Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
684 Elemer E. Rosinger
Distributions and Nonlinear Partial Differential Equations
Springer-Verlag Berlin Heidelberg New York 1978
Author Elemer E. Rosinger Department of Computer Science Technion City Haifa/Israel
A M S Subject Classifications (1970): 35Axx, 35 Dxx, 46 Fxx
ISBN 3-540-08951-9 ISBN 0-387-08951-9
Springer-Verlag Berlin Heidelberg NewYork Springer-Verlag NewYork Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
to my wife
HERMONA
PREFACE
The nonlinear method in the theory of distributions presented in this work is based on embeddings of the distributions in
D' 0{n)
into associative and commutative algebras
whose elements are classes of sequences of smooth functions on fine various distribution multiplications.
R n. The embeddings de-
Positive powers can also be defined for car
rain distributions, as for instance the Dirac 6 function.
A framework is in that way obtained for the study of nonlinear partial differential equations with weak or distribution solutions as well as for a whole range of irregular operations on distributions, encountered for instance in quantum mechanics.
In chapter i, the general method of constructing the algebras containing the distributions and basic properties of these algebras are presented. The way the algebras are constructed can be interpreted as a sequential completion of the space of smooth functions on
R n. In chapter 2, based on an analysis of classes of singularities of piece
wise smooth functions on
R n, situated on arbitrary closed subsets of
R n with smooth
boundaries, for instance, locally finite families of smooth surfaces, the so called Dirac algebras, which prove to be useful in later applications are introduced.
Chapter 3 presents a first application. A general class of nonlinear partial differential equations, with polynomia~ nonlinearities is considered. These equations include among others, the nonlinear hyperbolic equations modelling the shock waves as well as well known second order nonlinear wave equations.
It is shown that the piece wise
smooth weak solutions of the general nonlinear equations considered, satisfy the equations in the usual algebraic sense, with the multiplication and derivatives in the algebras containing the distributions.
It follows in particular that the same holds
for the piece wise smooth shock wave solutions of nonlinear hyperbolic equations.
A second application is given in chapter 4, where one and three dimensional quantum particle motions
in potentials arbitrary positive powers of the Dirac 6 function are
considered. These potentials which are no more measures, present the strongest local singularities studied in scattering theory.
It is proved that the wave function solu-
tions obtained within the algebras containing the distributions, possess the scattering property of being solutions of the potential free equations on either side of the potentials while satisfying special ~unction relations on the support of the potentials. In chapter 5, relations involving irregular products with Dirac distributions are proved to be valid within the algebras containing the distributions. lar, several known relations in quantum mechanics, involving
In particu-
irregular products with
VI
Dirac and Heisenberg distributions are valid within the algebras.
Chapter 6 presents
the peculiar effect coordinate scaling has on Dirac distribution derivatives. That effect is a consequence of the condition of strong local presence the representations of the Dirac distribution satisfy in certain algebras. In chapter 7, local properties in the algebras are presented with the help of the notion of support, the local character of the
product being one of the important results. Chapter 8 approaches the
problem of vanishing and local vanishing of the sequences of smooth functions.which generate the ideals used in the quotient construction giving the algebras containing the distributions. That problem proves to be closely connected with the necessary structure of the distribution multiplications. The method of sequential completion used in the construction of the algebras containing the distributions establishes a connection between the nonlinear theory of distributions presented in this work and the theory of algebras of continuous functions.
The present work resulted from an interest in the subject over the last few years and it was accomplished while the author was a member of the Applied Mathematics Group within the Department of Computer Science at Haifa Technion. In this respect, the author is particularly glad to express his special gratitude to Prof. A. Paz, the head of the department, for the kind support and understanding offered during the last years.
Many t h a n k s go t o t h e c o l l e a g u e s a t T e c h n i o n , M. I s r a e l i reference
indications,
als positive
respectively
and L. Shulman, f o r v a l u a b l e
for suggesting the scattering
powers o f t h e D i r a c 6 f u n c t i o n ,
problem in potenti-
s o l v e d i n c h a p t e r 4.
The author is indebted to Prof. B. Fuchssteiner from Paderborn, for his suggestions in contacting persons with the same research interest.
Lately, the author has learnt
about a series
the Institute
P h y s i c s a t Aachen, p r e s e n t i n g
for Theoretical
proach to the problem of irregular
o f e x t e n s i v e p a p e r s o f K. K e l l e r , a rather
operations with distributions.
from
complementary a p
The a u t h o r i s v e r y
g l a d t o t h a n k him f o r t h e k i n d and t h o r o u g h exchange o f v i e w s . A special gratitude and acknowledgement is expressed by the author to R.C. King from Southampton University, for his generosity in promptly offering the result on generalized Vandermonde determinants which corrects an earlier conjecture of the author and upon which the chapters 5 and 6 are based.
All the highly careful and demanding work of editing the manuscript was done by my wife Hermona, who inspite and on the account of her other much more interesting and elevated usual occupations found it necessary to support an effort in regularizing
VII
the irregulars ..., in multiplying the distributions
...
By the way of multiplication: Prof. A. Ben-Israel, a former colleague, noticing the series of preprints, papers, etc. resulted from the author's interest in the subject and seemingly inspired by one of the basic commandments in the Bible, once quipped: "Be fruitful and multiply ... distributions
..."
E. E. R.
Haifa, December 1977
CONTENT
Chapter
I. §I. §2. §3. 54. §5. §6. § 7. §8. §9. §i0. §Ii. §12.
Chapter
2. §i. §2. §3. ~4. §5. §6. §7. §8. §9.
§i0. Chapter
3. §i. §2. §3. §4.
Chapter
4. §I. §2. §3. §4. §5.
Chapter
5. §i. §2. §3. §4. §5. §6. §7. §8.
Associative, Commutative Algebras Containing the Distributions
. .
Nonlinear Problems ........................ Motivation of the Approach .................... Distribution Multiplication . . . . . . . . . . . . . . . . . . . . Algebras of Sequences of Smooth Functions . . . . . . . . . . . . . Simpler Diagrams of Inclusions .................. Admissible Properties . . . . . . . . . . . . . . . . . . . . . . . Regularizations and Algebras Containing the Distributions . . . . . Properties of the Families of Algebras Containing D'(R n) . . . . . Defining Nonlinear Partial Differential Operators on the Algebras Maximality and Local Vanishing .................. Stronger Conditions for Derivatives . . . . . . . . . . . . . . . . Appendix ............................. Dirac Algebras Containing the Distributions
.
............
3 3 5 7 9 12 13 14 18 22 23 28 29 33
Introduction ........................... Classes of Singularities of Piece Wise Smooth Functions . . . . . . Compatible Ideals and Vector Subspaces of Sequences of Smooth Functions . . . . . . . . . . . . . . . . . . . . . . . . Locally Vanishing Ideals of Sequences of Smooth Functions . . . . . Local Classes and Compatibility . . . . . . . . . . . . . . . . . . Dirac Algebras .......................... Maximality ............................ Local Algebras .......................... Filter Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . Regular Algebras .........................
33 33 3h 39 42 h5 47 49 52 55
Solutions of Nonlinear Partial Differential Equations Application to Nonlinear Shock Waves ...............
60
Introduction ........................... Polynomial Nonlinear Partial Differential Operators and Solutions . Application to ~nlinear Shock Waves ............... General Solution Scheme for Nonlinear Partial Differential Equations
60 60 65 66
Quantum Particle Scattering in Potentials Positive Powers of the Dirac ~ Distribution . . . . . . . . . . . . . . . . . . . .
70
Introduction ........................... Wave Functions, Junction Relations ................ Weak Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . Smooth Representations for 6 . . . . . . . . . . . . . . . . . . . Wave Function Solutions in the Algebras Containing the Distributions ...................
70 70 72 78
Products with Dirac Distributions
85
.................
Introduction . The Dirac Ideal "I~ i ~ i i i i i i i i i i i i ~ i i i ~ i ~ i i i Compatible Dirac Classes TE . . . . . . . . . . . . . . . . . . . Products with Dirac Distributions . . . . . . . . . . . . . . . . . Formulas in Quantum Mechanics . . . . . . . . . . . . . . . . . . . A Property of the Derivative in the Algebras ........... The Existence of the Sequences in Z o . . . . . . . . . . . . . . . Stronger Relations Containing Produc£s ~ith Dirac Distributions . .
82
85 86 86 89 97 98 705
6,
Chapter
§I. §2. §3. §4. Chapter
7. §I. §2. §3. 54.
Chapter
8.
§1. §2. §3.
Reference
Linear Independent Families of Dirac D i s t r i b u t i o n s
........
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . C c ~ p a t i b l e Algebras and T r a n s f o r m a t i o n s . . . . . . . . . . . . . . Linear Independent Families of Dirac Distributions ........ G e n e r a l i z e d Dirac Elements . . . . . . . . . . . . . . . . . . . . Support,
Local Properties
111 111 111 113 115
. . . . . . . . . . . . . . . . . . . . .
120
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . The E x t e n d e d Notion of Support . . . . . . . . . . . . . . . . . . Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . The E q u i v a l e n c e b e t w e e n S = 0 and supp S = ~ .........
120 120 12~ 131
Necessary Structure o f the D i s t r i b u t i o n M u l t i p l i c a t i o n s
132
......
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Zero Sets and Families . . . . . . . . . . . . . . . . . . . . . . Zero Sets and Families at a Point . . . . . . . . . . . . . . . . .
132 132 13~
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
XI
NOTE
The
Reader
first
interested
lecture
Chapter
PARTIAL
on the
following
DIFFERENTIAL
E~UATIONS , may
at a
sections:
1 -
9
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pp.
3 - 23
§§ 1 -
6
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33 - h7
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pp.
55 - 59
pp.
60 - 69
2
§ 10
Chapter
concentrate
in N O N L I N E A R
1 §§
Chapter
mainly
.
.
3 §§ 1 - 4
. . . . . . . . . . . . . . . . . . . . . . . . . .
"Never
forget
the beaches
of
ASHQELON
... "
Chapter
1
ASSOCIATIVE, COM~TATIVE ALGEBRAS CONTAINING THE DISTRIBUTIONS
§I.
NONLINEAR PROBLEMS
The theory of distributions has proved to be essential in the study of linear partial differential equations. The general results concerning the existence of elementary solutions, [103], [34], P-convexity as the necessary and sufficient condition for the existence of smooth solutions, [103], the algebraic characterization of hypoellipticity, [64], etc., are several of the achievements due to the distributional approach, [154], [63], [64], [153], [156], [33], [114].
In the case of nonlinear partial differential equations certain facts have pointed out the useful role a nonlinear theory of distributions could play. For instance, the appearance of shock discontinuities in the solutions of nonlinear hyperbolic partial differential equations, even in the case of analytic initial data, [62], [89], [113], [50] [70], [51], [24], [25], [26], [31], [32], [52], [58], [71], [79], [84], [90], [91], [133], [1493, [163], indicates that in the nonlinear case problems arise starting with a rigorous and general definition of the notion of solution. Important cases of nonlinear wave equations, [5], [9], [i0], [Ii], [121], prove to possess distribution solutions of physical interest, provided that 'irregular' operations, e.g. products, with distributions are defined. Using suitable procedures, distribution solutions can be associated to various nonlinear differential or partial differential equations, [I], [2] [30], [42], [43], [45], [80], [92], [94], [117], [118], [119], [120], [122], [138], [146], [147], [159], [160], [161]. In quantum mechanics, procedures of regularizing divergent expressions containing 'irregular' operations with distributions, such as products, powers, convolutions, etc., have been in use, [6], [7], [12], [13], [15], [19], [20], [21], [29], [54], [55], [56] [57], [60], [76], [77], [78], [112], [143], [151], suggesting the utility of enriching in a systematic way the vector space structure of the distributions. A natural way to start a nonlinear theory of distributions is by supplementing the vector space structure of
D'(Rn)
with a suitable distribution multiplication.
Within this work, a nonlinear method in the theory of distributions is presented, based on an associative and commutative multiplication defined for the distributions in D'CRn), [125-131]. That multiplication offers the possibility of defining arbitrary po-
e.g. the Dirac 6 function, [130],
sitive powers for certain distributions,
The definition of the multiplication
rests upon an analysis of classes of singulari-
ties of piece wise smooth functions on Rn
with smooth boundaries,
in
R n (chap. 2, §3).
[1513.
Rn, situated on arbitrary closed subsets of
for instance locally finite families of smooth surfaces
Several applications are presented. First, in chapter 3, it is shown that the piece neral class of nonlinear partial differential
wise smooth weak solutions of a ge-
equations satisfy those equations in
the usual algebraic sense, with the multiplication containing the distributions.
and derivatives
in the algebras
As a particular case, it results that the piece wise
smooth nonlinear shock wave solutions of the equation,
[903, [713, [1333, [52], [323
[1313: ut(x,t
)
u(x,o)
where
a
+ a(u(x,t)) = Uo(X)
,
• Ux(X,t ) x
c aI
= 0
, x
c R1
,
t
>
0
,
,
is an arbitrary polynomial
in
u , satisfy that equation in the usual al-
gebraic sense. Second, in chapter 4, quantum particle motions in potentials arbitrary positive powers of the Dirac d distribution are considered. est local singularities [115], [1163, [1403. The
~2"(x)
+
These potentials present the strong-
studied in recent literature on scattering,
[273, [33, [283,
one dimensional motion has the wave function ~ given by
(k-a(~(x))m)~(x)
= 0
,
x
e RI
, k,c¢
c R1
, m ~ CO,~]
while the three dimensional motion assumed spherically symmetric and with zero angular momentum has the radial wave function (r2R'(r)) ' + r2(k-~(6(r-a))m)R(r)
R
given by
= 0 , r ~ (0, °°) , k,~ c R 1 , a,m ~ (0, ~) .
