertion (with x = y) follows from the last two estimates. Thus we have completed the proof of Lemma 2.1. In a similar manner one one can can get get the the following following Lemma Lemma liar manner Lemma 2.2. Let Let the the O.N.P.S. O.N.P.S. P P= = iv„\. {pn}, Q = th„\ {hn} (n (n G Z+) satisfy n}, H O= = {q faA. H= £ Z_^ the conditions (1.3), (1.16)^(1.18). (1.16)-(1.18). Then the following estimate I

n+1 I1 ; vnnlnnl n ^ ± l JV N •y/{l ( l - -x ax2))(l ( l --y ay2)) Nnn ' (riGZ+; - K a : , y < l )

\Vnn{x^z)\dz
/J-i- . * •

(2.10)

6.2. A Generalized Product Formula

245

holds, where the constant C > 0 is independent o/n £ Z+ and x,y G (—1,1). Now, one can prove the main result of a present paragraph. Theorem 2.3. Let the O.N.P.S. P = {pn}, Q = {qn}, H = {hn} (n € Z+) satisfy (1.3), (1.16)-(1.18), and for the weight w3(x) fulfills 2

dx < oo. 02(x)wz(x) (x)u;3(x)cfo 00.

/

/ >

«/—1

Suppose that for the sequence {7 n } (n(n€€ Z+) Z+) (see (see ^(1.13)) ce 77== {7n} J . l ^ ; the condition n-l

n—i fc + 1 iIna«2! ±L±t l+J £ ^|A(T*)|MkIn < CC( (n n = 1,2,...) l,2,...) |l7n|W |A(7fc)|Mk In n ^ 7„|JV»nln n N^ kf c n fc=o fc=0 "

^r~ Ei

(2.11)

fc=0

ZioZds. ho/ds. 7%en Then tfie the generalized product formula lm.Pm(x)qm(y) == / /:
(m is fixed m = 0,1,2,...)

L

(2.12)

is valid, where oo 00

E'

A(7fc)Pfc(x, y, y,z) 2) (( - 1K< x, x,y, y,zz << 1). 1). tf(x,y,z;7) # ( * , y, *; 7) = Y, ;^(7ik)^fc(x,

(2.13)

fc=0 fc=0

Moreover, there exists a constant C > 0 independent ofn£ Z+ and x,y € (—1,1) such that f

1 I

( (- -KKxx, ,yy<
(2.14) (2.14)

( - K x, y, z < 1). < - ! < « . * * < *>•

(2.15) (2-15>

Va-^a-w')=_

|A-(x,j/,z;7)|^
/ >

N

while 1 K(x,y,z;7) =: K{xAT(x, y Z y, z; 7) K(*>y>*->7)=v(x)rp\y)9(z) ^ ( y ) ^ ( z ) ^>( X> ' ^

Proof. It is easy to see from (2.13) and (2.15), that oo

A>(7fc)2>fc(x,y,z) K(x,y,z;1) == YlM'Tk)i>k(x,y,z) K(x,y,z;7)

=fc=0 E (->

( -- lK < xx,y,z , y , * <
(2.16)

fc=0

iRrom the estimate (2.3) and from the condition (2.11) one can obtain for the weight p(x) the following estimate OO

(x, y, z)\p(z) dz <j'j\V {x,y,z)\p(z)dz

V I A|4("»)l (7fc)| / fc=0

k

L

xy

246

The Representations

of the Trilinear Kernels

and, consequently, by virtue of B. Levi's theorem the series is absolutely convergent - K <x x, , yy < < 1), i.e. almost everywhere (x, y are fixed,I, —1 oo OO

00. Y, ">(7 \A(-yk)\\T> Fxy(z) e Lj(-1,1), k(x,y,z)\ -+ *".»(*) £IA( L\(-i,i), n -> oo. 7 It )ll*k(*,v,*)l

fc=0

Thus the partial sums of the series (2.16) have an integrable majorant. On the other hand, the series (2.16) in view of Lemma 1.4 converges for all z e (—1,1) (z ^ C±) to OO

¥>(*W(V)*(*) rtxMvMz) g ; IIA(7 A fc ^ )|[ifc(x, ) ! ^ ^y,)z) ++ B Bkfc{x,y,z) (a:,y, z) ++ E£hfc(x,y,z)), (x,y, z)], 7^ ^ (* C-)(C+ ( 2-- c - ; ( C- + -*); fc=0

(2.17)

where the functions Ak(x, y, z), Bk{x, y, z), Ek(x, y, z) are defined by (1.22)-(1.24), and the series on the right-hand side of (2.17) uniformly and absolutely converges for x,y, z € (—1,1) in accordance with (2.11). By Theorem I. 2.1 (for p(z) =

e2(z)w3(z))

I>

0 00 0

hm(z)J2^('yk)'Dk(x,y,z)w3(z)dz -► / /^(zJ.KXx, hm{z)K{x,y,z;~i)wz(z)dz, ^ A(7fc)Pfc(x,y,z)u;3(2) dz -► y, z; 7)w3(z) dz, 1 f ♦ / > fc=0 nn -> —¥oo. oo. Therefore (7m -- ln)Pm(x)qm(y) (7m 7n)Pm(x)qm(y) Since

Kh (z)K(x, (z)K(x,

*L

-> -> /

rn m

y, z\ -► oo. z; i)wz(z) 7)^3(2) dz nn-+ 00.

lim 7 n = 0,

n—>oo

we have a generalized product formula (2.12). Taking into consideration (2.10), (2.11), (2.13), (2.15), the estimate (2.14) follows from Fatou's Lemma I. 2.2. We have thus completed the proof of our theorem. If the orthonormal systems {p n }, {qn}, {hn} (n e Z+) coincide, i.e. w(x) = wi(x) = w2(x) = w3(x) (x e (-1,1)), then we obtain Corollary 2.4. Suppose that the O.N.P.S. P = {pn} (n e Z+) satisfies (1.3), (1.16) and

£•

2 ip2(x)w(x)dx I

(2.18)

Further, let the condition (2.11) hold for a sequence 7 = {7 n } (n G Z+) (cf. (1.13)). Then the product formula lmPm(x)pm(y) = 7mPm(^)Pm(y) =

(z)ifi(x, y, 2; // pmm(z)Ki(x, z; 7)w(z) j)w(z) dz / :

(m 25 m == 0,1,2,...) 0,1,2,...) 25 fixed, /ixed, m

6.2. A Generalized Product Formula

247

25 valid, where oo oo

A A: k:

E

5>

x = ^2 A(70] &(Kk)Y^Pi( )Pi(y)Pi(z) E^i^)^^)^^)

K 1{x,y,z;-y) Ki(x,y,z\~i)

(( _- 1K<x,y,z< x , y , z < l )1). .

ii=0 =0

fc=0 fc=0 fc=C

One can apply Lemma 2.2 to the pointwise estimate of the first n t h partial sums of the following orthogonal expansions oo oo

-1

E<

Y]c]c (x)q kpkkp k(x)q k(y), k(y), fc=o fc=0

ck = ckfc(f) f(z)hkk(z)w33(z)dz (/) = / f{z)h / J-i ;

and oo OO oo

7.7 .1

E

y2ckkCfcfcPfc(z), pk(x),

: {k (fG c €ZZ++))

(2.19)

i

cCfcfc kk=

fc=0 k=0 fc=0

=

.1

/ f(y / ( yJz)q , z)q k(y)h k(z)w 2(y)w 3(y)dydz k(y)h k(z)w 2(y)w 3{y) .1 J-iJ-i

dydz

(fc ffc G Z+). Z.A

W-l (2.20) (2.20) T h e ou rri vee m 2.5. Let the P = satisfy e O.N.P.S. {{»„) p n/}*, Q Q— = {q {q }, H = {h {/»„} (n € Z+) f ) , satisfy T m O.N.P.S. = nn}, JH n} (n G Z+) xh u ve o m 2.5. ^ . v . Let x>ot the wot- vx.^».i .^. P ^ = — {p u ^nn}, sc = \Hnj7 -L — X'^nj (1.3)7 (1.16)-(1.18). Then the following statements are valid: (1) at every x,y G (—1,1) the following estimate n

IE-

\^2ck(f)p Cfc(/)Pfc(x)gfc(y)| k(x)qk(y)\ fc=o fc=0 fc=0 71. 4- 1

¥>(*) y ( y ) =JV AT ln ^ i = =, - 7^= =^= =n i Vln n n +) '-TvT 1 2 22 V I1 --Xa: V yj\1 - y iVn 25 va/id, tuyere the constant C C > 0 is independent off f,n G e Z+ and x,y x,y Ge (( -- iM . i ) ;;

fi^ a£ everT/fx,y x,y €G (—1,1) ( - 1 , 1 ) the estimate n

\^2ckkpk(x)^ fc=0 o

n+1 AT In [\f(y,zM(y)e(z)w =iV ln-2 ± i N sup [|/(y, z ) | ^ ) 0 (2*(y)w W3(z)} 0 is independent of the function f and of the variables n G Z+, x G (—1,1). - _ ^ 5 )
7r^t i V nn Jnin

It can be seen without difficulty that nn

»i

y2ckpckp{x)q k(y) k{x)q k(y) fc=0

and

n

E<

7.

y^CfcfcPfcOr) ^CjfcfcPfc(x) == / fc=0 fc=0

-J>

= /

f(z)V ,f(z)Vn(x,y 1z)w 3(z)dz 3(z)dz n{x,y,z)w

/ / ( y , z)Vn(x, f(y,z)V y,n(x,y,z)w 2f)u;2(j/)ti;3(l/) dycfe 2(y)w3(y)dydz v-i

from which these results follow by the Lemma 2.2.