The wave function solutions obtained possess a usual scattering property, namely they consist of pairs
~ ,~+
of usual
C~
solutions of the potential free equations,
each valid on the respective side of the potential while satisfying special junction relations on the support of the potentials. Third, it is shown in chap. 5, §5, that the following well known relations in quantum mechanics,
[108], involving the square of the Dirac 6 and Heisenberg
butions and other irregular products hold:
6+,6_
distri-
(6)2 _ (llx)2/ 2 = _(i/x2)/ 2 6 • (I/x) = -D6/2 [~+)2 = -D6/4~i - (i/x2)/4:2
(~_)2 = D 6 / 4 ~ i
where
§2.
- (1/x2)/4~ 2
6+ = ( 6 + ( 1 / x ) / ~ i ) / 2
, 6_ = ( 6 - ( 1 / x ) / ' r r i ) / 2
.
MOTIVATION OF THE APPROACH
The distribution multiplication, defined for any given pair of distributions in D'(Rn), could either lead again to a distribution or to a more general entity. Taking into account H. Lewy's simple example, [93] (see also [64], [155], [48]), of a first order linear partial differential operator with three independent variables and coefficients polynomials of degree at most one with no distribution solutions, the choice of a distribution multiplication which could in the case of particularly irregular factors lead outside of the distributions, seems worthwhile considering. Such an extension beyond the distributions would mean an increase in the 'reservoir' of both data and possible solutions of nonlinear partial differential operators, not unlike it happened with the introduction of
distributions in the study of linear partial differential operators,
[is4].
One can obtain a distribution multiplication in line with the above remarks by embed*) D'(R n) into an algebra A . It would be desirable for a usual Calculus if the
ding
algebra
A
were associative, commutative, with the function
~(x) = i ,
V x ~ Rn ,
its unit element and possessing derivative operators satisfying Leibnitz type rules for the product derivatives. Certain supplementary properties of the embedding
D,(R n) c A
concerning multiplication, derivative, etc. could also be envisioned.
There is a particularly convenient classical way to obtain such an algebra
A , namely,
as a sequential completion of
D'(R n) .
D'(R n)
or eventually, of a subspace
F
in
The sequential completion, suggested by Cauchy and Bolzano, [158], was employed rigorously by Cantor, [22], in the construction of
R 1 . Within the theory of distributions
the sequential completion was first employed by J. Mikusinski, [105] (see also [ii0]) in order to construct the distributions in functions on
*)
D'(R I)
from the set of locally integrable
R 1 , however without aiming at defining a distribution multiplication.
All the algebras in the sequel are considered over the field numbers.
C1
of the complex
Later, in [106], the problem of a whole range of 'irregular' multiplication
The method of the sequential
completion possesses
First, there exist various subspaces ciative, commutative
F
in
Starting with such a subspace
completion
A
A , one obtains
which are in a natural way asso~(x) = 1 ,
F , it is easy to construct commutative
a sequential
algebra with unit ele-
ple characterization
Choosing a suitable
completion
W
is an associative,
subalgebra
A
in
W
of 'regular'
results in a constructive in
A
is obtained.
functions
in
much in the spirit of various
of partial differential
in
F .
commutative
al-
and an ideal
completion
a stage in a succession
Rn
a sim-
Indeed, these elements will (in this work,
'weak solutions'
of Calculus,
the distributional
of a function space, [105],[110],[4]
F = C~CR n)
used in the study
approach
from numbers to numbers.
even nonmeasurable
functions,
definition
That extension
- essenti-
- can be viewed as
of attempts to define the notion of function.
as an analytic one was extended by Dirichlet's
valent correspondence
way. Further,
equations.
Within the more general framework ally a sequential
A
of the elements
be classes of sequences will be considered)
on sequences,
A = A/I .
Second, the sequential
encompassing
advantages.
W = N ÷ F , that is, the set of all sequences with elements
gebra with unit element.
function,
- among them, completion.
is - from purely algebraic point of view - the following
With the term by term operations
in
D'(R n)
which will also be an associative,
Indeed, the procedure
one. Denote
two important
algebras, with the unit element the function
W x ~ Rn .
ment.
operations
- was formulated within the framework of the sequential
Euler's idea of
accepting any uni-
although significant
-
provided the Axiom of Choice is assumed,
[491 - failed to include certain rather simple important cases,
as for instance, the
Dirac 6 function and its derivatives.
It is worthwhile mentioning that the distributional tain approaches
in Nonstandard Analysis.
ned by a sequential dard
completion
to numbers
can be paralleled by cer-
In [1341, a nonstandard model of
of the rational numbers was presented.
R 1 , the Dirac ~ function becomes
(nonstandard)
approach
a usual univalent
obtai-
correspondence
from numbers
(nonstandard).
The notion of the germ of a function zation of the notion of function,
at a point which can be regarded as a generali-
since it represents more than the value of the func-
tion at the point but less than the function on any given neighbourhood is related both to the distributional
The variety of interrelated lus is still
R1
In that nonstan-
approaches
'in the making'.
approach and Nonstandard Analysis,
of the point, EI09],[97].
suggests that the notion of function in Calcu-
The particular
success of the distributional
approach
in the theory of linear partial differential equations (especially the constant coefficient case, otherwise see [93]) is in a good deal traceable to the strong results and methods in linear functional analysis and functions of several complex variables. In this respect, the distributional approach of nonlinear problems, such as nonlinear partial differential equations, can be seen as requiring a return to more basic and general methods, as for instance, the sequential completion of convenient function spaces, which finds a natural framework in the theory of Algebras of Continuous Functions (see chap. 8).
The sequential completion is a common method for both standard and nonstandard methods in Calculus and
its theoretical importance is supplemented by the fact that it
synthetizes basic approximation methods used in applications, such as the method of 'weak
solutions'. The nonlinear method in the theory of distributions presented in
this work is based on
the embedding of
D'(R n)
into associative and commutative al-
gebras with unit element, constructed by particular sequential completions of C=(Rn), resulting from an analysis of classes of singularities of piece wise smooth functions on
Rn , situated on arbitrary closed subsets of
Rn
stance locally finite families of smooth surfaces in
§3.
with smooth boundaries, for inR n (see chap. 2, §3).
DISTRIBUTION MULTIPLICATION
The problem of distribution multiplication appeared early in the theory of distributions, [135], [81-83], and generated a literature, [9], [11-21], [35-41], [46], [5357], [61], [66-69], [72-74], [76-78], [85], [106-108], [112], [125-132], [134], [136] [137], [148], [151], [162]. L. Schwartz's paper [135], presented a first account of the difficulties. tive algebra
A
Namely, it was shown impossible to embed
a)
the function
b)
the multiplication in
~(x) = 1 ,
1,x,x(ZnlxI-1) is c)
identical
there
exists
D'(R I)
with
A
V x ~ R 1 , is the unit element of the algebra
~ c°(R I)
the usual
a linear
D
satisfies D(a.b)
c.2) D
on = (Da)
A ;
of any two of the functions
multiplication
mapping
in
(generalized
O ° ( R 1)
;
derivative
operator)
of product
derivative:
such t h a t : c.1)
into an associa-
under the following conditions:
A
the Leibnitz
• b + a • (Db)
applied to the functions
rule ,
V a,b
~ A ;
D : A ÷ A ,
l,x,x2(~nlx]-l)
c
CI(RI)
is identical with the usual derivative in d)
there exists
~ e A ,
6 # 0
CI(R I) ;
(corresponding to the Dirac function) such that
x • ~ = 0 .
The above negative result was occasionally interpreted as amounting to the impossibi lity of a useful distribution multiplication. That could have implied that the distributional approach was not suitable for a systematic study of nonlinear problems. However, due to applicative interest (see §i) various distribution multiplications satisfying on the one side weakened forms of the conditions in [135] but, now and then also rather strong and interesting other conditions not considered in [135], have been suggested and used as seen in the above mentioned literature. In this respect, the challenging question keeping up the interest in distribution multiplication has been the following one: which sets of strong and interesting properties can be realized in a distribution multiplication?
There has been as well an other source of possible concern, namely, the rather permanent feature of the distribution multiplications suggested, that the product of two distributions with significant singularities can contain arbitrary parameters. However, a careful study of various applications shows that the parameters can be in a way or the other connected with characteristics of the particular nonlinear problems considered. The complication brought in by the lack of a unique, so called 'canonical' product, and the 'branching' the multiplication shows above a certain level of singularities can be seen as a rather necessary phenomenon accompanying operations with singularities.
The study of the literature on distribution multiplications points out two main approaches. One of them tries to define for as many distributions as possible, products which are again distributions, [9], [11-21], [35-41], [46], [53-57], [61], [66-69], [72-74], [78], [106-108], [112], [132], [136], [137], [148], [162]. That approach can be viewed as an attempt to construct maximal
'subalgebras' in
D' (Rn) ,
using various regularization procedures applied to certain linear functionals associated to products of distributions. Sometimes, [9], [ii], [78], the regularization procedures are required to satisfy certain axioms considered to be natural. A general characteristic of the approach is a trade-off between the primary aim of keeping the multiplication
within the distributions and the resulting algebraic and to-
pological properties of the multiplication which prove to be weaker than the ones within the usual algebras of functions or operators. The question arising connected with that approach is whether the advantage of keeping the product within the distributions compensates for the resulting restrictions on operations as well as for the lack of properties customary in a good Calculus.
The other approach,
a rather complementary
ic structure with suitable derivative strictions,
[81-83],
enabling a Calculus with minimal re-
[76], [771, [851, [125-1311,
seen as an attempt to construct
The p r e s e n t
one, aimes first to obtain a rich algebra-
operators,
embeddings of
work b e l o n g s t o t h e l a t t e r
A more fair comment would perhaps
[134~,
[151~. That approach can be
D'(R n) into algebras.
approack,
say that within the first approach,
one knows what
he computes with, even if not always how to do it, while within the second approach, one easily knows how to compute,
even if not always what the result is. However,
the
second approach seems to be more in line with the initial spirit of the Theory of Distributions,
aiming at lifting restrictions,
ges of operations
§4.
in Calculus,
simplifying
rules and extending
even if done by adjoining unusual
the ran-
entities.
ALGEBRAS OF SEQUENCES OF SMOOTH FUNCTIONS
The set
(1)
W = N + C~(R n)
of all
t h e s e q u e n c e s o f complex v a l u e d smooth f u n c t i o n s
quel the
s ( v ) (x)
general
framework.
If
s • W ,
For
~ ~ 6~°(Rn) • W
and
denote by u(~)(w)
u(~)
= ~ ,
{u(0)}
Denote by by
V
(2) where
o
the constant
give in the se-
s(v) • C~ (Rn )
sequence with the terms
algebra with the unit element
of sequences,
u(1)
and
~
, then
W
is an associati-
; the null subspace of
W
is
.
S
the set of all sequences s • W , weakly convergent o the kernel of the linear surjection: SO ~ s
>
<s,.> • D' (Rn)
<s,~> = lira f s(v)(x)~(x)dx v+oo R n
,
V , • D(R n)
.
Then
(3)
will
V ~ • N .
With the term by term addition and multiplication ve, comutative =
Rn
• C1
u(~)
o
on
x • Rn , then
v • N ,
So/Vo ~ (S+Vo)
is a vector space isomorphism.
>
<s,'>
• D ' ( R n)
in
D ' ( R n)
and
10
Since
W
is an algebra, one can ask whether it is possible to define a product of any
two distributions s + V
o
and
<s,.>
t + V
,
~ D'(R n)
by the product of the classes of sequences
o
A simple way to do it would be by constructing I
> A
diagrams of inclusions:
> W
(4)
V with
A
~S
o
subalgebra I n S
(4.1)
o
in = V
W
o
and
I
ideal in
So/V °
w
<
s+V
isom
(Vo • V o) n S o 4
v2 ~ S o
if
n = 1 , take
and
into an associative
A/I
> s+/ o
lin,inj
diagrams of type (4) cannot be constructed,
(S)
D'(R n)
w
<s,'>
Indeed,
linear embedding of
algebra with unit element:
D'(R n)
However,
, satisfying
o
which would generate the following and commutative
A
v 2 ~ Vo
since
Vo
v(w)(x)
, since
= cos(~+l)x
,
Vw
¢ N ,
x ~ R 1 , then
v £ V
o
= 1/2 .