248

6.3

The Representations of the Trilinezr Kernels

A construction of t h e "double-humpbacked majorant". A Generalized Translation Operator in orthogonal polynomials

Our main goal is to obtain the estimate of the Generalized Translation Operator in general orthogonal polynomials. We begin with the construction of the majorant. Nonnegative function *$;(£,r/) ;«,i|)

( n e6 ZZ++ ; 7q 7€G ( c( c, d, d) )cc((--ll,,ll ) )

is called a "humpbacked majorant" for the sequence $n(£,*/) in the variable rj at the point £ if the following conditions are satisfied: (1) for all n € Z+ and £, ry G (c, d)

\*n&v)\£, then

|^/(0»(OrfMO|
" / P'

Proof. Let, as usual, \E denote the characteristic function of the set E. If g is a simple function (see Ch.I, §2), then except possibly for a finite number of points it can be written in the form

,a*x(fai,f]) + *

>>x(fc,*>]), 3

A Construction of the "Double-Humpbacked Majorant"

249

where the a* and bj are positive. Substituting this for g shows

i/ r i_ ,*]«<[i/i s/w{E*.*.[ <-L» [? '[M^.C]/ S ja/>«'] \L/ l/M/t ,Dr i +y>*/ ZMK» il)hj7 [a if" ]) |jf/(0»«><W«)| V(Off(0
d/i|

5ZM^»€]O» z

\f\dfi\

|M&*j]/g

3

<»»^o[i;wfc.(])tEwH''4 [£ <M./(«)[ £ 3 <*£])l) + 5M >M ) ] t

3

The right-hand side of the last relation, however, is just ||<7||i,MMM/(f). For the general case choose a monotone increasing sequence of nonnegative simple functions gn, which are monotone increasing for 77 < f, monotone decreasing for n > f and converge to g pointwise. Then using the result just proved and the monotone converge Theorem I. 2.1 twice gives \fg\dfi= lin limG I^ \f\g \f\9ndfl \jjf9dn\<S <J\f9\* j j \fg\ dn = Jiim n d/i

I/H J

n—K Jl n-+ooJ

M ^/(oOUfcllirf, <<. lim M^ttxngnh^ =M MMM/(x)|| /(*)y|i,„ 5 || liM n—►oo

which is precisely what we wanted to show. Corollary 3.2. Let \i be an absolutely continuous finite measure in the interval I and Fn(£,n) has a "humpbacked majorantn F*(£,n) is an uintegrable humpbacked majorant". Then, for

-1,

QiO^lmFni^d^n) 9(0 = mFnti^dfiiv) the following statement are valid: I: 1) if f isinL^I),

then

\g(£)\ << CM^(0 CM^fiO \g(0\ and

d

Mtt ls(OI > A}) < C7±[ ^jVWdtiv), |/(,,)| dnirj), MttlA}) 0 is independent of g and X > 0; (/) (r > 1),, then 2) iff GiftiffeL] t{I)(r>l),then

N|r,„M < < Cr||/||r, Cr\\f\\Mr.^, || 0 is independent of f.

250

The Representations

of the Trilinear Kernels

In fact, one can observe that M(|/|)(f) is the same as M/(f), and < \m\\Fn{^n)\dn: W)\ <J\f{n)\K&ri)d*/'• Then the Corollary 3.2 follows from Lemma 3.1 and Theorem I. 4.10. We say that the function f)n(x,y,z) (-1 < x,y,z < 1; n e Z+) (cf. (2.1)) has on (—1,1) a "double humpbacked majorant" X>£(x, y, z) in the variable z at the points £± (cf. (1.6)) if on each of the intervals (—l,xy) and (xy, 1) it possesses a "humpbacked majorant" at the points C->C+> respectively, i.e. for fixed x and y, P* (x, y, z) is nondecreasing on (—1, C_), nonincreasing on (£_, xy), nondecreasing on (xy, £+), and nonincreasing on (£+, 1). The following assertions play a significant role in our estimations. Lemma 3.3. Let the O.N.P.S. P = {pn}, Q = {qn}, H = {hn} (n e Z+) (orthonormal with respect to the weights w\(x) = w{x), W2(x),W3(x), respectively) satisfy the conditions (1.3) and (1.16)-(1.18). Then the function X>n(x,y, 2) has on (—1,1) in the variable z a "double humpbacked majorant" P*(x,y, z) at the points^-, C+ and, furthermore, the estimation n+1 =iV ln£,*U**->**C^_^_^!±l a V* {x,y,z)dz l v7 V^l-^Xl-y*) I

l

1

n

n

1

(31) (3.1)

i ^x ,«.y-., z < 1) i\ ( n € Zrz+ ;. - K

holds, where Nn is defined by (1.14) and the constant C > 0 is independent of n € Z+ and x, y e (—1,1). Proof. We show, that there exist "humpbacked majorants" for the function Z>*(x,y,z) on ( - l , x y ) and (xy, 1) at the points C- and C+, respectively. We con struct the "humpbacked majorant" for X>*(x,y,z) on (xy, 1) at the point C+; on the interval (—1, xy) the construction can be deduced in a similar way. At first, consider the case (+=xy+

y/(l - x 2 )(l - y2) ^ 1, i.e. x ^ y.

2 Put, as in (2.6), S <>nn = ^(1 V ( l --* x2 ) )(l ( l - -y 2y*)N ) ^ nn//(n ( n + 1), s£nn = (1 - (+)Nn/(n + 1), then by the relation (1.15) ion

Sn -)► 0, en -> 0 (n -> oo).

A Construction of the "Double-Humpbacked Majorant"

251

Using the estimates (2.4) and (2.5), we define the "humpbacked majorant" by 1

1

2

2

V(i-^ *~)(i-2/ / \if )C ) C+ + - * '

if x y < z < C+ - <*n

3

V*n(x,y,z)

I C(n-f ^(n + l )(- l - x 2 )1( l - y 2 ) '

if C+ - Sn < z < C+

= < 1 1 | C(n + 1 ) if C+ < ^ < C+ + £n 22 22 VV((il --xx ) ((li -- y )) i 1- -cC++' l l CNn C inV- n -m—lrrr=^T^r' \cN if C+ +£n ifC+ + e n<
where the constant C > 0 are independent of n G Z+ andl xx,, y?/,, z GG (—1,1). (-1,1). In fact, when xy < < z < < 1, 1then

\(z |(z - c_)(c+ C-XC+ - *)l *)| > V(i > / ( i - **22))(i ( i --v 2v)2)\z l * - C+lC+|. So, for all x,y e (—1,1) and n G Z+, the estimate | p^ (xl x, y, 2, /z, )z|)<|
(nG e Z ++ ; xy < z < 1)

holds. Next, by the defining relation df)*n(x,y,z)

^

f

> 0 °fr% dz * * > > 0

(xy ( * y<
dV* (x,y,z) ^ < 0 (C+ feM
and

"+1 p ;n((x,y,(+-6 V* x , y , c n+)- ^ ) ^=cC-: (l_x 2+l _2/2), '(l-x2)(l-y2)' 1 V*n(x,yX++en) = C2 2 x )) (( ll -- yy 22 )) (( ll -- C x, // (( ll -- z C ++ )) ' n

1

Consequently, the function >n P T)*(x,y, * ( x , y , z) is nondecreasing nor onn (x,y,C+) T l (x,y,z) on creasing on i n ((£+,1). C + , l ) . SSo o PX>*(x,y,z) ; ( x , y , z ) i £is a "humpbacked majorant" n O] l{x,y,z) (xy, 1) at the point £+. Now consider

■A

J „ ( x , y ) == / In{x,y)

Jxy

V*n(x,y,z)dz

[n G Z+; --1K<x x,y (n , y << l 1). ).

This integral can be estimated in the following way: in virtue of the definition of

252

The Representations of the Trilinear Kernels

the majorant

V*n{x,y,z)

1 In(x,y) (x,y)
/

vor-^)(i-» )/J y yj{l-X )(l-y ) 2

dz

2 1 Jxy

2

r

C+-z

{+-Z X 1 fC+ dz + C{n + 1) 2 2 ' ( l - x* 2))((ll--2y/ * ) . Jc+_Sn

+ C(n+1)

2

1

2

i1

/-C++en

\/a-x )a-i/ )i-c+/ + C(n+1) , 7- / 1

+ CNn-

dz ^

dz

V a - ^ 2 ) ( i - 2 / 2)) l+en ^ , ** - c + '

+ CNn—=======

I

.

Hence, using (2.6), one obtains Hence, using (2.6), one obtains

/ „ ( x , j / ) < C i VTnn -

11

V^=

= ln 11

]

y/(l-X2)(l-y2 x2))(l-y2)

7(1 -z 2 )(l-y 2 ) A

1

1

+ C ( n + 1i ;) -rr-Jn+C^+l)(l-x»)(l-y2)° ' V ( l - *2)(1 - 2/2) r) + CAT,, CJV„AT„ .

1 1

=

1

0^a £ n

1-C+ In In ^—^i

£71 ^ V((ll--l^2()1( -l -2y/ 22 )) £n 11 rc+1 ■ + MnlLLl 1
. „

- v(l-* )(l-» )" ^ T '

where the constant C > 0 is independent of n 6 Z + and x,y € (—1,1), in accordance with our statement. We still have to consider the remaining case We still have to consider the remaining case C+ = 1, C+ = 1,

i.e., x = y, C+ = 2x22 - 1. i.e., x = y, C+ = 2x - 1.

In this case we define the "humpbacked majorant" by In this case we define the "humpbacked majorant" by 1 1 (CATT^T^~z' if x 2
C(n C + 1; ) - 2 222 ( l - x(1-X) ) '

if 1 - £ n < z < 1,

where C > 0 are the absolute constants and iVn *„ l - : r 22 )). <^n = = ;- ^ ( r(i-* + 11 n+ Obviously, the function P * (x, y, z) is nondecreasing oni {x (x222,l-S ,1 ,l — — 5nn)< ) and and nonincreasnonincreasn) 5 i n g o n (1 - J -6 ). n ,nl,l).

A Construction

of the "Double-Humpbacked

Majorant"

253

Furthermore, as above, we obtain by straightforward calculation

' £ *«<*• *• *> * «ih? C" T ^ rh+c{n+x) (T=W) J /.*.dz (1-^ ln T + c (1n) — + L1 ) ^ C *r^ " r 6 l D x C(n ^ 2

Jx

dz

CNi v.-i_

V*n(x,x,z)dz
n

rl-Sn6n

~

9 ^2 x~

dz

1 2

1 1 —- x Jx / 2 , 1 x _1 n — — ++C

++ C(n + 1L)) -

2 2

Jl-6n

1 1-x2)2^

+

On

1

By the defining relation

v^_, n2 Il±i ^ ^ ' 1- x N

/ 2 V* Vn{x,x n(x,x,z)dz
l-x

l n

2

z

Nnn

where the constant C > 0 is independent of n G Z+ and x G (—1,1). It coincides with (3.1) as x = y. We have completed the proof of our assertion. Remark. By virtue, of "symmetry" of the function X>*(x,y,;z) we can construct "double-humpbacked majorants" in the variables x and y. Let oo oo

£=o■ k

n

fc=0J

- 1L

= , c<*(/) k(f)

^2clck(f)p k(f)pk(x)q k(x)qk(y), k(y),

■

J 1i

~/

/(z)/ifc(2)w f(z)hk(z)w 3 (z)dz 3(z)dz >

(3.2) V3-2)

(-1 < x , y <1)

be a quasipotential function, (3.2):

and Sn(f; x, y) (n G Z+) are the n t h partial sums of

n

5 n ( / ; xa;, , yy)) == ^ c f cCk(f)Pk(x)q ( / ) p f c ( x ) 9 kf {y) 5n(/; c(y)

(nG e ZZ++ ; x,y a ; , ?G / €((--11,,11))))..

(3.3)

fc=0 fc=0

Put

\Sn(f;x,y)\

S . ( / ;x*,v) ,y) = = SU s Pu p ^ g ^ ^ s*(f;

(x,y ( i , yG e ((--l1,, l1)) ) .

n€Z V nT111n l n ^ ^y± ^ n 6 Z +++ iA

Corollary 3.4. Assume £/ia£ t/ie O.N.P.S. {pn}, {<7n}> {^n} (^ £ Z+) satisfy (1.3), (1.16)-(1.18). Then there exists the constant C > 0, independent of function f, such that

U":ij [J-iJ

rA

v)] [ [A(/;x,y)]

{£' U

1

Cl _

X r

2

W 2 (1 l

V? (x)

dx

v2)r/2

.^ Vr(2/) N

I

/ >i

fxdy > dxc j

, X' » 7TT?j /7f*

>

x

x

| / ( x ) | ^ ( x K ( xc) )y

,

dx

ll//rr

,

= < c00. .2

l/r

(3.4) 3 4 ( - )

254

The Representations

of the Trilinear Kernels

Proof. Evidently, 5„(/,x,y) Sn(f,x,y) = J

f(z)Vn(x,y,z)w33(z) (z)dz dz (x,y, € (-1,1); n e Z+).

It follows from Lemma 3.1 and .KX Lemma - L J C l l l H i a . 3.3, «J.lJ, that t ip(x)ip(y) S*(f;x,y)

)(1

y

(3.5) (3-5)

)

r)

Here the constant C > 0 doesn't depend on / and x,j/6 (—1,1), C± a r e defined by (1.6), and Mf is a maximal function for / , where f(x) f{x) = /(x)d(i)t03(i) /Or)0(x)w3(x)

(-K x< < 1).

(3.6)

Hence

j j\J\[S.U\x,y)\

[S,{f;x,y)}ri-r

1

2 22 2 (l-x (l -x-y'-(i-y-y-2p2(l-y x2yi y/(i-y (\ Y r

v l/r

v

^ dxdy >

o

lT r 1/r p r 1/r < C{ [ / J _[M/(C-)] [Mf(C-)] dxdy] ^ + [J {Mf{C JjMf{<+)Ydxdy] ' ). da;dy] +| +)} dxdy\ }

-{[//'

[/;

Changing variables x = cosrj, cosr;,

yy = cos^, cos£,

and, consequently, C± C± == COS(T7=F£), cos(r)TO>

we have

£/> fj {/-'./ cc * il? {f'

Mf;x,y)]r{-

(1 2_Yl3 . 2 ) r / 2(l ( 1 _ y 2 ) r2/ r/2 J\-x - y )2 ^ ^•(x)^(y) ^^

1 l/r

I*}

c

dt r [M f (cos u)]rdu\ < C{ jT'[M/(co8«)]d u } V == c { f[Mf{t)Y [Mf{t)Y ^ = =^ f }

{iy

^/^ t2J

•

Using the weighted norm estimated for the maximal function M (see Theorem I. 4.12) with u(x) = 1/Vl - x2 and (3.6), we obtain (3.4) in accordance with our statement. Our main aim is to investigate the Generalized Translation Operator (G.T.O.) associated with the O.N.P.S. P = {pn(w;t)} (n e Z+) with respect to the weight u>{x) (-1 < x < 1).

A Construction

of the "Double-Humpbacked

Majorant"

255

From the point of view of the abstract theory we define the Generalized Translation Operator (formally) by oo

ikYktt)Pk(y)Pk(x), 7 k Ck(f)Pk(y)Pk(x), = E^° )c { ) /

T»/(x) Tyf(x) =

fc= fc=0 ° fc=0

/I

<*(/) == / <*(/)

(3.7) ( --1K< xx < < 1; A;G k G Z+),

f{z)pk(z)w(z)dz,

where ?/ G (—1,1) is fixed and 7 (f» := n

11

'

ma£ x _ i < x 00 ((nr w->• oo). oo).

(3.8) (3.8)

2/0 < 1). 1)( - 1 < 3/o

(3.9)

Suppose also that

We consider the partial sums of the series (3.7) n

c kU)Pk(y)Pk(x) (( --1K< «.» x,y < < i; 1; « n Ge z+) Z+) ?£/(*) vim == 5>fcikO] W)Pk(v)p*C«0

fc=0

and introduce the majorant

(-Kx,y
II?/(x) 7/(x) = == sup |T\1*f(x)\ n V(x)| n€Z+ neZ+

A "double-humpbacked majorant" gives an opportunity to estimate T*f(x) by the aid of the Hardy-Littlewood maximal function. L e m m a 3.5. Suppose that the O.N.P.S. P = {pn} (n € Z+) iu#/i respect to the weight w(x) (—1 < x < 1) satisfies (1.3), (1.16) and (3.9), and for the sequence 0) 7 (°) = { 7 £ } (n G Z+) (cf. (3.8)) the condition n+ 1

n-1

0) | 7 (°)|iV„ln |N n In = ± 1 + g l7i

fc=0

fc+i

| A ( 7 ^r)i^i» ) | A T , Ini —* r±r —1 iA( 7 Nk

*< CC(

~

(n n == 1, l , 22,...) ,...)