An other way could be given by diagrams of inclusions: I
>A
Vo
> SO
> W
(6)
with
A
subalgebra
(6.1)
V
(6.2)
g
o o
in
W
and
I
ideal in
A
, satisfying
n A = /+ A
= S
o
which would generate the following linear injection of an associative algebra onto
D' (Rn)
:
and commutative
11
D'(R n)
So/V °
W
W
<S,'>
~
S+~ ~ < O lin,inj,sur
isom Here the problem
arises
diagrams
(6) with
of type
[4], [35-41],
[53],
connected
[68],
A/I
A
with
(6.2).
containing
[69],
Indeed,
s+l
it is not possible
some of the frequently
[105-110],
[136],
[137],
[162],
used
to construct
'6 sequences',
as results
from
0 c R n , when
v ~
(see
the proof in §12):
Lemma 1 Suppose
given
s • W
such that supp s(~)
shrinks
to
Then i)
In case perties I.I) 1.2)
2)
If
s
s • S
and
o
lim f
of nonnegative
and
o
<s,.>
=
<s,'>
the following
two pro-
(the Dirac distrihution)
= 6 , then
of the above type of
they generate
functions,
s(~)) (x)dx = 1
'~Rn s c S
The importance
is a sequence
are equivalent
for functions
s2 % S
'6 sequences'
f E L1 (Rn) Ioc
o
is due to the smooth approximations
through
the convolutions
fv = f * s(V)
~) g N .
In [125-1313 complicated
(7)
it was shown diagrams
(see also Theorem
of inclusions
I
~
A
V
->
s
->
S
V
o
with
A
tisfying (7.1)
subalgebra
in
the conditions:
I n S = V
W ,
i, §7) that the following,
slightly more
can be constructed: ->
W
~ u(1)
o
I ideal
in
A
and
V , S
vector
subspaces
in
So , sa-
12
(7.2)
V
(7.3)
V
o o
n S = V + S = S
o
and thus generating the following linear embedding of
D' (Rn)
into an associative and
commutative algebra with unit element:
So/V °
D ' ( R n)
S/V
A/I
(8) <s,'>
<
s+V isom
The intermediate
<
s+V
o
quotient space
s+l
lin,inj
has the role of a regularization of the repre-
S/V
sentation of the distributions in
>
isom
D'(R n)
by classes of sequences of weakly conver-
gent smooth functions, given in (3).
§5.
SIMPLER DIAGRAMS OF INCLUSIONS
In constructing diagrams of inclusions of type (7), the main problem proves to be the choice of the regularizing quotient space lem to the choice of
S
only, since
V
. One can think of reducing that prob-
S/V
can be obtained from (7.2). However, it will
be more convenient to consider (see (20.2) and Remark D in ~7):
(9)
s = vGs'
with
S' vector subspace in S and to replace the problem of the choice of S by o the one of the choice of the pair (V,S') . In that case, the conditions (7.2) and (7.3) become
(lo)
s
o
=
v° Q s ,
Now, the difficult task remains to fulfil (7.1). Obviously, if any ideal (7.1) then the smallest ideal containing
V
respect, taking in (7) the smallest possible
(ii)
u(1) ~ S
satisfies
will still satisfy that relation. In this I
will be convenient when one constru-
cts the algebras containing the distributions. The smallest since
I
I
can easily be obtained
in (9). Indeed, denoting
I(V,A) = the ideal in
A
one obtains the smallest ideal in
A
vector subspace generated by
V • A
into the structure of the algebra
Therefore, the diagrams
generated by which contains in
V V . Moreover,
I(V,A)
being the
A , one obtains a particularly useful insight
A/I(V,A)
containing the distributions.
(7) will be considered under the particular form
13
I (V,A)
A
l
(12)
V
V
W
> V(~S'
-> S
o
~ u(1)
o
with (12.1)
VoQS'
= SO
(12.2)
I(V~A) n ( V Q S ' )
= V
It will be useful to notice that (12.2) can be written under the equivalent simpler form: (12.3)
1(v~4)
n s,
= o
In the case of the diagrams (121, the embeddings C8) will obtain the particular form
D'(Rn)
So/V°
W
W
(13)
<s,'>
<
Besides the problem of choosing sing
A/I(V,A)
W
s+V isom
V(~S'/V <
s+V
o
(V,S')
>
isom
, it apparently remains the problem of choo-
A • However, that latter problem will be solved in §7, in an easy way. There-
fore the problem of embedding the distributions in
D'(R n)
into algebras will be re-
duced to the problem of constructing suitable regularizations
§6.
s+I(V,A)
lin,inj
~MISSIBLE
(V,S')
PROPERTIES
Several properties of the algebras
A/I(V,A)
, such as the existence of derivative o p
erators on the algebras, the existence of positive powers for certain elements of the algebras, etc. will depend on corresponding properties of the algebras
A . A uniform
approach of these properties can be obtained with the help of the following definition. A property P , valid for certain subsets (14.1) (14.2)
W
has the property
H
in
W
is called admissible, only if
P ,
an intersection of subsets in will also have that property.
W , each having the property
P ,
14
Suppose,
P
and
and denoted
Q
are admissible properties,
P ~ Q , only if each subset
Q . Obviously,
if
PI,...,Pm
P = (PI and ... and Pm) with the above partial
Denote by
P
then
in
W
P
is called stronger than
satisfying
are admissible properties,
is also an admissible property
P
Q
will also satisfy
then their conjunction and
P = max {Pl,...,Pm}
order ~ .
the property of subsets
the strongest
H
admissible
Three of the admissible
H
in
W
that
H
= W , then obviously
P
is
property.
properties
of subsets
~ c W
encountered
in the sequel
are
defined now. (15)
~
(15.1)
DPH c H
where
is derivative ,
V
Dp : W ~ W i s
that is,
invariant,
p ~ hn
the term by term p-order
(DPs) (W) (x) = (DPs(~))(x)
(16)
~f
(16.1)
V s e ~
only if
,
derivative
V S c W ,
is positive power invatiant,
of the sequences in
~ ~ N ,
W ,
× ~ Rn .
only if
: \
*)
s(w) (x) -> 0 ,
**)
where
s~(v)(x)
s~
~ W,
= (s(V)(x)) ~
(17)
~
(i7.1)
V t 6 W :
is sectional
S¢V
,
t(v)
A reiated, (17')
:
H
e N
invariant,
:
(s ~
~ H
,
~
~
(0,~))
x c Rn only if
~EN:
s(~)
to
only if any subsequence H
AND ALGEBRAS CONTAINING
ruct the corresponding
The construction
invariant,
is also belonging
REGULARIZATIONS
P
J
~ (0,~)
Vw
is subsequence
The aim of this section
petty
x e Rn I
i n a way stronger admissible property is defined in :
s E H
§7.
V~
V w c N ,
of a sequence
.
THE DISTRIBUTIONS
is to define general classes of regularizations algebras
of the algebras
was specified
t
containing
D'(R n)
and to const-
.
is carried out assuming that a certain admissible
in advance.
pro
15
Suppose now, given any admissible porperty
For
(V,S')
Q , such that
given in §5, the choice of the subalgebra
Q ~ P .
A
needed in the diagram (12)
in order to obtain the embedding (13), will be (18)
AQ(v,s ') = the smallest subalgebra in and containing
V Q
W
with the property
Q
S' •
The notation (see(ll))
~9)
I(v,AQ(v,s'))
IQ(v,s ') =
will be useful.
In Theorem I, below, it is proved that one can reduce the construction of diagrams (12) to the choice of pairs given in the following:
Definition A pair
(V,S')
of vector subspaces in
V°
respectively
So
is called a P-regu-
larization, only if (20.1)
s o = vo ®
s'
(20.2)
g c Y(p)
Q
(20.3)
IP(v,s ')
n S' = 0
S' ,
Vp
~ ~n
*)
where (21)
U = {U(~)
I ~ e C°°(Rn)}
is the set of constant sequences of smooth functions, and (22)
V(p) = {v ~ V I V r ~ Nn ,
Denote by If
Q
r _< p : Dry ~ V} ,
R(P) the set of all P-regularizations
is an admissible property and
(19) and (18). Therefore,
R(P) c R(P)
Q ~ P
then
V p c ~n .
(V,S')
•
R(P) c R(Q) , due to (20.3),
for any admissible property
P , since
is the strongest admissible property (see §6). A pair
(V,S')
e R(P)
will be called regularization.
Thus, a
(V,S')
rization, only if it is a P-regularization, for any admissible property
*)
~ = ~ u {=}
is regulaP .
16
Remark 1
l)
The above condition
(20.1) is identical with (12.1) while
(12.3), assuming that one takes in (12), stronger version of the relation
u(1) ~ VC)S'
to secure the fact that the multiplication on
6~
the usual multiplication
(20.3) is equivalent with
A = AP(v,s ') . The condition
in the algebras containing
of functions.
The presence of
V[P~
below, used in order to define proper derivative
If
(V,S')
is a P-regularization
(20.1) and (20.3) result in therefore
then
D'
induces
• wit~
p c ~n , in (2o.2) is connected with the family of algebras mentioned
2)
(20.2) is a
in (12) and it is needed in order
in Remark D,
operators.
V ~ V o • Indeed, assume
V = Vo , then
IP(v,s ') n S o a V ° . But (19) implies
V o c IP(v,s ') ,
(5) is contradicted.
Remark D It is important operators
to point out the necessary connection
volving important distributions. a)
between the way the derivative
are defined on the algebras and the validity of certain basic relations
the existence
in-
Indeed, as seen in §Ii, assuming:
of an algebra
A = D'(R I) possessing
a derivative
operator
D : A ÷ A , and b)
the validity of the relation
one necessarily terpretations x • 6 = 0
obtains
of
x • 6 = 0 ,
82 = 0 , a relation not always in line with the possible
62 (see chap. 4 and [11], [21], [108], [1511)
in-
(the relation
is important in that it gives an upper bound of the order of singularity
the Dirac ~ function exhibits at A way out is to embed ing derivative
D'(R n)
operators
Dq : A
0 c R I) .
into a family of algebras
,
V
q ~ Nn ,
P ~ ~n
p
From here the presence of the vector subspaces
V(p)
, with
(20.2). However,
that method can still lead to the situation
that
V q c N n , in which case the algebras
Dqv c V ,
p ~ A n , possess-
(see Theorem 3, §8): > A
P+q
Ap , with
Ap
p ~ A n , in the condition in a) above, provided with
P ~ ~n
will be iden-
tical.
And now, the basic result in the present chapter. Theorem 1 R(P)
is not void.
Suppose given
(V,S') ~ R(P)
and an admissible property
Q , such that
Q ~ P .
17
Then, for each
P e ~n , the diagram of inclusions holds
IQ(v(P) ,S')
(23)
> AQ(v(P),S ')
V(P)
V(P)
V0
>
(~S'
"> W
<"
U
So
and (23.1)
IQ(v(p),S ') n (V(p) Q s ' )
= V(p) ,
or, equivalently (23.2)
IQ(v(p),S ') n S' = 0
Proof R(P)
is not void due to Theorem 4 in chap. 2, §6.
The inclusions in (23) result easily and we shall only prove (23.1). Obviously, V(p) c V , hence
AQ(v(p),S ') a AQ(v,S ') . Noticing that due to
AQ(V,S ') c AP(v,s ')
holds, one obtains
IQ(v(p),S ') n (V(p) ~)S') c IP(v,s ') n ( V ( ~ S ' ) from (20.3). Now, obviously
Q ~ P , the inclusion
IQ(v(p),S ') a IP(v,s ') . Therefore, a V ° , the last inclusion resulting
IQ(v(p),S ') n (V(p) Q S ' )
c V ° n (V(p) Q s ' )
= V(p)
and
the inclusion a in (23.1) is proved. The converse inclusion resulting from (23), the proof of Theorem 1 is completed VVV
And now, the definition of the family of associative and commutative algebras with unit element, each containing the distributions in
D' (Rn) , family associated to any
given P-regul arization. Suppose
(V,S') E R(P)
(24) The algebras
and
Q
is an admissible property, with
AQ(v,S',p) = A Q ( v ( p ) , S ' ) / I Q ( v ( P ) , S ' AQ(v,s ' ,p)
) , with
Q < P . Denote then
p ~ ~n .
will be called derivative algebras, positive power algebras,
sectional algebras or subsequence algebras, only if the admissible properties (iS), (16), (17) or (17').
Q
is respectively stronger than
18
§8.
PROPERTIES OF THE FAMILIES OF ALGEBRAS CONTAINING
D'(R n)
The next three theorems present the main properties of the embeddings D'(R n) c AQ(F,S',p) . Theorem 2
(V,S') ~
Suppose given
R(P)
and an admissible property
Q , such that
Q <_ P •
Then i)
AQ(v,s',p)
is an associative, commutative algebra with the unit element
u(1) + IQ(v(p),S ') , for each 2)
P ¢ ~n .
The following linear applications exist for each
C~p
S°/Vo with
<
~(s+V(p))
bij
V(P)
P ~_ ~n :
Bp inj
(~S'/V(P)
> AQ(v's' 'p)
= s + V°
Bp(S+V(p)) = s + IQ(v(p),S ') 3)
For each
(2s)
p E ~n , the linear injective application (embedding) exists: Cp D'(R n) "> AQ(v,s ',p)
with 4)
-I -I ~p = Bp o ap o ~0
For each
(see (3))
P ~ ~n , the multiplication in
AQ(v,S',p)
induces on
oo n C (R)
the
usual multiplication of functions.
s)
For each
p , q , r e ~n , p s q ~ r , the diagram of algebra homomorphisms is
commutative: Yr ,p
AQ(v,s',r)
> AQ(v,s',q)
Yq,p
Yr,q with
6)
yq,p(s+IQ(y(q),s'))
= s + IQ(F(p),s ')
etc.
P,q e ~n • P _< q , the diagram is commutative:
For each
> AQ(v,s',q)
Sq T Eq
v(q)
Qs'/v(q)
~O~q I D'(R n) <
with
,
-> AQ(v,s' ,p)
> AQ(v,s ' ,p) <
T Bp
Yq'P n
" v(p)
q'P id
Qs'/v(p) l~O~p
~ D'(Rn)
nq,p(S+V(q)) = s + V(p)
therefore,
yq,p
restricted to
Eq(D' (Rn))
is injective.
E
P
19
Proof 4) results from (20.2). The rest follows from Theorem 1
WV
The existence of derivative operators on the algebras as well as their properties are established now.
Theorem
3 In the case of derivative algebras
admissible property
i)
(see §6], suppose given
Q , such that
p e Nn
q ~ A n , the following linear mapping
rivative)
exists (see Remark: D, §7):
D pq+p : AQ(v,S',q+p)
and an
Q _< P . Then
For each
(26)
and
(V,S') ~ R(~)
÷
(p-th order de-
AQ(v,s',q)
with (26.1)
DP+p(s+IQ(Y(q+P),S'))
and the
restriction of
= DPs + I Q ( v ( q ) , s ' )
D pq+p
to
co
n
C (R)
is the usual p-th order derivative
of functions. 2)
The relation holds Dp 2
(27)
D~PI+P 2 3)
For
each
= Dp l + p 2
q+P2
p e Nn , q , r
e I~n
q+PI+P2 , q _< r
DP r+p
AQ(v,s ' ,r+p)
V
p l , p 2 e Nn
' , the
~ln
, diagram
is
~ c
commutative:
> AQ(v,s,,r)
I Yr+p,q+p
I Yr,q
AQ(v,s',q+p)
~ AQ(v,s ' ,q) DiP q+p
4)
The mapping
product (28)
D pq+p , with
derivative D~+p (S.T)
in particular,
(28.1)
: ~ keN n k~p
Yq+p-k,q
q+p
" Yq+k,~
if [Pl = 1 , the relation holds:
D~+p(S-T) where
p ~ N n , q ~ A n , satisfies the Leibnitz rule of
= D pq+p S • Yq+p,q T + Yq+p,q S • D pq+p T ,
S,T ~ AQ(v,S',q+p)
in both of the above relations.
q+p
'
20 Proof i) First, we prove (26). Obviously (29)
V p • Nn ,
DPv(q+p) c V(q) ,
q~¢.
Now, we show that (30)
DPAQ(F(q+p),S') c AQ(F(q),s')
Indeed,
(18) r e s u l t s
gp
,
n
qe~n t
in
DPAQ(v(q+p),S ' ) c n DPA where the intersection is taken over all subalgerbras ty
Q
and contain
V(q+p~)s'
each of the subalgebras
A
A
in
W
which have the proper-
. Since we are in the presence of derivative algebras,
above satisfies the condition
DPA ~ A , therefore,
one ob-
tains D P A ~ ( 7 ( q + p ) , S ' ) c ,~, A with the intersection
h a v i n g t h e same r a n g e as b e f o r e ,
boticing that
V(q+p) c V(q)
,
one o b t a i n s DPAt~(V(q+p),s ' ) c n A here the intersection perty
Q
obtains
being taken over all
and c o n t a i n
V(q) @ S '
subalgebras
. Taking now i n t o
A
in
W w h i c h have t h e p r o -
account the definition
in (18),
one
(30).
The r e l a t i o n s (31)
AQ(v(q+p),S ') c AQ(v(q),S ')
result
easily
noticing
Comparing ( 2 9 ) , (32)
that
DPIQ(v(q+p),s')
The s e c o n d p a r t 2),
it
follows
~n
q
that
c IQ(v(q),s')
,
g p • Nn ,
q e ~n .
imply (26).
o f 1) r e s u l t s
3) and 4) r e s u l t
V p • Nn ,
V(q+p) c V ( q ) .
(30) and ( 3 i ) ,
Now, (31) and (32) w i l l
,
from (20.2).
from 1) and Theorems 1 , 2
WV
Now, positive powers will be defined for certain elements of the algebras. Denote *) **)
n C~(Rn) = {~ e C ( R ) Obviously, ist
if
~ • c~(R n)
to £ C°°(Rn) ,
to(X) = x ~ . . . X n 2 ,
and
t0(x) >_ 0 ,
~(x)
~(x) > 0 , V x • R n I ~ • c~(R n) g a e (0,~)
> 0 ,
V x e R n , then
V x e R n , such that
V x = (x 1 , . . . ,
to ~ C+(R n) . But there ex-
to 4 C+CR n] , for instance,
Xn) ~ Rn . However, d e f i n i n g
21
exp
~(x) =
(=(i/x I + .... + i/Xn) )
0 one obtains ~ ~ C + ( R n)
w+: It follows that, (33)
for
sa(V)(x)
and one obtains
(34)
T
V l~i~n
'
Denote further
[ s(v)
~ C~(R n)
s e W+
,
one can define
= (s(V)(x)) a
~ N }
vv
any positive power
V ~ ~ (0, ~)
J
W e N
,
s
, by
x ~ Rn
•
,
in (16) for a positive p o w e r invariant
subset
H
in
W
can he re-
as follows V
Suppose
i
s ~ e W+
Now, the condition formulated
x. > 0
otherwise
.
{s ~ w
if
s c H n W+
,
a c (0, ~)
is a vector subspace
in
: sa e H
S
and denote O
D' T,+
(Rn) = { [ t ¢ T n W
The distributions
in
D' T,+
(Rn)
}
+
will be called
T-nonnegative.
Theorem 4 In the case of positive p o w e r algebras, missible porperty
Q , such that
suppose
Q ~ P . If
given
T
(V,S')
e R(P)
is a vector subspace
and an adin
SO ,
satisfying the condition (35)
U n W+
c
T c S'
then
l) 2)
cT(Rn) c D~,+ (R n) For given
p e ~n
and any
a ~ (0, ~)
, one can define
a mapping
(positive
power) :
D~,+ with
(Rn ) ~ T
T =
ons w i l l
Tm e A Q ( v , s ' , p )
÷ Ta = t a + I Q ( v ( p ) , S ' )
, where
t
e T n W+
and t h e r e l a t i
result:
(36.1)
T1 =
(36.2)
T (~+8 = T (~ • T 8 ,
(36.3)
(T(~)TM = T e'm ,
3)
~
T V (z,8 e 09, °o)
V ~ e (0, ~)
,
For any
p e ~n • the mapping
positive
power o f functions.
m e N\(0}
.
in i) is identical
on
CT(R n)
with the usual
22 4)
Suppose in addition the case of derivative algebras• then the following re lations hold in the algebras
(37)
Dq T ~ = ~ • T -iDq T p+q p+q v
T ¢ D'
AQ(v,S',p)
, with
p e An :
•
(Rn)
~ ~ (i,~)
q
e
Nn
[ql
= i
Proof i)
It follows from (35).
sults from i) and 2).
2)
It results from (35) and 2) in Theorem 2.
Finally, 4) is a consequence of (28.1)
3)
It re-
VVV
Remark 2 The condition (35) can be easily fulfilled in case (20.2) is replaced by the stronger condition (see (25) in chap. 2, §6):
§9.
U c S'
DEFINING NONLINEAR PARTIAL DIFFERENTIAL OPERATORS ON THE ALGEBRAS
As stated in §i, one of the main aims of the nonlinear method in the theory of distributions presented in this work is to offer a framework for the study of the nonlinear partial differential equations. The embedding of the distributions in algebras
AQ(v,s',p)
D'(R n) into the
(see Theorem 2, §8) creates the possibility of studying nonline-
ar partial differential operators of the general form
T (D)u(x) =
Z ci(x ) l~i~h
DPiJu(x)
(or with
ci
smooth and
•
x e Rn ,
l-<j-
~ c Rn
~ $
open)
Pij e Nn "
In order to define these operators on the algebras containing the distributions, one has to take into account the following two features of the distribution multiplication presented in this work: I)
The algebra homomorphisms
(see 5) in Theorem 2, §8)
AQ(v,S ',q)
~ AQ(v,S ',p) Yq,p
need not generate algebra embeddings. 2)
The derivative operators (see pct. 1 in Theorem 3, §8) Dp : AQ(v,S ' •q+p) q+p
+
AQ(v•S ' ,q)
act between different algebras in case
q c ~n
has finite components.
Suppose, we are in the case of derivative algebras. Given the above operator define its order by
T(D) ,
23
= max { Pij ] l-
q ~ ~n
T(D)
+
: AQ(F,S',q+p)
the mapping AQ~/,S,,q)
by T(D)u :
Z c. l~i~h i
I I Yq+P-Pij l~jKki
where the additions and multiplications AQ(v,S',q)
,q
DPi~~ u q+p
u c AQ(v,S',q+p)
i
in the right term are effectuated in
. The commutativity of the diagrams in 5) in Theorem 2 and 3) in Theorem 3,
58, grants the consistency of the above definition. §3, the above definition of u
V '
T(D)
Moreover, due to 4) in Theorem 2,
coincides with the usual one in the case of smooth
.
An important
particular case is when
in Theorem 2, §3, one can consider with
r E h~ ,
u
T(D)
acts upon
u ~ D'(R n) . Then, due to 3)
belonging to any of the algebras
AQ(v,S',r)
,
r ~ p , and apply the above definition.
Due to 5) in Theorem 2,§3, the same happens when
T(D)
acts upon
u c
AQ(v,s', ~) .
In chapter 3, the above definition will be used in the study of piece wise smooth weak solutions of the so called polynomial nonlinear partial differential nonlinear hyperbolic partial differential
ticular cases of polynomial nonlinear partial differential
§i0.
operators.
The
equations modelling the shock waves are parequations.
MAXIMALITY AND LOCAL VANISHING
There exists an applicative interest in constructing in a way that the ideals
AQ(v,S',p)
, p c ~n
IQ(v(p),S ') , p ~ ~n , are large, possibly maximal.
the algebras
Indeed,
according to (24), the larger these ideals, the more equality relations are obtained in the corresponding algebras. To construct the
aigebras
P-regularizations
(V,S')
the ideals
AQ(v,S',p) .
and the ideals
IQ(V(p),S ')
According to (ii) and (19), the larger
IQ(V(p),S ') , p ~ A n , will be. Therefore,
maximal ideals, the problem of
(V,S')
~
R(P)
means to choose V
is, the larger
as a first approach in securing
, with maximal
V , will be studied in
the present section. An alternative approach to the problem of maximal ideals iQ(v(p),S, ) ,
P ~ ~n , will be given in chap. 2, §7.
And now, several results on the structure of (38)
V
(V,S') ~ R(P)
R(P)
• In addition to the relation
: V c V°
obtained in 2), Remark i, §7, the following two simple results will be useful.
24
Lemma 2 Suppose
(V,S')
~ R(p)
. If
S''
is a vector
subspace
in
S
satisfying
o
the
conditions
v(Ds"
(40)
U ¢ V(p)(7)S"
,
(V,S")
.
then
:
vGs'
(39)
c R(P)
V
p e An
(see
(20.2))
Proof Since
V c V ° • the relations
Now, it suffices (19) and
(39) and (20.i)
to show that
(18), one obtains
(V,S'')
will obviously
satisfies
(20.3).
IP(v,S '') = IP(V,S ')
imply
Indeed,
S O = VoQS" taking
into account
VVV
Lemma 3
Suppose
, (V 2 , S½) c R(P)
(V1 , S i )
S 1, c S 2'
. If
then
S 1' = S 2' .
Proof It follows
easily
from the fact that both
(V1 , S~)
and
(V 2 , S½)
satisfy
(20.1)
WV
The relations ideals
(ii),
(19) and Lemma~2,3
suggest
e R(P)
Define on
with
R(P)
(v I The admissible
V
the
in finding
maximal.
a partial
,s{)
way in enlarging
V . From here the interest
IQ(v(p),S ') , P e ~n , is to enlarge
(v,s')
that an appropriate
<_ (v 2
property
P
order -< by
,s½) ~ v l ~ w 2
and
is called
only if each chain
regular,
Si = S½ in
(R(P)
, ~ )
has
an upper bound. We recall sets
H
Theorem P
(see §6) that the strongest in
W
that
admissible
H = W , was denoted by
property,
namely
the property
of sub-
P .
5 is a regular
admissible
property.
Proof Assume
((V h , S')
prove that
(V,S')
1%
~ A)
~ R(P).
is a chain in
(2(P)
, ~) - Denote
V =
u
leA
V X . We shall
25
Obviously, prove
V
is a vector subspace in
(20.3). First, we notice that
Therefore,
IP(F,S ') = I(V,W)
Assume now
s e I(F,W) n S'
(41)
s
Since
=
I ~ c A)
and (20.1) and (20.2) hold.
(20.3) becomes
s e I(V,W) with
is a chain,
It remains to
') = W , since P holds only for
and the condition
. Then,
~ v i • wi , l~i~k
((Yl , S')
V l~i~k . Now,
V
AP(F
results
v. e V 1 .
I(V,W) n S' = O •
in
wi e W , . .
there exists
A = W .
~oe
V
l
.