(3.10) (3.10)

holds. If I

\f(x)\ip(x)w{x)dx \f{x)\ip{x)w(x)dx

/ > then the following

< oo,

/

/><

(3.11)

estimate

f, < C-%&=^L={Mfo /(C-)} {M/(C+) +M M/(C-)} T*f(x) < C-
Tyf(x)

' VI y/l=X* - x i/l y/T=y* - y

(3-12) (3.12)

256

The Representations

of the Trilinear Kernels

is true, where the constant C > 0 is independent of the function f and variables ,V € (—1> 1); the function f(x) is defined by (3.6). Proof. It is easy to see that X

>-0

T n y /(z) = where

/(z)il„(i,y,2;7(0)M«)dz

(x,y€(-l,l); n€Z+),

n

> = £->Tk

fl„ (*,!/, z;7 (0) ) =

(x,y,z€ (-1,1); n € Z + ) .

Pk(x)Pk(y)Pk(z)

fc=0

In view of Abel's summation by parts n-l

fl„(x,y,z;7(°>)

=7£°>Z>n(x,y,z)H

£•A(7i

0)

)^n(x,y,z)

k=0

(x,y,z<= (-1,1); n s Z + ) . (0) Since the sequence 7 7^) = = { U0)}^ (n € Z+) (cf. (3.8)) satisfies (3.10) we see by Lemma 3.3 that the functior

Rn{x,y,z;i<®):=-

1

- T fi B (x,y,z;7 ( 0 ) )

(n€Z+;

-Kx,y,z
has on (—1,1) in the variable z the "double-humpbacked majorant"•fl;(x,y,z;7 ( 0 ) ) at the points £- and C+: n-l

K(x)y,z;7(0))=|7i0)l^(^y^) +

|A( 7 < 0 ) )|^(z,y,z)

k=0

(-1 < x,y,z < 1; n € Z+), moreover, the estimate

1 i$(x,y,z;7(0))
/ >

(n £ Z+; - 1 < x,y, z < 1)

holds, where the constant C > 0 is independent of n G Z+ and x,y,z G (—1,1). Further, by Lemma 3.1 one obtains (3.12), as stated above. In the following assertion we obtain weak and strong type estimates for the maximal operator

!?:/-»• T?/. L e m m a 3.6. Assume that the hypotheses of Lemma 3.5 are satisfied for some r, 1 < r < 00, the inequalities

/;•

fr(z)V)

r w w (z) w= .az < oo • z2

(3.13)

A Construction

of the "Double-Humpbacked

and

Ma.jora.nt"

257

dz j \ |\f(z)\ / ( Zr

/ >1

'

are fulfilled. Then the following statements

oo :< < OO

(3.14)

Vl-z2

\>

n vo valid: nirtlnrl' are

1) there exists C > 0 such that for each function f and variable ye (—1,1) the estimate

1

[-
( / :-l [

y

-i r

T -TSf(x) J

r

*f

i

>{k

x) l1/r /r

cefx >

)

(3.15) (3 15) '

^| l '/ rr

dx

r |/(x)|V(xK(x) x VxWrx\l\-x*

)

holds; £,) there exists a constant C > 0 independent y G (—1,1) and £ > 0, such that 2

of the function

f and variables

Vl-z ™ y \ Lx,x x e€ (_!,!) T f{x)>A\ (—1,1) : — :£E^ X w (*)>o 1 ' v(*) 1 <-

JI

ij

fr(y) f(2/)

^ 1 < J- C— e r

?/(*)>*}

(3.16) (3.16)

w r ( x ) / ' |/(*)IV(*) |/(x)| W ( x> ) -A. ^ M==2rnrffdxx ax 2 x t (VT^Fr J L Vl — x

Vr

25 va/id; w/iere |J4| is £/ie Lebesque measure of A. Proof. In order to prove the first inequality (3.15), we observe, that in view of (3.12) and Minkowski's inequality one obtains

w

VjE**-T

L ^w

r i

N

+ V(x)

1/r

) dx\

'1

y>(y) < C -, ¥>(y) 1 — ^x/r /— ^

.{[/.'.' -1

~

i |l l// rr

rr M Mf((+)dx\ /(C+)^] 1/r J

/.I

r + [J M f(C-)dx] ix\ 1/r}, ), Mrf(C-)dx\ T Ly.i"

♦[/> where the function / is equal to / = f(x)
Consider the integral

H■[/;

r r /r I-=[J_ M Mrf(c+)d f(C+)d Xyxy

(the second term can be estimated in the same way). We introduce the new variable v / ( 2l)(l-y - xx 222))))(l | ( i --yy2 2)) ( (- -KKx(-Kx,y
258

The Representations

of the Trilinear Kernels

and obtain (for example, in the case 0 < y < 1)

-{^

/ =

S

/ ;

C

^r*

nyT^^l ■ du + y M / ( « ') 2/ 2/ + r

■{£' L

^/^^2 J

M f(u)-v

x 1/r 1/r

du

r

2 Wl-u V

r

M f(u) *)\y-

_ uy/l

-y2\I

1/r

1 1/r

v/T^I 2 J

j 2

Since the weight (1 — x ) - 1 / 2 belongs to the A r -class of Muckenhoupt for r > 1 (see Ch. I, §4), then we have

,|l/(«)N 1

^vT^f} H/> •}''' V. ^M \ 1/r

dM

,.11

/
whence follows directly (3.15). The estimate (3.16) can be deduced from (3.15) by Chebyshev's inequality (Theorem 1.2.12)and .lzjanu the following relation

.^ZZ |{ _ £}|I x ; x e ( 1 ) 1 ) : ^T T v■I?/(x) / ( x ) >>e41 If ! evnz\\ <||,;, 6 (-l f l):M/(C-)>^^}| lx;xe

(-1,1)

(z)

ll

*|{-

< L < x; ; xxe €( - l , l ) :

M/(C-)>^ 2C
If

f^/l-j/O I

y^-y" 1 + \lx; x ; x €e (( -l,l):M/(C - l , l ) : M / « + ) >+g)>^^}|. + ||x;, 2CV>(y) / r

We have thus proved our statement. Now one can prove T h e o r e m 3.7. Assume that the polynomial system P = {pn} {jt 6 Z+) orthonormal with respect to the weight w(x) (—1 < x < 1) satisfies (1.3), (1.16), (2.18), (3.9), and suppose that the sequence 7 0 = {7^ ^} (n G Z+) (cf. (3.8)) satisfies (3.10). If for some r, 1 < r < 00, the assumptions (3.13) and (3.14) are fulfilled, then the following statements are valid: 1. the orthogonal expansion (3.7) converges almost everywhere on (—1,1) for every fixed y € (—1,1); 2. the estimate

f{/- 1 [w / ;1 [

vr=z |T,,/(x)l r f(x)\\r -\T -

2

Vi^X

] }

u>

^(x)

x 1/r

-1 r

y

\ 1/r 1/r

dx \f(x)\
r

rr

r

r

(3.17)

A Construction

of the "Double-Humpbacked

Majorant"

259

is valid, where the constant C > 0 is independent of the function f and the C>0 variable y G € ((—1,1). -i,i). We introduce the weight u wrr(x) (x) === ip ^rr{x)w (x)wrr{x)/y/T^x (x)/y/l 1 - x 2 (see (3.13)). Since the function / satisfy (3.14), then for a given e > 0 there exists a polynomial II(x) G 7rm such that

[/: [ /

L

Let n>m

-,1/r 1/r

r x ] 1 A *< £e.. |\f(x) / ( x ) - -n (n(x)| x ) r c jcj ) < fdxX\ r ( rx(x)

(3.18) (3.18)

-1

and n' n ; > m. By (II. 2.4) m

n(x) n ( z ) == a<m)p,(z), =££< j7=0 j=0 =0

then

m m

rvn(z) = = 1;4m)7fp,(x)Pi(y)

(n > m).

1

3J=U

Hence ||T 2 *n V(x) / ( x ) - Z*,/(x)| C77?[/ 7*/(x)| < C 7 ? [ / - n](x). ^From inequalities (3.16) and (3.18) follows

If

^r " 41 [(i-y ) l1 l/(x) n(x)ra;r(s)dx C£r_1 ¥/--/-i e
y(y) « / (nx V ) | >(£ 1,/ )' -| :i x : ,e(-l,l): x G ( - 1 , 1 ), :: ^ | T| r„| TVn V( (xx ))- -T T ||x: > -3/2)1/2 ^ ^ ^ }j || c

\/T^2._

2 1 /722

1/r

<pr{y)

l

| / ( x ) - n(x)| r o; r (s)dx <

Cer~l

from which we obtain convergence almost everywhere of the orthogonal expansion (3.7). The estimate (3.17) may be derived from 3.15 and the Fatou Lemma I. 2.2. This shows the validity of Theorem 3.7.