A
such that
v i c Vlo ,
(41) will imply
s ~ I(V l
, W) n S '
= 0
0
the last equality resulting
from the fact that
(V h
, S')
~ R (P)
WV
O
Based on Zorn's Lemma, Theorem i, §7 and Theorem 5, one can define for a given regular admissible property
P
the nonvoid set of maximal
Rmax(P)
=
{ (V,S') e R(P)
The fact that the strongest the possibility perty
Q , with
V
maximal.
and the construction The above remark of
V , with
(19)
and
The following
Indeed, since
(see Theorem 5) offers
, p e ~n , for any admissible pro(V,S') c Rmax(P)
c Rmax(P)
(38) generate an interest in a possible upper bound
, which would give an insight into the necessary structu-
multiplications
defined according to the relations
Theorem 6 gives an upper bound of the mentioned Y
V QS'
contains
[68], [69], [105-110],
has to satisfy.
[136], [137], [162])
0 e D'(R n)
which constitute
in points arbitrarily near to each point in p c A n , denote by
Wp
Namely,
type under the form of a
it is proved for instance,
(see Lemma 1 in §4 and [4], [35-41], [53], the sequences of smooth functions weakly V , have to vanish infinitely many times Rn .
the set of all sequences of smooth functions
satisfy the local vanishing property:
(42)
V
G c Rn .
G #. ~
~
x ¢ G ,
w ¢ N ,
. open.
q ~ Nn
q<-P,
~cN
:
w >-P :
Dq~(v) (x) = 0 or, written simpler V (42')
(24), (23),
(ii).
'5 sequences'
For
, ~) }
Q ~ P , one can choose
local vanishing property which
to
is regular
A Q(V,S',p)
that in case
convergent
P
in (R(P)
of the algebras will proceed according to (24) and (23).
re of the distribution (18),
the algebras
a n d the relation
~')
I (V,S') maximal
admissible property
to construct
P-regularizations:
x ~ Rn
,
qc
Dqw(v) (y) = 0
Nn
,
q_
:
for infinitely many and
Y E Rn
v ~ N
arbitrarily near to
x
w c W
which
26
A P-regul arization
(43)
(V,S')
V
G c Rn ,
3
SG~V+S'
is of local type, only if
G $ ~ ,
(43.1)
<s G , "> $ 0 ~ D ' ( R n)
(43.2)
supp SG(~ ) e G ,
for a c e r t a i n
open
:
:
x~¢N,
V
~>-~
,
~ ¢ N .
Theorem 6
(V,S')
Suppose given a local type r e g u l a r i z a t i o n sectional
c R(P)
such
that
vQs'
is
invariant.
Then V(p)
c Wp
,
V
p ~ ~n
.
Proof Taking into account
(22) and (42), it suffices to prove the inclusion
Assume it is false and @
G c Rn
V
x ¢ G ,
v ~ V ~ W ° . Then ,
V c W°
only.
(42) implies
G ~= ~ ,
open
~ ¢ N ,
~ >- ~'
I~' ¢ N :
,
:
v ( . ) (x] ~ 0 Now, d u e
to
(43.2),
V
one obtains
V ¢ N ,
~ -> ~''
:
supp SG(~ ) c G Define
w c W
by 0
w(~) (x)
if
x~
G
: SG(~ ) (x)/v(~) (x)
whenever (44)
~ c N , v(~)
~ -> ~ = max {~'
" w(~)
= SG(~ ) ,
, ~''} V
if
x E G
. Then
v c N ,
~ ->
therefore (45) since
v " w c VQS' sG
(46) since Now,
c ~)S'
and
VQS'
v • w ¢ I(V,W)
is sectional invariant.
But
= IP(V,S ')
v c V . (45)
and
(46)
together
with
(20.3)
will
imply
v
• w £ V
which
due t o
(44)
re-
27
sults in
<s G , "> = 0 £ D ' ( R n)
, since
V c Fo
(43.1) was c o n t r a d i c t e d .
• Therefore
mY
Remark 3 There exist relevant in T h e o r e m 6 above. whenever
Y
instances of regularizations Indeed, one can notice that
has that property.
Theorem 4, chap. ideal obtained
Indeed,
T~ c T
6, chap.
V QS'
which meet the conditions will be sectional
(V,T(~SI)
the regularizations
2, §6, can be chosen with
in Proposition
V
sectional
invariant,
2, §6 is obviously
sectional
invariant obtained
in
since any Dirac invariant.
Final-
(T/,T(~)SI) considered in chap. S, §4, are of local type.
ly, the regularizations
open,
Further,
(V,S')
where
~ = (sx
one can take in (43)
] x ~ R n) ~ Z 6 , therefore
s G = sx
, provided
that
given
G c Rn •
G # ~ ,
x e G .
With the method used in the proof of Theorem 6 one obtains
the following more general
result. Theorem
7
Suppose given a regularization
(V,S')
such that
V(~)S'
is sectional
ant. Then each
v e V
V
satisfies
t ~ V(~S'
,
v ( ~ ) (x)
t ~ V,
~ ~ p ,
~ ~ N , :
the vanishing
condition
p ~ N ;
x E supp t(v)
:
o
Proof
v ~ V , t ~ V(~S'
Assume it is false and V
~
E
v(~) Define
N
,
, t ~ V
and
"o>_p :
~ 0
on supp t (~)
w ~ W by 0 w
(~) (x)
if
x ~ supp t(V)
= t(V)(X)/V(V)(x)
whenever (47)
9 E N , v(,)
~ a p • w(,)
if
x E supp t(V)
. Then = t(V)
,
therefore (48)
since
p ~ N
v • w ~ vC)s'
t £
V~)S'
and
VQS'
is sectional
invariant.
such that
invari-
28
But
(49)
~_ i ( v , w )
V " w
since
v e V .
The relations
(48) and
= 0 ¢ D ' ( R n) dict
i P ( v , s ,)
:
t
~ V
(49) together with
since
V c V
(20.3)
. It follows
o
imply
that
v • w ¢ V . Then
t ¢ V
o
. l~w,
(47) will give
C20.I) will contra-
~7V
In the same way, one can prove: Theorem 8 Suppose
(V,S')
is a regularization,
then each
v~
V
satisfies
the vanishing
condi-
tion V
t
c V@$'
~ c N ,
,
t
~ V :
x ¢ supp t(v)
v (v) (x) = o
§ii.
STRONGER CONDITIONS
FOR DERIVATIVES
It will be shown that even in the one dimensional on derivatives
mentioned
vial distribution Suppose
A
nomials
on
n = 1 , the stronger conditions to a particular,
rather tri-
multiplication.
is an associative R1
case
in Remark D, §7, lead necessarily
and commutative
as well as the distributions
algebra containing in
D ' ( R I)
the real valued poly-
with support
a finite number
of points. Suppose also that (50)
the m u l t i p l i c a t i o n the polynomial
(51)
there exists
(51.1)
D
~(x)
D
A
induces the usual multiplication
= 1 ,
V
a linear mapping
is identical
distributions (51.2)
in
satisfies
x ¢ R 1 , is the unit element D : A ÷ A
A
the Leibnitz
D(a'b)
= (Da) • b + a • (Db)
(X-Xo)
" ~x
= 0 ~ A o
,
V
A ,
when applied to polynomials
a finite number of points
,
rule of 'product derivative' V
a n d finally
(52)
in
such that
with the usual derivative
with support on
on the polynomials
x°
c
R1
a,b ¢ A
or
and
2g
Theorem
9
W i t h i n the a l g e b r a (53)
(X-xo)P
A
• Dq8 x
the r e l a t i o n s = 0 c A ,
hold:
V
x° c R1 ,
p,q c N ,
p > q
o
(54)
(p+l)
+
• DP8 x
" DP+I6 x
(X-Xo)
o
= 0 c A ,
¥
xo c R1 '
P ~ N
o
(55)
(X-xo)P
• (DP8 x Iq o
= 0 E A ,
(56)
• D8 x = 0 6 A , (~x )2 = ~x o o o
V
x° ~ R1 , V
q > 2
p,q c N ,
xo c R1
Proof Applying
D
C57)
to 6x
o
(52) and t a k i n g
into a c c o u n t
+ (X-Xo)
= 0 E A
which multiplied V
by
(X-Xo)
X 0 c R1 . A p p l y i n g
one o b t a i n s
peating
the
• D6 x
D
to
The relation
V x° ~ R 1
g i v e s due t o the
latter
one obtains
(54) r e s u l t s
Now, m u l t i p l y i n g
(54) b y
(p+l) (X-Xo)P
one o b t a i n s
o
(52) t h e r e l a t i o n
relation
in the s a m e w a y the r e l a t i o n
procedure,
(51),
(X-Xo)3
D
to
that relation
by
= 0 ~ A o
'
¥ x
o
'
(X-Xo) c R1
"
Re-
(57).
• DP+I6 x
o Multiplying
x
by
= 0 e A O
, one o b t a i n s + (X-Xo)P+l
" DP~x
• D28
multiplying
X
(53).
applying repeatedly (X-xo)P
and then,
(X-Xo)2 • D8
= 0 ¢ A ,
¥
x o ¢ R1 ' p c N
o
(Dp~ x )q-i o
and t a k i n g
into a c c o u n t
(53), one o b t a i n s
(ss). Taking D
p = 0
and q = 2
to t h a t r e l a t i o n ,
§12.
in (55) ' one o b t a i n s
the p r o o f o f
(56)
(~x)2 = 0 c A o is c o m p l e t e d WV
APPENDIX
The proof
of L e m m a 1 in §4 is g i v e n here.
I)
It f o l l o w s
2)
For
easily.
a ~ R1
E(a,v)
and
~ ~ N
= { x
¢ Rn
denote
] s(V)(x)
~ a )
,
¥ x ° ~ R I . Applying
30
First,
we prove
(58)
the relation
lira x)-~oo
[
s(~) (x)dx >- 1 ,
a¢
¥
R1
J E(a,V)
Assume i t i s f a l s e .
Then
a e R1 ,
~ > 0 ,
~) ~ N
x) _> ~'
I
,
~' e N : :
s ( V ) ( x ) d x -< 1 -
E(a,v) But,
s ~ S O , <s,'>
assuming
~ c D ( R n)
i
:
¢(0)
= ~
and
and
~ = 1
on a neighbourhood
f
s('O) (x)qJ(x)dx = xt+oo lim I
:
supp s(~)
shrinks
to
Rn It follows
0 e R n , when of
~ ÷ ~ . Therefore,
0 ~ R n , one obtains
s('o)(x)dx
Rn
that 11'' ~ N "
x) ~ N ,
~) -> II'' : f
1 - ~/2 -<
[
S(~)) (x)dx
J
Rn
Now, for
c N , the relations
hold
f
I s(~)(x)dx = I s(~)(x)ax +
1
s(v) (x)dx
J
Rn
E(a,~)
I
supp s(~)\E(a,~)
s(9)(x)dx
+ a
E(a,9)
Therefore,
one obtains
for
J
dx
supp s(v)
~ ~ N ,
v ~ max {~'
, ~''}
the inequality
(
1 - E/2 ~ 1 - z + a
]
dx
supp s(~) which is absurd since
leted.
supp s(9)
shrinks
to
0 c R n , and the proof of [58) is c o m p -
31
We prove now that there exist (59)
lim a~ = ~
and
a
¢ [0, ~) , with
lim
f
~ ¢ N , such that
.~(V) ( x ) d x z 1
E(a v ,v) Indeed, according to (58), there exist
(60)
v ° < ~1 < . . .
~
£ N , with
~ ¢ N
such that
< x.)P < . . .
and
(
1
(61)
l/(p+l)
-
-<
J/
s(vv) (x)dx ,
p ¢ N
V
E( v , v ) Define now Then
a
av = inf { p ¢ N ] v ~ v
~ a +I ,
(62)
av
=
V v ¢ N V ,
V
} ,
with
x)¢N
.
and V ¢ N
due to (60), hence, the first relation in (59) is proved. Taking into account (61), the second relation in (59) follows from (62). Finally, we prove (63)
xr+~lim
I
E(a Indeed, V
(s(~)(x))2dx
=
+ ~
,~)
(s(v)) 2 ~ a s(~)
on
E(a
V ~ c N , since
,~) ,
sO)) >- %
~ ¢ N . Therefore
I
(s(v) (x))2dx
>
I
aM
-
E(% ,~)
s (v) (x) dx ,
I
~ ¢ N
E (av ,v)
The relation (63) will result now from (59). Obviously, lim
V
(s (v) (x) ) 2dx
=
(59) implies
+ ~
Rn Then
s2 ~ S
o
since
supp s 2 ( ~ ) = supp s ( ~ )
shrinks
to
0 ¢ Rn
when
a o on E(%
,~))
32
Remark 4 The condition red in special
of nonnegativity cases.
s(v)(x) where
@ ~ D(R n)
For instance,
= a~ ~(bux) av E C 1
see that the equivalence valid.
of the sequence
,
assume V
bv ~ R 1
between
s
s
given by
~ c N , and
in i) in Lemma i, §4, can be remo
lim
x ~ Rn , I b~
I = + ~
i.I) and 1.2) in the mentioned
Then, lemma,
it is easy to will
still be
C h a p t e r
2
DIRAC ALGEBRAS CONTAINING THE~DISTRIBUTIONS
§i.
INTRODUCTION
In chapter i, diagrams of inclusions of the general type (23) were constructed in order to obtain the algebras (24) containing the distributions in
D'(R n) . The const-
ruction of diagrams (23) was based on the presumed existence (Theorem I, chap. I, §7] of ~regularizations
(F,S')
, for a given admissible property
P .
In this chapter two results are presented. First, specific instances of the diagrams (23), chap. I, §7, are constructed, leading to so called Dirac algebras in which
nonlinear operations of polynomial type can be
performed with piece wise smooth functions on
Rn
and their distributional derivati-
ves. The nonlinear operations considered, cover the ones encountered in the nonlinear partial differential operators introduced in chap. i, §9. In that way, the Dirac algebras prove to be useful in chapter 3, in the study of nonlinear partial differential equations with piece wise smooth weak solutions. The class of the piece wise smooth functions admitted in the nonlinear operations is rather wide, their singularities being situated on arbitrary closed subsets of locally finite families of smooth surfaces in
Rn
with smooth boundaries, for instance, Rn
As a second result, based on the existence of Dirac algebras, one can prove the existence of the regularizations
(V,S')
used in chapter I, and therefore validate the
general method of embedding the distributions into algebras, presented there. For an alternative validation, not using Dirac algebras, see §§8 and 9.
§2.
CLASSES OF SINGULARITIES OF PIECE WISE SMOOTH FUNCTIONS
When performing nonlinear operations with piece wise smooth functions on
Rn
and their
distributional derivatives, a problem arises in the neighbourhood of the singularities. The classes of singularities, concentrated on arbitrary closed subsets of
Rn
smooth boundaries, for instance, locally finite families of smooth surfaces in
with Rn ,
are defined now. A set
F
of mappings
generator on
y : Rn ÷ RmY , y ~ C ~ , with
Rn . The closed subsets in
Rn
my ~ N
is called a singularit X
84 m
Fy = { x ~ Rn ] y(x) = 0 ~ R Y } defined
by t h e m a p p i n g s
The set F F
of all
y • r
will
represent
the basic
sets
of possible
singularities
u F where A c F and F A is closed, will be called the yea Y ' class of singularities associated to F . Obviously, if A c F and A is finite or
more generally,
FA =
(Fy I Y • A) is locally finite in
Rn , then
F A e F F . Therefore, we
shall in the sequel be able to consider singularities concentrated on arbitrary locally finite families of smooth surfaces in Denote then by nite in
FF,Io c
Rn .
the set of all
FA
with
A c F
and
(Fy ] y • ~
locally fi-
R n . It follows that P F , l o c c F F .
Remark I The subsets
y(x I Then ever,
. . . . .
F
can be fairly complicated. For instance, suppose m = 1 and Y Xn) = exp ( _ 1 / x 1 2 ) s i n ( l / x 1 ) if x 1 # 0 , while y(x) Y = 0 otherwise.
F
i s an i n f i n i t e set of hyperplanes Y obviously Fy • F F , l o c .
The piece wise smooth functions on
Rn
in
Rn
which is not localIy
finite.
How-
considered will be those in
6~F(Rn) = { f : Rn ÷ C 1 [ @
F • F F : f • 6#°(Rn\F) }
thus, having the singularities concentrated on arbitrary closed subsets of smooth boundaries, for instance locally finite families of surfaces from The nonlinear operations on functions in
6~F(Rn)
Rn
with
F •
and their distributional derivatives
will be of the following polynomial type (I)
where
T(f I . . . . .
c.x • C~(Rn)
'
fm )
=
Pij • Nn
Z c. ] I DPijgij l~i~h i l~j~ki and
gij e {fl , .... fm } c C~(Rn).
The actual range of the nonlinear operations (I) will be the set of distributions
(C~fR 1 n )) + D~CRn) . n) n ClocCR where D~(Rn) = { S • , ' (Rn)
§3.
] ~
F • fr
: supp s c F } .
COMPATIBLE IDEALS AND VECTOR SUBSPACES OF SEQUENCES OF SMOOTH FUNCTIONS
The construction of the Dirac algebras will proceed through ending with Theorem 4 in §6.
§§3-6 in several stages,
35
Given a regularization ing embeddings of
(V,S') , one obtains (see Theorem 2, chap. i, §8) the follow-
D'(R n)
into algebras
So/V o
D' (Rn)
Y(p) C)S'/V(p)
AQ(V,S ' ,p)
(2) <s,'> <
s+V < bij
where
Q
S+V(p)
o
> s÷IQ(v(p],S ' )
bij
is any admissible property and
inj
P e ~n .
It follows (see also Theorem 3, chap. l, §8) that the nonlinear operations of type (i) co n CF(R ) - are effectuated
when applied to distributions - in particular, functions in within the algebras, according to the relation T (<s I ,.> , ... ,
(s)
=
where
sm . > )
Z c. i ~- DPiJs.. + IQ(F(p),S ') ~ AQ(v,S',p) l~i~h i l~j~ki l]
sij ~ { s I ,...,s m } c V(p) Q S '
in (3) since
=
V(p) c V °
appear only. Therefore,
. One can always assume that
and in the left term, the distributions S'
s I .... ,sm c S'
<s I ,.>,...,<s m ,.>
has a particularly important role, since the nonlinear
operations (I) and (3) when observed from
S'
become the corresponding classical ope-
rations applied term by term to sequences of smooth functions. The role is to generate ideals
IQ(V(p),S ')
V
will have
which annihilate within the embeddings (2) the ef-
fect the singular distributions in
D~(R n)
cause in the nonlinear operations (i) and
(3). In this respect, the regularizations a)
V
will be a vector subspace in
(V,S')
will be chosen as follows:
I n V
where I is an ideal in W of sequenco ' es of smooth functions vanishing on certain singularities F c F F , as well as on
neighbourhoods of points outside of those singularities.
b}
S'
will be split into
functions in
T
T Q S 1 , where the sequences of weakly convergent smooth
represent the distributions in
The main part of the construction, both theoretical (in chapters 3, 4 and 5) rests upon tlhe ideals The final choice of the ideals
I
D'(R n) 1 (in this chapter) and applicative
I .
and vector subspaces
T
and
S1
obtained in §6,
will evolve in several steps. It is particularly important to point out that the above way of choosing a regularization
(V,S')
belongs to a natural, general framework presented in Theorem 1 below,
where a basic characterization of regularizations is given. That characterization will be used throughout the chapters 3-7, when constructing algebras containing the distributions needed in applications to nonlinear problems or in theoretical developments.
36
An ideal
I
(see Fig.
i.):
in
W
and a vector subspace
(4)
I n T = V
(s)
z n s o ~ VoG)Tv
T
in
SO
are called compatible,
only if
n T = 0
o
Theorem i Suppose
the ideal
subspace
in
I
and
(6)
Vo(gr(9s
(7)
U c V(p)(~TOs then
and vector subspace
I n V°
S1
T
are compatible.
is a vector subspace
in
S°
If
V
is a vector
satisfying
I = So
(V,T(~S1)
1 ,
c R(P)
V
p ~ ~n
f o r any a d m i s s i b l e p r o p e r t y
C o n v e r s e l y , any r e g u l a r i z a t i o n
(V,S')
P . (see Fig.
2)
c a n be w r i t t e n u n d e r t h e above form.
Proof S' = T(~)S 1 . It suffices
Denote (s)
I (v,w)
n s'
First, we notice that (9)
I(V,W)
to show that
(see
(20.3) in chap.
i, §7)
= o
I(V,W) c I
since
V c I
and
I
is an ideal in
W . Therefore
n S' c I n S'
But (i0)
I n S' = O
Indeed,
(5) results
(Ii)
in
I n S' a
(InSo) n S' c
the last inclusion being lations
(9) and
assume given
I = I[V,W)
therefore,
(V,S') ~ R(P) I
The proof is completed,
(i0) follows
and denote I
is an ideal in
one can choose a vector subspace I n T c I n S' = 0
a T from
(ii) and
(4). The re-
(8).
ce~ there exists a vector subspace
imply
n (TQSI)
implied by (6). Now,
(I0) imply
Conversely,
(Vo(~T)
while
noticing
S1
in
Vo(~S'
g • But
T c S'
such that S'
so that
(20.2) will result that
V c I(V,W)
= IP(v,s ') . Then, obviously
in
= I
= S o , due to (20.1). Hen-
I n S o c VoQT S' = T C ) S
. Obviously,
1 . Now, (20.3) will
U a V (p) + ~ T ~ ) S , since
u(1)
~ W
1 ,
V p E
~n
VVV
Remark 2 Theorem 1 gives an affirmative tions
answer to the question of the existence
(V,S') provided one can prove the existence of:
of regulariza-
37
I
S 0
V
0
L~
W
Fig. 1
38
I
S 0
V 0
T
S
Fig. 2
39
a)
compatible
b)
vector
ideals
subspaces
These two problems
§4.
I
and vector
subspaces
S1
satisfying
(6) and (7).
will be solved
LOCALLY VANISHING
IDEALS OF SEQUENCES
A first specialization of locally vanishing For
p c ~n
in Theorem
of the ideals
I
T , as well
as of
2, §5, respectively
Corollary
2,
§6.
OF SMOOTH FUNCTIONS
in Theorem
i, §3 is given here, under the form
ideals.
denote by
W
the set of all sequences
of smooth
functions
P fying the local vanishing V
x ~ Rn
¥
~c
•
w c W satis-
property Nn
qc
,
q-
:
(12) N,
~->~
:
Dqw(~)) (x) = 0 or, formulated (12')
in a simpler way
Dqw(w)(x)
Obviously
Wp,
= 0
with
for each
p c Nn,
x c Rn ,
are ideals
in
q c Nn , W
and
q -< p ,
if
~
is big enough
p c Nn (_see chap.
Wp c W p ,
1
§i0) An ideal (13) Given
/
in
W
I c W
is called
a singularity
(14)
generator now.
For
by all sequences @ G c G
(14.1)
only if
o
als is constructed generated
locally vanishing,
V
r
on
~ c FF
R n , a class of associated and
of smooth
P e ~n , denote
functions
w c W
: q c Nn ,
q -< p :
~i c N : V
"~ c N , Dqw(v)
(14.2)
g
x c Rn\G V
simply:
on
= 0
: C
:
neighbourhood
w(~) or, formulated
"o -> ~i
= 0
on
V
of
x,1~2¢
N:
by
locally vanishing
IG,p
satisfying
the ideal
in
ideW
40
(14')
@ G ~ G :
(14'.1)
Dqw(w)
(14'.2)
w(w)
In case
G = {G}
Proposition
= 0
= 0
on
G ,
for
q c Nn
on a neighbourhood
IG,p =
, the notation
IG,p
q ~ p
of each
and
x • Rn\G
~
big enough
, if
w
is big enough
will be used.
i
Ig,p c
Wp , therefore
IG, p
is a locally vanishing
ideal.
Proof w c W whenever w c W satisfies (14). Assume w • W saP (14) for a certain G ~ G and take x c R n . If x e G then (14.1) will imply
It suffices tisfies (12).
to show that
In case
Denote
J(: p. iG, p
by
Obviously,
Two e x a m p l e s
Suppose
x • Rn\G
of
, (12) will be implied by
the
set
of
is
the
set
of
in
JG,p
elements
w ~ W ,
all
y ¢ F, a ¢ O
sequences all
of
finite and
(Rmy)
(14.2)
smooth
sums of
thus,
and define
wy,~(w) (x) = ~((w+l)y(x))
function elements
IG, p ,
in
• w(~)(x)
WV
are
wy, a ~ W ,
V
w ¢ W in
satisfying
(14).
JG,p "
presented
in
Lemmas 1 a n d
by
~ • N ,
x • Rn
Lemma 1 If
~ ¢ D ( R n~)
Dq(O) then
wy,e
and satisfies
= O, e gG, p
v ,
for a given
r ~ NmT, V
k ¢ N
the condition
I r I -
G c EF ,
G 9 Fy
,
p c Nn ,
] p
] -~ k
.
Proof It can be seen that
Suppose now surface
F
w
y e F , with . Suppose
a,B c C~(R ~)
define
and
y,a
y
my = 1
Sy E S o Sy,q c W
Sy,q(V)(x)
F
• G
satisfy
(14)
and denote by
such that
<sy
6y
WV
the Dirac 6 distribution
, • > = 6 y . For
q ~ Nn
of the
and
by
= ~((v4)X(x))
• 8(X(x))
" Dqsx(V)(x)
,
V
w ¢ N , x ¢ Rn
and
B
satisfies
Lemma 2 If
~ • D(R I) , ~ = 1
given
k • N
in a n e i g h b o u r h o o d
the condition
of
0 ~ R1
for a
2
41
Dr~(o)
= 0 ,
V
r E
N ,
r -< k
then i)
S y , q ~ JG ,p n S o '
V
G c FF ,
2)
Sy,q ¢ Vo provided t h a t
]ql
G ~ FX ,
pc
~n
,
I;l
k ,
< k
Proof
i)
It c a n b e s e e n t h a t The relation
2)
It f o l l o w s
An i m p o r t a n t Proposition
Sy,q
Sy,q e S o
easily
and
Fy ~ G
results
satisfy
(14), t h e r e f o r e
S y , q e JG,p
"
easily.