NOTES In this notes we give additional comments about the results in the text and bibliographies references. C h a p t e r I. Most of the material of this chapter are classical. §2 contains the standard results from the university course of Function Theory. We bring some classical results and indicate their authors. The standard references are the books: Edwards R. [1982]; Kantorovich L., Akilov G. [1959]; Kolmogorov A., Fomin S. [1975], Natanson I. [1965]; Stein E. [1970]; Stein E., Weiss G. [1971], Zygmund A. [1959]. Theorems 2.17 and 2.18 were given by Jackson D. in 1911. The proof of Theorem 2.15 has been borrowed from the book of Freud G. [1971]. For information about Gamma and Bessel functions see Lebedev N. [1965]; Tolstov G. [1962]; Whittaker E., Watson G. [1933-1934]. A detailed exposition of the theory of divergent series will be found in Baron S. [1977], Hardy G. [1949]. Theorem 2.20 is due to Toeplitz O. [1911], and conditions (i), (ii), (iii) of regularity are sometimes called Toeplitz conditions. §3 contains the standard results from the university course of Functional Analysis (see, for example, Edwards R. [1982]; Kantorovich L., Akilov G. [1959]; Kolmogorov A., Fomin S. [1975]. Uniform Bounded Principle, Resonance and Banach-Steinhaus Theorems were obtain in Banach S., Steinhaus H. [1927]. §4. A theorem equivalent to Theorem 4.5 was first proved by Riesz M. [1926] and Thorin G. [1939]; Theorem 4.7 was discovered by Marcinkiewicz J. [1939]. These results are stated in details in a lot of books: Edwards R. [1982]; Garnett J. [1981]; Stein E. [1970]; Stein E., Weiss G. [1971]; Zygmund A. [1959]. The conception of maximal function M was introduced by Hardy G. and Littlewood J. [1930]. The weighted estimates (weak and strong types) for M have been discovered by Muckenhoupt B. [1972]. The proof of Theorem 4.12 in the text follows Coifman R., Fefferman Ch. [1974], and, especially, Garnett J. [1981]. Theorem 4.15 is due to Gehring F. [1973]. Two-weight estimates for the maximal function M were proved by Muckenhoupt B. [1972] (weak-type estimate) and Sawyer E. [1982] (strong type estimate). More detail information about these problems one can get

261

262

Notes

from the survey of Dyn'kin E. and Osilenker B. [1985] (see also Kokilashvili V. [1985], Garsia-Cuerva J., Rubio de Francia J. [1985]; Torchinsky A. [1986]). Chapter II. §1. The conception of orthogonal system is based on the orthogonality in an abstract Hilbert space (see, for example, Kantorovich L., Akilov G. [1959]; Kolmogorov A.,Fomin S. [1975].) Theorem 1.1 is due to Rellich F. [1934]. The procedure of orthogonalization of linear expressions by the aid of which we constructed the orthogonal polynomials was given by Gramm J. [1883] and Schmidt E. [1907]. §2. The foundation of the general theory of orthogonal polynomials are established in the works of a Russian mathematician Chebyshev P. L. In a series of works connected with the problem of parabolic interpolation he introduced not only the orthogonal polynomials with respect to the discrete measure Chebyshev P. [1855], [1858], but considered the continuous case passage to the limit Chebyshev P. [1859]. Independently of Chebyshev P. a Dutch mathematician Stieltjes T. J. introduced the polynomials orthogonal with respect to the general measure. This approach is connected with his deep investigations in the theory of continued fractions Stieltjes T. [1884], [1894], [1895]. The modern conception of orthogonality belongs to Stieltjes T. [1884] (and independently to Sonin N. Ya. [1887]). For the extremal properties see Szego G. [1975], Alexits G. [1961], Freud G. [1971]. Christoffers formula was discovered by Christoffel E. [1858]. The recursion formula (2.17) was deduced by Christoffel E. [1858] for Legendre polynomials and by Darboux G. [1878] in the general case, but the Christoffel-Darboux formula has been already published by Chebyshev P. [1855], [1858], Rahmanov E. A. [1977], [1983] has proved the following result about recurrence coefficients: if the distribution d\i satisfies Erdos-Turan condition (i.e. \J > 0 almost everywhere), the dfj, belongs to Nevai .M-class (this class was defined in Nevai P. [1979a]). Theorem 2.14 usually goes under the name Favard's Theorem Favard J. [1935], but it is older Stone M. [1932]. Theorem 2.15 is due to Stone M. [1932]. An extensive discussion of the recurrence formula in the general context of discrete boundary value problems is given by Atkinson F. [1964], Berezanskii Yu. [1961]. More detail information on the fundamental spectral theorem and classical moment problem see Akhiezer N. [1965], Chihara T. [1978]. Most of the general results concerning the zeros given in this chapter go back to Stieltjes T. More detailed theorems about the zeros can be found in Szego G. [1975], Freud G. [1971], Chihara T. [1978], Nevai P. [1979d], Gilewicz J., Leopold E. [1985], Van Doom E. [1984]. §3 of Chapter II contains the properties of Jacobi polynomials. The Jacobi polynomials have been introduced in the work Jacobi G. [1859]. The results of §3 are classical (see, for example, Szego G. [1975], Suetin P. [1979], Erdelyi A. [1953]). In the book we follow Suetin P. [1979] with slight modifications. Lemma 3.9 is due to Polya D. Besides the Jacobi polynomials there exist the Hermite and Laguerre polynomials which are orthogonal on noncompact intervals. The Jacobi, Hermite and Laguerre polynomials together form so-called "classical orthogonal polynomials".

Notes

263

Now, there exists an extended conception of a "classical hypergeometric orthogonal polynomials" introduced by Askey R.; Askey R., Wilson J. [1985]. The conception of classical orthogonal polynomials was extended by Szego G. [1975] and Bernstein S. N. [1930], [1931]. On the other hand, Pollaczek F. [1949], [1956] (see also Szego G. [1975]) introduced polynomial system {p n (x;a, &)} orthogonal with the weight w(x\a, b), when the Szego's condition is violated. Recently besides traditional problems of the orthogonal polynomials, investigation of the properties of the orthogonal polynomials defined by recurrence formula (Jacobi matrices) attracts considerable attention (see, for example, Dombrowski J., Nevai P. [1986]; Geronimo J., Case K. [1980]; Geronimo J., Van Assche W. [1986], [1988], [1991]; Gilewicz J., Leopold E. [1985];IsmailM., Mulla F. [1987]; Lubinsky D., Nevai P. [1992]; Nevai P. [1979a], [1979b], [1979c], [1979d], [1986], [1992]; Nikishkin E. [1986]; Osilenker B. [1993]; Szwarc R. [1992], [1993a]; Van Assche W. [1987], [1990]; Voit M. [1990a], [1990b]), as well as the orthogonal polynomials in Sobolev space (see, the survey of Marcellan F., Alfaro M., Rezola M. L. [1992] and the orthogonal polynomials with the degenerated and indefinite weights (see, for example, Aptekarev A. [1984]; Barkov G. [I960]; Geronimo J. S., Van Assche W. [1991]; Mingarelli A., Krall A. [1983], Peherstorfer F. [1988], [1990], [1991]). Among the important topics omitted are the orthogonal polynomials related to g-extensions (see the book of Gasper G., Rahman M. [1990]). Paragraph 4 contains some estimates of orthogonal polynomials. Theorem 4.1 is due to Peebles G. [1940], Theorem 4.2 was proved by Korous J. [1938]. Theorem 4.3 and 4.5 are established by Freud G. [1952], [1954] (see, also, Freud G. [1971]), Theorem 4.4 has been discovered by Price J. [1963] (see also some papers about exponential growth of orthogonal polynomials: Lubinsky D., Nevai P. [1992]; Nevai P., Totik V., Zhang J. [1991]; Szwarc R. [1993b]). Theorem 4.7 has been shown by Nevai P. and Geronimo J., Case K. [1980], Theorem 4.9 is due to Nevai P. [1979b], [1979c] (our presentation of the proof of the statements 4.7-4.9 has been borrowed from the work of Van Assche W. [1990]). V. Steklov has proposed the following conjecture in 1921: let w(x) be a weight on [—1,1] such that for some S > 0 w(x) >6>0,

x e [-1,1];

then for each closed subinterval [a, 6] of (—1,1) there exists C > 0 such that \pn(w;x)\
(n e N; x e [a, b}).