VVV
property
o f the s e q u e n c e s
of smooth functions
s ¢ IG, p n So
is g i v e n
in:
2 s e IG, p n S o
Suppose
@ G 1 , ...
then
, Gh c G
supp <s,.> afr Therefore
:
G 1 u...u
int supp < s , ' >
fr G h
= ~ .
*)
*)
Proof s ~ IG, p , t h e r e
Since (15) and
exist
Wl
''''' W h ~ JG,p
and
G 1 ,..., G h E G
such that
s = w I + ... + w h w.1 , G i , w i t h
lations
1 <- i <- h
s e S O , (15) and
(16)
, satisfy
(14.2)
s u p p < s , ' > a G1 u...u
Take now
1 -< i -< h
and
(14).
Since
G 1 ,..., G h
are c l o s e d ,
the re-
imply Gh
x e int G. 1
and d e n o t e
I = { I _< j _< h
l x ~ int G. } , 3
K = { 1 -< j <- h
] x * G. } J
J = { I <- j -< h
I x ¢fr
G. } , J
Obviously I n J = I n K = J n K = ~ , For
j c I
take
the n e i g h b o u r h o o d
*)
fr A
and
I u J u K = ( 1 .... ,h }
V. c G. , Vj n e i g h b o u r h o o d of x . For j ~ K take V. c Rn\G. , V. J J J J J of x r e s u l t i n g f r o m (14.2). Denote V = n V. . T h e n (14.1) j~IuK J
int A
denote respectively
the f r o n t i e r
and i n t e r i o r
of a s u h s e t
A c Rn
42
applied to
V. with J through (15) in
j c I
and (14.2) applied to
V. J
with
3 c K
jcJ If
J = ~ , the above relation implies
then
x Efr
x ~ supp <s, "> . Assume
will resNlt
J J # 6
Go . Taking now into account (16), the proof is completed J
and
j ( J ,
WV
Corollary i Suppose
G c FF,Io c (see §2),
p e ~l
and
s ~ IG
,p
n
S
o
then
As c F : i)
(Fy ] y ~ As)
2)
supp <s, "> c
locally finite U fr Fy YeA s
Proof For each
Gi e G
in Proposition 2, there exists
locally finite and Choosing
§5.
A.I c F
such that
(Pv [ x ~ a i)
Gi =
A = A 1 u...u
u Fy , therefore fr G. c u fr Fy . yeA. 1 yeA. 1 1 i h , the proof is completed WV
LOCAL CLASSES AND COMPATIBILITY
A specialization of the vector spaces A vector suhspaee
T
{17.1)
Tn
(17.2)
%/ t E T
V
XE V
o
in
SO
T
in Theorem i, §3, is given now.
is called a local class, only if
=0 t~O
:
Rn :
]/~ N
:
t(V) [x) ~ 0
Proposition 3 A locally vanishing ideal I n S o c V° Q
f
and a local class
T
are compatible, only if
T
Proof It suffices to show that
f
and
V
satisfy (4). Taking now into account C17.I) it
43
remains Then
to p r o v e
(17.2)
results ]
But
(13)
that
cal
result,
classes
R1
v e I n T
, v % 0
.
will
ends
in t h e
c N
imply
that
v ~ ~'
:
the proof
following
C ° ( R n)
, V
~ ~
:
v(v)(x)
v(v)(x)
~ o
= 0
VV
of c o m p a t i b l e proposition
locally
whose
vanishing
proof
uses
ideals
a n d lo-
the c a r d i n a l
equi-
.
4
So Q T
c
~ w
the existence
and
For any vector
J
:
V v £ N ,
implying
is g i v e n
Proposition
q = 0
obtained
between
. Assume
V ~ < N
c N :
The c o n t r a d i c t i o n
valence
:
(12) w i t h
~'
= O
in
x c Rn
and
A basic
I o T
subspace
J
in
We n S o
there
exist
local
classes
T
such that
.
Proof Assume
(e i [ i ~ I)
e i = s. + 1
(J n % )
V x ~ Rn
. One
is a H a m e l
, where
can assume
base
s. ~ J 1
in the v e c t o r
. Assume
the existence
~
space
e C ~ ( R n)
o f an i n j e c t i v e
J/(J n
E = such
go)
. Then
that
~(x)
~ 0 ,
a
: I ÷
(-i,I)
mapping
. In-
deed
j c w c (N+
C°(Rn))
and
car
C°(R n)
= car
R1
*)
therefore
car car E ~ car J ~
Now,
define
v i e So
(18)
' with
Vi(~) ) (x) =
Denote by ve that First,
T T
the
the is
sought
inclusion
J c
ing
Z c i ( s i + v i) ieJ
t =
since
*)
car
I
,
certain
v. c V i o
X
J
denotes
after
finite
with
,
subspace
J c V
+ T
o
and
IV
~) e N
,
in
So
generated local
x e Rn
by
s e J
c. c C1 1
. Hence
that
t c T
then
number
J i c I } . We p r o -
+ vi
of the
s + J
n V
= o
s -
and
set
Z c. s. = v i 1 icJ s = v -
.
the cardinal
{ si
class.
. Assume
it f o l l o w s i e J
= car R 1
i e I , by the relation
(a(i))~)~(x)
vector
the
N
(car R I)
X
Z c. e. iEJ i i ~ J
n V
o
, for
. Denot-
~ c. v. + t ~ V + T 1 i o icJ
,
44
It only remains to prove that t•
Y
o
n T . The r e l a t i o n
(19)
t =
F
is a local class. First, the r e l a t i o n
t ¢ T
(17.1). Assume
implies
ci(si+vi )
Z
ieJ
where j c I , J
finite and
c. ~ C 1 . Now,
t ~ F
i
since
v i e Y° , w i t h
gives
ci = 0 ,
i • J . But
V i ¢ J . Then,
holds for any
t c T ,
t ~ O
(20)
~
V
~
for
t
tion
(12) w i t h
• N
:
V
(21)
~
The relations (22)
E c i si ¢ J , hence i•J
(19) will imply
and
J
Z c. s. E F ieJ i I O
'
E c i ei = 0 c E , which i•J
t ~ O . We prove now that
(17.2)
x E Rn . Indeed, assume that
¢ N ,
given in (19). Since
results in O
~
-> D
:
t(V) (x) = 0
is finite and
s i c J c W° , with
i ¢ j , the rela-
q = 0 , will imply ~'
c N :
V
~
E N ,
V
->~'
,
i • J
:
si(~)(x)
= 0
(20), (21) and (19) give
~
~''
• N :
T a k i n g into account
V
v
E N
,
V
-> ~''
(18) and the fact that
:
E c i vi(V) (0) = 0 icJ
~(x) # 0 , we obtain from (22) the relati-
ons (23)
Z c i(a(i))~ = 0 , iEJ
V
~ E N ,
~ ~ ~''
Since to
a is injective, (23) implies c. = 0 , Y i ¢ j , therefore i (20). The c o n t r a d i c t i o n obtained ends the p r o o f W V
t ¢ 0 , according
Now, the answer to the first p r o b l e m in Remark 2, §3. Theorem 2 For any locally v a n i s h i n g ideal
I
there exist c o m p a t i b l e local classes
T
.
Proof Assume space in
Z
is a l o c a l l y v a n i s h i n g ideal. Denote W ° n So
local class
T
J = I n SO
then
J
is a v e c t o r sub-
according to (13). Now, P r o p o s i t i o n 4 will imply the existence of a
such that
J c
Vo(~T
. Taking into account the r e l a t i o n
J = Z n SO
and P r o p o s i t i o n 3, the above inclusion is the n e c e s s a r y and s u f f i c i e n t c o n d i t i o n for the c o m p a t i b i l i t y of
I
and
T . WV
45 §6.
Re
DIRAC ALGEBRAS
solution of the second problem in Remark 2, §3, n ~ e l y ,
s~spaces
S1
in
SO
~
satisfying (6) and (7)
the existence of vector
is obtained in Corollary 2, below.
Proposition 5 Suppose
V
and
T
are vector subspaces in
V°
respectively in
S
'
and the O
conditions
V
i)
nT=O O
2)
u n (vQT)
~ u n (v(p) C)~5
,
v
S1
in
pen
n
are satisfied. Then, there exist vector subspaces
S
so that (6) and (7) hold. O
Proof Denote
U 1 = U n (VoQT)
U = U I C ) U 2 . Then in
So
such that
VoQTQU2QS
A local class
T
(24)
t c T
V
and assume
U2
vector subspace in
U
such that
U 2 n ( 7 o G T 5 = 0 , therefore, there exist vector subspaces
2 = S o . One can take now
S 1 = U2QS
2
S2
VVV
is called Dirac class, only if :
int supp =
Corollary 2 Suppose
T
is a Dirac class, then there exist vector subspaces
S1
in
So ,
satisfying (65 and the following stronger version of (7): (25)
U c S1
Proof Due to (17.15 and (24) it follows easily that
U n (VoQT
ditions in Proposition 5 are satisfied. One can choose now the mentioned proposition
5 = 0 , therefore, the con. S1
The problem of finding Dirac classes
T
is solved in Proposition 6, below.
A locally vanishing ideal
I
is called Dirac ideal, only if
(26)
:
int supp <s~-> =
V
s ~ I n S 0
as in the proof of
VVV
46
Theorem 3 For any Dirac ideal
I
there exist compatible Dirac classes
T
.
Proof T
constructed with
J = I n S
o
according to the procedure
in the proof of Proposi-
tion 4, will be a Dirac class. Moreover, due to Proposition compatible
(see the proof of Theorem 2, §5)
3,
I
and
T
will he
WV
The last problem, namely to secure Dirac ideals is solved in Proposition
IG
6 is a Dirac ideal, for any
G c FF
and
P ~ ~n .
,P
Proof It follows from Proposition
2, §4
VW
Now, one can sum up the previous results of P-regularizations
(V,S,)
and obtain the final answer on the existence
for any admissible property P.
Theorem 4 For any Dirac ideal class
T. Further,
I
(see Proposition
6) there exists a compatible Dirac
there exist vector subspaces
S1
in
S o , satisfying
the
conditions: (27)
VoQTQS 1 =
(28)
U c SI
Choosing
So
any vector subspace
for all admissible properties
V
in
I ~ V o , one obtains
( V , T G S I) c R(P)
P.
Proof Assume given a Dirac ideal
I , for instance,
according to the method in Proposition
Then Theorem 3 grants the existence of a compatible Dirac class Corollary 2 ,
one can obtain a vector subspace
S1
in
Taking into account Theorem I, §3, the proof is completed
The algebras used in the applications
presented
So
T
satisfying
(27) and (28).
VV?
in chapters 3, 4 and 5 can be defined
now. Suppose given a Dirac ideal in
I1
and a compatible Dirac class
W , I ~ I i , compatible vector subspace
T
in
6
. Now, according to
T 1 . For any ideal
I
S O , T = T 1 , vector s~hspace
V
47
in
I n Vo
and vector subspace
SI
in
So
satisfying (6) and (7), the algebras (see
(24) in chap. I): A Q ( v , T ( ~ S 1 , P) • where
Q
Nn
P
is a given admissible property, will be called Dirac algebras.
Remark C The basic method whithin the present work, in constructing regularizations - and therefore, algebras containing the distributions - has been given in Theorem i, §3. That method rests upon the notion of compatibility between an ideal
subspace
T
in
S
I
in
W
and a vector
o
I t i s worthwhile mentioning the b o t t l e n e c k f e a t u r e the notion of c o m p a t i b i l i t y e x h i b i t s Namely, g i v e n t h e i d e a l
I
, the compatible v e c t o r subspace
T
has t o be small enough
in order to satisfy (4) but in the same time, big enough in order to satisfy (5). In that respect, Theorems 2 and 3 are nontrivial. Both of them are based on Proposition 4, which rests upon the cardinal equivalence between
R1
and
C°(R n) , an essential
characteristic of the set of real numbers. Alternative ways, not depending on Dirac ideals and classes, but still within the framework of Theorem 1 in §3 of constructing regularizations will be given in §§8 and 9.
§7.
MAXIMALITY
Taking into account §I0 in chap. 1 as well as §3 above, it follows that there exists an applicative interest in constructing algebras containing the distributions, based on large, possibly maximal compatible ideals Denote by
C
the set of all pairs
subspaces
T
in
So
(I,T)
I
and vector subspaces
of compatible ideals
satisfying for a certain vector subspace
tions (see Theorem i, §3):
(29)
Vo (9 T (!) Sl : So
(30)
U c V(p)
where
V = I n V
Q
T (~
,
S1 ,
V
P ¢ ~n
o
Define a partial order ~ on
C
by
(Ii ' TI) ~ (12 ' T2) ~=~ Ii c 12
and
Lemma 3 Each chain in
(C,~)
has an upper bound.
TI c T2
T .
I
in
W
S1
in
So
and vector the condi-
48
Proof
i~
~s= T =
A)
c
is a chain in
U T X , then obviously X~A
they are compatible.
I
(C,~)
is an ideal
It only remains
and denote
u I X and leA is a v e c t o r subspace in S
W, T
to show that
I =
and
o
I
and
T
satisfy
(29) and (30).
Denote
u o
= un
n (v(p)(~)r) p¢~n
and assume
U1
is a vector subspaee
(31)
(Vo(~T)
in
Denote
(32) But,
S2
in
S O , such that
®Vl ® $ 2 : s o
S 1 = UI(~S
there exists
1 . If
U = Uo@U
n UI = 0
then, one can choose a vector subspace
Vo®
U , such that
2 , then
X E A
(29) and
(30) hold obviously.