Rahmanov E. disproved it in 1979 (see Rahmanov E. [1980], [1982]). We formulate the following result (Ambroladze [1991]): let XQ e (—1,1), then there exists a positive continuous weight w(x) such that lim \pn(w;x)\ = +oo.

n-»oo

264

Notes

Chapter III. §1. The materials of this paragraph are standard (see, for example, Kantorovich L., Akilov G. [1959]; Kolmogorov A., Fomin S. [1975]). §2. The standard references are Alexits G. [1961]; Kashin B., Saakyn A. [1989]; Kaczmarz S., Steinhaus H. [1951]. The statements 2.1-2.4 are the consequence of the corresponding assertions of §1. Theorem 2.5 is a well-known result of Menshov D. [1923] and Rademacher H. [1922] (our proof follows Kashin B., Saakyn A. [1989]). Theorem 2.8 has been proved originally for the Fourier trigonometrical series and An = logn by Kolmogorov A. and Seliverstov G. [1925] and simultaneously by Plessner A. [1926]. The generalization for general orthogonal series is due to Kaczmarz S. [1929]. As an application of Menshov-Rademacher Theorem, it has been shown by Kolmogorov A. [1934], / € Lipa (a > 0) implies the convergence of the Fourier polynomial series almost everywhere. This results is contained in Corollary 3.4 of §3 below. §3. Theorem 3.1 is a consequence of the results of §2; Theorem 3.5 was discovered by Natanson I. [1934]. Lemma 3.7 and Theorem 3.8 are due to Sz.-Nagy B. [1951] (see also Ratajski H. [1966]). Theorem 3.8 was established by Alexits G. [1944]. §4. Theorem 4.1 and 4.2 were obtained by Kaczmarz S. [1929] (we follow Alexits G. [1961] with an insignificant refinement). One observes that the methods (C, a) (a > 0) are equivalent to the Poisson-Abel means in Ljj; (see, for example, Alexits G. [1961]). The investigation of the (C, l)-sums of general series of the orthogonal polynomials was begun of Tandori K. [1952], [1954]. An improvement of Tandori's result is given by Freud G. [1952] (see also Ratajski H. [1967]). The method of the proof of Theorems 4.4 and 4.6 has been developed in connection with Fourier trigonometrical series by Carleman T. [1923]. Another approach to the definition of (C, l)-means is due to Nevai P. [1979a]. Chapter IV. §1. The main result of this paragraph (Theorem 1.7) is due to Nikolaev V. [1948]. Conception of the polynomial operator and the lower estimate of the norm of this operator are obtained by Lozinskii S. and Kharshiladze F. (see, Kantorovich L., Akilov G. [1959]). The important representation (1.2) is the BermanMarcinkiewicz-Zygmund formula (Berman D. [1952]) (see for details Kantorovich L., Akilov G. [1959]). §2. Theorems 2.1, 2.5, 2.6 and Corollary 2.7 were discovered by the author who published them in his works Osilenker B. [1968], [1969]. The statements 2.22.4 were proved by Young W. [1913] and Hausdorff [1923] (trigonometric case), and by Riesz F. [1923] for bounded orthogonal systems (the local variant Tandori K. [1954], see also Freud G. [1971]), and for unbounded orthogonal system these results were first given by Marcinkiewicz J. and Zygmund A. [1937] (the local variant - Theorem 2.2 - is new). The condition (2.36) was first used by Fomin G. A. [1967] and Steckin B. for Fourier trigonometrical series. The statements 2.8-2.12 were discovered by Nakhman A. [1991]. The condition (2.48) was first used by

Notes

265

Sz.-Nagy B. [1950] for Fourier trigonometrical series (see also for details Bary N. [1964]). §3. Theorem 3.1 and 3.2 are new and were discovered by the author. Note that for Legendre polynomials {pk } we have ||j4 |U ~ Cinn (n —> oo). Theorem 3.3 was obtained by Tandori K. [1954]. §4. Linear method of summability of trigonometrical Fourier series were investigated by Efimov A. [1960], Fomin G. [1967], Hille E., Tamarkin J. [1933]; Nikolskii S. [1948], Telyakovskii S. [1964]; Trigub R. [1974] (and bibliography). Theorem 4.1 can easily be inferred from the results of §2 (Osilenker B. [1968]; Nakhman A. [1991]). Lemma 4.2 is a particular case of a corresponding result of a singular integral (see Alexits G. [1961). Theorem 4.4 is due to the author (Osilenker B. [1968]). The results of the end of this paragraph (the statements of 4.5-4.10) are established by Nakhman A. [1991]. The Fourier series in classical orthogonal polynomials are investigated very extensively. The problem of pointwise convergence of such series (and their generalizations) was resolved by Szego G. (see book of Szego G. [1975]) and Bernstein S. [1930], [1931] by means of using the results on equiconvergence with the trigonometric series (see also Badkov V. [1974], [1978]; Kalnej S. [1984], [1991]; Natanson G. [1956]). The mean convergence of the Fourier-Jacobi series (and their generalizations) was studied by Pollard H. [1947], [1948]; Wing M. [1950]; Newman J., Rudin W. [1952]; Muckenhoupt B. [1969]; Badkov V. [1974], [1978]; Osilenker B. [1988]; Guadalupe J., Perez M., Ruiz F., Varona J. [1991], [1992]; Kerman R. [1992], Xu J. [1993]. The mean summability of this series was investigated by Askey R., Hirschman I. [1963]; Bonami A., Clerk J. [1973], Connett W. Schwartz A. [1979]; Chanillo S., Muckenhoupt B. [1993]. Necessary conditions for weighted mean convergence of Fourier series in orthogonal polynomials are given by Mate A., Nevai P., Totik V. [1986]. Nevai P. [1979a] proved that the Fourier-Pollaczek series is convergent in the space Ll{x:ab)[-1,1] if and only if p = 2. Chapter V. §1. The results of this paragraph follow directly from the Theorem I. 3.6. For Fourier-Jacobi series Meaney Chr. [1983] and Badkov V. [1974] established more precisely statement. For example, there exists / € L4/3[—1,1] such that the Fourier-Legendre series of / is divergent almost everywhere. Recently Guadalupe J., Perez M., Ruiz F., Varona J. [1992] have established the essential generalization of this result. §2. The problem of summability of the orthogonal polynomial series of / € L* almost everywhere was raised by Alexits G. [1961]. The main result of this paragraph - Theorem 2.10 - was proved by the author in the works Osilenker B. [1972], [1973], [1974]. The proof of Theorem 2.10 is bounded on a representation of the de la Vallee-Poussin kernels (Lemma 2.1) follows Osilenker B. [1973], [1974]. An important formula (28) is a consequence of Lemma 2.1 (it was independently proved in Dombrowski J., Fricke G. H. [1975]; Dombrowski J. [1985]). Condition (2.28)

266

Notes

was introduced in Osilenker B. [1972] (as well Dombrowski J. [1985]; Mate A., Nevai P. [1983]). Paragraphs 3 and 4 present the generalization of the well-known Fatou Theorems for Fourier trigonometrical series (see, for example, Bary N., v.I [1964]; Zygmund A., v.I [1959]) and are obtained by the author B. Osilenker [1975]. The statements of §4 are new. Chapter VI. §1. The representations of the trilinear kernel in general orthogonal polynomials were obtained by author in the works Osilenker B. [1988a], [1991], [1993]. An infinite series with products of Jacobi polynomials were calculated by Dougall J. [1919]; Rahman M., Shan M. [1985] and Van Haringen H. [1986]. §2. Product formula for classical orthogonal polynomials was studied by numerous authors, we will cite the names Gegenbauer L.; Bochner S. [1954]; Askey R., Wainger St. [1969]; Gasper G. [1971], [1972]; Koornwinder T. [1974a], [1974b]; Laine T. [1980]; Connett W., Schwartz A. [1990a], [1990b]. Many works are devoted to the applications of the product formula for Jacobi polynomials to the problems of Approximation Theory: Jidkov G. [1966]; Connett W., Schwartz A. [1977], [1979]; Rafalson S. [1983]; Potapov M., Fjodorov V. [1985]; Yadav S. [1989]. Theorem 2.3 is due to Osilenker B. [1988a], [1993]. §3 is devoted to the Generalized Translation (Shift) Operator (GTO) in general orthogonal polynomials. This operator was introduced by Delsart J. in 1938 by analogy with usual shift operator, acting upon the trigonometric system. Some important ideas and a series of initial results belong to this scientist. Fundamental investigations of GTO were obtained by Levitan B. In the books Levitan B. [1964], [1973], the surveys Litvinov G. [1987]; Vainerman L. [1988], and in Schwartz A. [1988]; Connett W., Schwartz A. [1990a], [1990b]; Connett W., Markett C , Schwartz A. [1991]; Markett C. [1984]; Bavinck H. [1972] the results related with the abstract theory of GTO and their applications to the GTO, generated by Sturm-Liouville differential and difference operators, are presented (see also Contemporary Mathematics, 1994). The discrete hypercomplex systems, associated with the general orthogonal polynomials, were introduced by Berezanskii Yu. in 1951. They were studied by Berezanskii Yu., Kalujnii A. [1986]; Lasser R. [1983], Szwarc R. [1992], [1993a], [1993b]; Voit M. [1990a], [1990b]. More details about Harmonic Analysis in hypercomplex systems will be found in the book of Berezanskii Yu., Kalujnii A. [1992] (and bibliography). The results of §3 were discovered by authors in the works [1988a], [1992b], [1993].

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[1990b] Central Limit Theorems for Random Walks on No that are Associated with Orthogonal Polynomials, J. Multivar. Analysis, 34, 2, 290-322. Whittaker E. T., Watson G. N. [1933-1934] A course of modern analysis. Cambridge, Univ. Press. Wing M. [1950] The mean convergence of orthogonal series, Amer. J. Math., 72, N4, 792908. XuY. [1993] Mean Convergence of Generalized Jacobi Series and Interpolating Polynomials, I, J. Approx. Theory, 72, 237-251. Yadav S. P. [1989] On the Saturation Order of Approximation Processes Involving Jacobi Polynomials, J. Approx. Theory, 58, Nl, 36-49. Young W. H. [1913] On the determination of the summability of a function by means of its Fourier constants, Proc. London Math. Soc, 12, 71-88. Zygmund A. [1959] Trigonometric series, Vols. I and II. Cambridge Univ. Press, London and New York.

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281

Author Index

Abel N. H. 23, 145, 176, 203, 212, 215, 224 Akhiezer N. I. 262, 267 Akilov G. P. 261, 264, 272 Alexits G. 262, 264, 267 Alfaro M. 263, 273 Ambroladze M. U. 263, 267 Aptekarev A. I. 263, 267 Askey R. 263, 265, 266, 267 Atkinson F. 262, 267 Badkov V. M. 265, 267 Baire R. 31 Banach S. 25, 27, 32, 160, 261, 267 Barkov G. I. 263, 267 Baron S. A. 261, 268 Bary N. K. 265, 266, 268 Bavinck H. 266, 268 Berezanskii Yu. M. 262, 266, 268 Berman D. L. 264, 268 Bernoulli J. 15 Bernstein S. N. 9, 101, 226, 263, 265, 268 Bessel F. 16, 97-100,102, 122, 123, 135, 153, 164, 166 Bochner S. 266, 268 Bonami A. 265, 268 Borel E. 42 Bunyakowskii V. Ya. 8, 9, 53, 65, 71, 106, 110, 127, 131, 133, 139, 146, 147, 148, 151, 153, 163, 165, 169, 173, 183, 189 Calderon A. P. 42, 46, 268 Carleman T. 264, 268 Case K. M. 263, 270 Cauchy A. L. 9, 25, 65, 71,110,127,153, 163, 165 Cesaro E. 19, 20 Chanillo S. 265, 268 Chebyshev P. L. 9, 36, 103, 116, 117, 138, 194, 258, 262, 268 Chihara T. 262, 269

Christoffel E. B. 67, 76, 107, 108, 136, 140, 141, 148, 152, 163, 168, 170, 173, 174, 180, 186, 187, 202, 206, 207, 233, 262, 269 Clerc J. L. 265, 268 Coifman R. 261, 269 Connett W. 265, 266, 269 Darboux G. 76, 107, 136, 140, 141, 148, 152, 163, 168, 170, 173, 174, 180, 183, 186, 187, 202, 206, 207, 233, 262, 269 Delsart J. 266, 269 Dini U. 12 Dirichlet G. L. 33, 75-78, 129, 135 Dombrowski J. 263, 265, 266, 269 Dougall J. 239, 266, 269 Du Bois-Reymond 33 Dyn'kin E. M. 262, 269 Edwards R. E. 261, 270 Efimov A. V. 265, 270 Egorov D. F. 6 Erd'elyi A. 262, 270 Erdos R 262 Euler L. 15, 16, 21 Fatou R 6, 246, 259, 266 Favard J. 262, 270 Fefferman Ch. 261, 269 Fejer L. 152, 175, 202, 204, 205, 219 Fjodorov V. M. 266, 276 Fomin G. A 261, 264, 265, 270 Fomin S. V. 261, 262, 264, 272 Freud G. 261, 262, 263, 270 Fricke G. H. 265, 269 Fubini G. 6 Garnett J. B. 261, 270 Garsia-Cuerva J. 262, 270 Gasper G. 263, 266, 270 Gauss K. 15 Gegenbauer L. 266 Gehring F. W. 261, 270 Geronimo J. S. 263, 270 Geronimus Ya. L. 271

282 Gilewicz J. 262, 263, 271 Gram J. 54, 56, 262, 271 Gronwall T. H. 117 Guadalupe J. J. 265, 271 Hardy G. H. 33, 40, 42, 189, 248, 255, 261, 271 Hausdorff F. 264, 271 Hermite Ch. 262 Hilbert D. 49, 51, 54, 71, 73, 121, 262 Hille E. 265, 271 Hirschmann I. I. 265, 267 Holder O. 7, 8, 37, 45, 48, 172, 180, 192, 211 l'Hopital 76, 78, 203 Ismail M. E. H. 263, 271 Jackson D. 12, 14, 137, 261, 271 Jacobi G. G. J. 70, 71, 80, 83, 85, 86, 91, 92, 101, 105, 117, 142, 177, 184, 222, 230, 262, 265, 266, 271 Jidkov G. V. 266, 271 Kaczmarz S. 264, 271, 272 Kalney S. G. 265, 272 Kalujnii A. A. 266, 268 Kantorovich L. V. 261, 262, 264, 272 Kashin B. S. 264, 272 Herman R. 265, 272 Kharshiladze F. J. 264 Kokilashvili V. M. 262, 272 Kolmogorov A. N. 261, 264, 272 Koornwinder T. 266, 272 Korous J. 263, 272 Krall A. M. 263, 273 Kronecker L. 1, 55, 61 Laguerre E. 262 Laine T. 266, 272 Lasser R. 266, 272 Lebedev N. N. 261, 272 Lebesgue H. 5, 10, 11, 35, 59, 60, 74, 114, 116, 133, 134, 139, 140, 143, 147, 157, 182, 201, 206, 209, 211, 212, 221, 222, 242 Legendre A. 104, 200, 265 Leibnitz G. 81 Leopold E. 262, 263, 271, 272

Author Index Levi B. 6, 26, 116, 128, 146 Levitan B. M. 266, 273 Lindelof E. 33 Liouville J. 17, 91, 95, 266 Lipschitz R. O. 12 Littlewood J. E. 33, 40, 42, 189, 248, 255, 261, 271 Litvinov G. L. 266, 273 Lozinskii S. M. 264 Lubinsky D. 263, 273 Magnus W. 270 Marcellan F. 263, 273 Marcinkiewicz J. 33, 38, 261, 264, 273 Markett Cl. 266, 269, 273 Mate A. 265, 266, 273 Meaney Chr. 265, 273 Men'shov D. E. 264, 273 Mingarelli A. B. 263, 273 Minkowski G. 8, 26, 181, 257 Muckenhoupt B. 43-45, 257, 261, 265, 258, 273 Mulla F. S. 263, 271 Nakman A. D. 264, 265, 274 Natanson G. I. 265, 274 Natanson I. P. 261. 274 Nevai R G. 117, 233, 262, 263, 265, 266, 269, 273, 274 Newman J. 265, 274 Nikishin E. M. 274 Nikolaev V. F. 264, 274 Nikol'skii S. M. 265, 274 Norlund N. E. 22 Oberhettinger F. 270 Osilenker B. P. 262-266, 269, 274, 275 Parseval M. A. 122, 135 Peebles G. 263, 275 Peherstorfer F. 263, 275 Perez M. 265, 271 Phragmen E. 33 Plessner A. I. 264, 275 Poisson S. D. 23, 215, 216 Pollaczek F. 263, 265, 275, 276 Pollard H. 265, 276 Polya D. 262

Author Index

Potapov M. K. 266, 276 Price J. J. 263, 273 Rademacher H. 264, 276 Rafalson S. Z. 266, 276 Rahman M. 239, 263, 266, 270, 276 Rahmanov E. M. 262, 263, 276 Ratajski H. 264, 276 Rellich F. 262, 276 Rezola M. L. 263, 273 Riemann B. 73 Riesz M. 23, 33, 35, 36, 167, 261, 276 Riesz F. 264, 276 Rubio de Francia J. L. 262 Ruiz F. J. 265, 271 Rudin W. 265, 274 Sawyer E. 261, 277 Schmidt E. 56, 262, 277 Schwartz A. 265, 266, 269, 277 Schwartz H. 8, 53, 100, 106, 131, 133, 146, 147, 148, 151, 169, 173, 189 Seliverstov G. A. 264, 272 Shah M. J. 239, 266, 276 Sobolev S. L. 263 Sonin N. Ya. 262, 277 Stahl H. 277 Stechkin S. B. 264 Stein E. 261, 277 Steinhaus H. 32, 160, 261, 267, 272 Steklov V. A. 95, 263 Stieltjes T. 35, 73, 262, 277 Stirling J. 15 Stone M. 262, 277 Sturm 91, 266 Suetin P. K. 262, 277 Szego G. 262, 263, 265, 277 Sz.-Nagy B. 265, 277 Szwarc R. 263, 266, 277, 278 Tamarkin J. D. 265, 271 Tandori K. 264, 265, 278 Telyakovskii S. A. 265, 278 Thorin G. O. 33, 35, 36, 167, 261, 278 Toeplitz O. 261, 278 Tolstov G. P. 261, 278 Torchinsky A. 262, 278

283

Totik V. 263, 265, 273, 274, 277 Tricomi F. G. 270 Trigub R. M. 265, 278 Turan P. 262 Vainerman L. I. 266, 278 de la Vallee-Poussin 175, 191, 202, 205 Van Assche W. 263, 270, 278 Van Doom E. A. 262, 278 Van Haringen H. 266, 278 Varona J. L. 265, 271 Vitali G. 40 Voit M. 263, 266, 278, 279 Voronoj G. F. 22 Wainger St. 266, 267 Watson G. N. 261, 279 Weierstrass K. 12, 27, 53 Weil H. 129 Weiss G. 261, 277 Wilson J. 263, 267 Whittaker E. T. 261, 279 Wing M. 265, 279 Xu. Y. 265, 279 Yadav S. P. 266, 279 Young W. H. 264, 279 Zhang J. 263, 274 Zygmund A. 42, 46, 261, 264, 266, 268, 273, 279

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285

Subject I n d e x

>i-summability 23, 24 Closure equation 122 >l*-summability 216, 221, 223 Comparison Theorems 105 Abel's transform 19, 78, 145, 176, 203, Complete space 25 212, 224, 256 Complete system 51, 123, 126 Abel-Poisson summation process 23, 215- Conjugate index 8, 35 222, 227 Convergence Baire Theorem 31 -almost everywhere of orthogonal polyBanach space 25, 27 nomials series 137, 138, 142 Banach-Steinhaus Theorem 32,160, 261 -almost everywhere of orthogonal series Berman-Marcinkiewicz-Zygmundformu- 127, 130 la 264 -in mean with index r 26, 265 Bernoulli numbers 15 -in norm (metric) space 25, 126 Bernstein's inequality 9, 226 Cyclic element 74, 75 Bessel's differential equation 16 De la Vallee-Poussin means 191 Bessel's function Differentiated Fourier expansions 223 -first kind 16, 97, 99, 102, 261 Dini-Lipschitz's condition 12 -second kind 16, 98, 261 Dirichlet's kernel 75-78, 129, 139 Bessel's inequality 122, 123, 135, 153, Divergence of orthogonal expansion 155164, 166 160, 197-201, 265 Calderon-Zygmund Decomposition Lem- Double-humpbacked majorant 248-253, ma 42, 46 256 Cauchy-Bunyakowskii's inequality 9, 65, Egorov's Theorem 6 71, 110, 127, 153, 163, 165 Euler's constant 16-21 Chebyshev's inequality 9, 36, 194, 258 Fatou's Lemma 6, 246, 259 Chebyshev's polynomials Fatou's Theorem 223, 266 -first kind 103, 138, 233 Fejer's means 152 -second kind 103, 116, 117, 233 First category set 30 Christoffel's formula 67 Fubini's Theorem 6 Christoffel's function 108 Function Christoffel-Darboux summation formu- -Beta 15 la 176, 107, 136, 140, 141, 149, 152, 163, -bounded (finite) variation 3, 4, 6 168, 170, 173, 174, 180, 183, 186, 187, -characteristic 2, 10, 148, 174, 189 202, 206, 207, 233, 262 -concave 3 Cesaro mean's 214, 264 -convex 3 -(C,a) (a > 0) 19-22, 24, 188 -Gamma 14, 15, 88, 261 -(C, 1) 20, 23, 143-154, 264 -monotone 2, 71 Classical hypergeometric orthogonal po- -//-integrable 4 lynomial 263 -/x-measurable 4 Classical orthogonal polynomials 262, 263,-simple 5, 35 265 Function classes

286 -Up a 12, 264 - DL 12 - I £ , Lrw, C 4, 5, 25 Fundamental sequence 25 Fundamental Spectral Theorem 72, 262 Generalized closure equation 124 Generalized Product Formula 241-247 Generalized Translation (shift) operator 231, 248, 254-259, 266 Generalized sum 18 Gram's matrix 54 Gram-Schmidt process 56, 58, 75, 262 GronwalFs inequality 117 (i/, g)-summability 150, 151 Hardy-Littlewood maximal function 40, 189, 248, 255 Hardy-Littlewood Maximal Theorem 41, 42 Hermite polynomials 262 Holder's inequality 7, 8, 37, 45, 48, 172, 180, 183, 192, 211 l'Hopital rule 76, 78, 203 Humpbacked majorant 248-251 Interpolation Theorem 33-48 -Marcinkiewicz 38 -Riesz-Thorin 36, 167 Jackson's Theorem 12, 14, 137 Jacobi matrix 70, 71, 263 Jacobi polynomials 80-104,105,119,177, 184, 214, 222, 230, 262, 265, 266 Kernel function -De la Vallee-Poussin 175, 202, 205, 265 -Dirichlet 75-78, 135 -Fejer 175, 201, 204, 205, 219 -A-kernel 212 -Poisson 216 -trilinear 231-240 Kronecker's symbol 1, 55, 61 Zr-space 26, 27 Laguerre polynomials 262 Lebesgue -constants 114, 157 -functions 129, 133, 139, 140, 175, 177 -inequality 134

Subject Index -points 9, 182, 187, 201, 206, 209, 211, 212, 214, 216, 221, 222 -quasifunction 242 -Theorem 5 (C, l)-functions 143, 147 A-function 161-178 Legendre polynomials 104,109, 200, 265 Levi's Theorem 6, 116, 128, 133, 146 Linear functional 128 Linear means 19 Linear methods of summability 184-195, 201-215 Linear operator 27-33 Linear span 50 Liouville formula 17 Liouville-Steklov method 95 Lipschitz's condition 12 .M-class 232, 233, 262 Mass distribution (distribution) 4 Mean Value Theorem 5, 138 Measure -absolutely continuous 4, 248 -Borel 42 -completely additive 3 -Lebesgue 4, 180. 257 -/z-measure 4, 5, 59 Menshov-Rademacher Theorem 264 Metric space 24, 27 Minkowski's inequality 8, 26, 181, 257 Modulus of continuity 12 Momentum problem 73, 214, 262 Muckenhoupt's condition 43-45, 257 Nontangential track 216, 221, 228 Norm -element 24, 50 -operator 28 Normed space 24 Operator -bounded 28 -continuous 27 -inverse 159 -isometry 73, 159 -polynomial 155, 156, 158, 160, 264 -self-adjoint 72

Subject Index

287

-symmetric 72 (C, l)-summation almost everywhere 145 -type (r, s) 35, 36 T-regularity 18, 19, 21, 22, 23, 152, 161, -weak-type (r,s) 35, 36 171, 183, 193, 194, 212, 213, 214 Orthogonality 51 Three-line Theorem 34 Orthogonal (orthonormal) polynomials Triangle inequality 24 -definition 59-63 Triangular matrix 19 -class Qo 215, 216, 220 Total set 52, 53, 59 Orthogonal system Uniform Boundedness Principle 31, 261 -definition 55, 58 Weight function (weight) 5, 62, 239, 263 Orthogonalization Theorem 55 Weighted norm inequalities 42-48, 261 Orthonormal basis 55 Weil's multiplier 129 Parallelogram identity 50 Weierstrass Theorem 12, 27, 53 ParsevaFs formula (equality) 122, 135 Variation of function 3 Perturbation of the Chebyshev polyno- Vitali's Lemma 40 mials 117 Voronoj-Norlund means 22 Phragmen-Lindelof Principle 33 Zeros of orthogonal polynomials 78-80 Poisson-Abel method (see also Abel-Poisson summation process) 23, 215-222, 227, 264 Principle of localization of singularity 32 Projection of element 51, 121 Quasipotential function 253 Recursion formula -three-term 68, 70, 72, 263 -five-term 70 Residue 31 Resolution of unity 72 Resonance Theorem 32, 261 Reverse Holder's inequality 47 Riesz weighted means 23 Riesz-Thorin Theorem (see also Interpolation Riesz-Thorin Theorem) 36,167 Schwartz-Bunyakowskii's inequality 8, 53, 100, 106, 131, 133, 139, 146, 147, 148, 151, 169, 173, 189 second category set 30, 199 separable space 127 spectrum 73 simple (Lebesgue) spectrum 74 Steklov's conjecture 263 Strong summability (see also (if, g)-summability) 150, 179-184 Sturm-Liouville singular operator 91, 266 Summation of the series 177

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