Now,
if (31) is false,
then
such that
(vo Q r X) n u I # o ~X
' TX) c C
, therefore
(33)
Vo(Dq
(34)
U c VX(P)~T
where
(DSIx
V X = I X n Vo
there exists
a v e c t o r subspace
SIX
in
SO
such that
= So X~)SIX
. However,
,
V
p ~ ~n
the relations
(33) and (34) contradict
(32).
Indeed,
denote UoX = U n
(35)
then, obviously such that
Uo X c U ° , therefore
U = UolQUix
(wo Q T X ) Assume
u I ( UIX
(36) But,
, u 1%
and
there exists a vector subspace
UIX ~ U I . Then,
UIX
in
U ,
(32) implies
n UlX ~ o 0
,
v ~ Vo
and
t ( TX
such that
uI = V + t (34) implies
(37) where
n
pc iin (VX (p) ~ ) r k )
u I = v X + t' + s 1 vX E
(33) result
n
pE~n in
together with u I c UIX
Vx(p)
,
t' ¢ T X
v = v X , t = t' u I E UIX c U
. The contradiction
and and
s I c SIX
s I c 0 . Then
and (35) will obtained
. Now, the relations
imply
completes
(37) gives
u I E UoX the proof
u I = v X + t'
• Therefore V~7
(36),
(37) and which
u I ¢ 0 , since
49
It follows, due to Zorn's lemma, that the set
Cma x = { (I,T) ~ C I (I,T) maximal in (C,_<) } is not void. Moreover,
V
(I,T) E C
: max
IcI,Tc~
§8.
LOCAL ALGEBRAS
In this section,
(V,S')
regularizations
dure in Theorem i, §3, for the biggest
will be constructed according to the proce-
locally vanishing
V c I n V ° . The resulting algebras however,
ideal
I = W
and for o will not be used in the present work and
their interest here is only due to the alternative proof they offer for the existence of regularizations.
Proposition
7
Suppose
I = W ° , then there exists a compatible
subspace
S1
in
So
satisfying
local class
T
and a vector
(6) and (25).
Proof The existence
of a compatible
local class
T
results from Propositions
The problem is the existence of a suitable vector subspace obtained through
Corollary 2
J = Wo n So
and
E = J/(J n Vo) . Then
in
3 and 4 in §5.
So . That will be
in §6.
In this respect, we recall the way Assume
S1
T
was obtained
(e i I i E I)
in
S
4.
is a Hamel base in the vector space
e i = s i + Wo n V ° , with
as the vector subspace generated
in the proof of Proposition
o
by
s.1 E WO n So . Now,
(si+v i I i E I}
where
T v. i
is obtained are given in
(18). We shall prove that
(38)
u
n
Indeed, assume
(VoQr)
= o
~ E C~(R n)
, v ~ V ° and
into account the definition of (39)
u(~)
= w
+
Z
c.
iEJ where
w E V o , J c I,J
z
T
t ~ T
such that
and the fact that
u(~) = v + t . Then, taking
v i c V o , one obtains
s.
i
finite and
c i E C 1 . But
s i E W o , Therefore,
applying
(12)
50
for
q = 0 , one obtains from (39) the r e l a t i o n V
x~
Rn
~ N
: :
w(v) (x)
~(x)
=
w h i c h a c c o r d i n g to Lemma 4 below, low, the r e l a t i o n
gives
tO = 0
R n , thus ending the p r o o f of (38).
on
(38) grants the existence of a vector s u b s p a c e
S2
in
So
such
that
Vo®T®u®s 2 = Taking
S 1 = U QS
s o
2 , the p r o o f is completed
VW
Lemma 4 Suppose
t~ • C ° ( R n )
ly convergent to x~R
n
]JcN
is a sequence of continuous functions on
R n , weak-
such that
:
"O ~ N ,
on
~ = 0
v
:
v(v)(x) then
and
o ~ D' (Rn)
"~ -> ~t :
= ~(x) Rn .
Proof Assume,
it is false and
B c Rn
is a n o n v o i d open subset such that
V x c B . But, according to Lemma 5 below, there exists c N
G c B ,
~(x)
~ 0 ,
G nonvoid,
open and
such that v(v)(x)
= ~Cx)
It follows that for any
,
V
x ~ G ,
X • D(Rn)
with
v c N ,
v ~
supp X c G , the r e l a t i o n holds
[ n ~ ( x ) x ( x ) d x = v+oo lira fRn v ( v ) ( x ) X ( x ) d x = = 0
w h i c h c o n t r a d i c t s the fact that
~(x) ~ 0 ,
V x c G
WV
Lemma 5 Suppose
E
is a c o m p l e t e m e t r i c space and
given the continuous functions that
f : E ÷ F
F and
is a topological space. Suppose f
: E ÷ F , with
~ ~ N , such
51
V
x~
E :
9
~c
N:
V
~) ¢ N ,
Then,
for
open
subset
each
f(x)
~) >- l/ :
= f(x)
f~(x)
nonvoid
G c H =
f(x)
closed
and ,
subset
~ c v
x
N ~
G
H c
such
that
,
~
v
N
E , there exists a nonvoid relatively
~)>-]J
,
.
Proof Given
H
and
p ~ N , denote
f(x) =f(x) , v v~ N ,
H~={x~H[
~p}
The hypothesis implies obviously M ~ ~£N
= H
Now, it is easy to notice that nuity of
f
and
fv " Since
~ H
, with
gory argument implies the existence of HDo
is not void
~ c N , are closed in
E
due to the conti-
is in itself a complete metric space, the Baire cate~o c N
such that the relative interior of
VVV
The alternative proof for the existence of regularizations,
not based on Dirac ideals
and classes is obtained in Theorem 5 There exist local classes S1
in
So
(40)
%®T®s
(41]
U c S1
T
compatible with
~
as well as vector subspaces
satisfying
I = so
Choosing any vector subspace
Y
in
W o n V ° , one obtains a regularization
(v,~ Q s 1) Proof It follows from Proposition 7 as well as Theorem 1 in §3
Suppose given an ideal For any ideal space gebras
V
in
I
in
I n V°
I.
in
W
and
I. c W
WV
o
W , I = I. , compatible vector subspaee and vector subspace
S1
in
So
T
satisfying
in
S o , vector sub-
(6] and (7), the al-
52
Aq(V,eQs I where
Q
, p) ,
~n
p
is an admissible property, will be called local algebras. Obviously, they
contain as p a r t i c u l a r cases the Dirac algebras.
Remark M T h e o r e m 6 in chap.
i, §I0, the inclusion
W
o
as well as §7 of the present chap-
c W°
ter rise the question: Are the local algebras obtained for V = W° n V°
I = I, = W °
is maximal according to chap.
maximal either in the sense that
I, §i0, or
I = W°
is maximal according
to §7? The algebras c o n s t r u c t e d in the next section give a n e g a t i v e answer.
§9.
FILTER ALGEBRAS
Given a filter base tions
w e W
B
on
WB
R n , denote by
the set of all sequences of smooth func
w h i c h satisfy the c o n d i t i o n BoB V
(42) V
:
xc
B :
Vc
N :
V e N ,
~-> p
w(~) (x)
=
:
o
or, u n d e r a simpler form B(B
(42')
:
w(~))(x) = 0 , Obviously, If
B1
WB
and
is an ideal in
B2
in
Rn
x c B ,
~ £ N,
big enough
W .
are filter bases on
Rn
and
B2
generates a larger filter than
B1 ,
WBI c WB2 .
then o b v i o u s l y A filter b a s e
¥
B
on
for each
is called s t r o n g l y dense, only if
Rn
B ¢ B
.
T h e following filters on
Rn
Fv
=
{R n }
Ff
=
{ F c Rn I Rn \ F
finite
F~,f =
{ F c Rn [ Rn \ F
locally finite }
~d=
{F~Rnl
n o w h e r e dense
Rn\F
}
Rn \ B
is nowhere dense
53
are examples of s t r o n g l y d e n s e filter bases on Moreover,
if
B
R n . Obviously,
is a s t r o n g l y dense filter b a s e on
R n , then
Fv c Ff c F%f c Fnd B c End
.
.
Due to the r e l a t i o n wo
=
v the algebras c o n s t r u c t e d in this section will contain as p a r t i c u l a r cases the local al gebras d e f i n e d in §8. And now the important p r o p e r t y of the ideals
WB
.
Proposition 8 Suppose
B
is a s t r o n g l y dense filter base on
subspaces
T
in
S
compatible w i t h the ideal
o S1
vector subspaces
Rn . Then, there exist vector
in
S
o
satisfying
I =
WB
.
Further, there exist
(6) and (25).
Proof W e shall adapt the p r o o f of Propositions 4 and 7. Assume
(e i I i c 15
is a Hamel b a s e in the v e c t o r space
E = (7 n So) / (I n Vo)
.
Then e. = s. + I n V , where s. ~ I n S . Assume ~ • C ~ ( R n) such that i i o i o ~(x) ~ 0 , V x c R n . Finally, assume a : I ÷ (-I,I) injective. Define n o w v i c V° , w i t h (43)
i c I , b y the r e l a t i o n
vi(w )(x) =
Denote b y p r o v e that
T
(a(i)) v ~(x)
and
T
(445 Assume
(455 with
w • N ,
I n S
remains
to
S
o
x ~ Rn . by
{s
i
+ v i I i • I}
"
We shall
are compatible.
+T o p r o o f of Proposition 4, in §5. only
V
the v e c t o r subspace g e n e r a t e d in I
The relations
It
,
c V
o
prove
and
V
n T = 0
o
result easily, as can be seen in the
that
I n T = 0 t c I n T t = j c I ,
, then
J finite and
x
¥
t • I = WB
implies
:
:
W •N, t(V5 i x )
In the same time,
c i • C 1 . Now
c B :
~cN
(46)
implies
Z c i (si+vi) iEJ
B•B
V
t c T
W->!J
:
= 0
si • I =
WB
,
with
i • J , and the finiteness o f
J
imply
54
B'~B V (47)
x
•
~ B':
~
~' c N :
V
~)
e N ,
v -> ~'
si(V)(x) The r e l a t i o n s
(48)
(46),
and
B''
eB
%;
xe
B v' "
V
9 e N ,
(48.1)
Z
i E J
:
= 0
(47)
~''~
,
(45) r e s u l t
in
•
N
: 9 -> ~''
c i vi(~)(x)
•
= 0
ieJ D u e to
(43), t h e r e l a t i o n
(48.2)
can be written
as
Z c. (a(i)) ~ = 0 iEJ i
since
~(x)
therefore proof
(48.1)
~ 0 ,
o f (44)
Now, we p r o v e suhspaces First,
in
B'' $ ~ • F u r t h e r ,
8, n a m e l y
give
the e x i s t e n c e
a
is i n j e c t i v e ,
t ~ 0 , ending
the
of s u i t a b l e v e c t o r
So
the r e l a t i o n
(VoQT)
U n indeed
implies
in
the s e c o n d p a r t of P r o p o s i t i o n
S1
(49)
B'' ~ B
c.z = 0 , V i ~ J . T h u s (45) will and e s t a b l i s h i n g t h a t I and T are c o m p a t i b l e .
we p r o v e
Assume
V x E R n , and
(48.2) r e s u l t s
=
~ e C°°(Rn)
0 , v e V
and
t ~ T
given by
(45) and s u c h t h a t u(~)
= v + t
Then (S0)
with
u(~)
w
= w +
Z c.s. ieJ i i
£ V O
hbw,
(50) and V
(47) w i l l x e B'
,
g i v e for a c e r t a i n ~ e N,
~) -> B'
B' e B
the r e l a t i o n
:
(Sl) (x) = w ( v ) (x) with B'
eB
p'
possibly
depending
on
x e B'
~ ~ ,
open
. But
.
Therefore V
( 82)
G c Rn , G
G' c G , G' ~ 0 , o p e n G' c B'
:
R n \ B'
is n o w h e r e
dense
in
Rn
since
55
Now, (51), (52) and Lemma 5 in §8 imply that One can conclude therefore that
~ = 0
on
~ = 0 Rn
on
B' , since
Taking
2
S 1 = U(~S
2
in (51).
and the proof of (49) is completed.
The relation (49) implies the existence of a vector subspace
Vo
w ~ VO
S2
in
S
such that
o
: s o
, the proof is completed
WV
Theorem 6 Given a strongly dense filter base in
So
compatible
with
WB
as well
B
on
R n , there exist vector subspaces T
as vector
subspaces
S1
in
ing (53)
v®
(s4)
UcS
®h
o
satisfy-
= so
1
Choosing any vector subspace
(V,TQS 1)
S
V
in
WB n V °
, one obtains
a regularization
.
Proof It follows from Proposition 8 and Theorem 1 in §3
Suppose given a strongly dense filter base that
B
on
WV
Rn
and an ideal
I,
in
W
such
I, c W B .
For any ideal space
I
in
I n Vo
W , I ~ I, , compatible vector subspace and vector subspace
S1
in
So
T
in
S O , vector sub-
V
in
satisfying (6) and (7), the al-
Q
is an admissible property, will be called filter algebras and they are with-
gebras
where
in the present work the most general instances of algebras given by a specific construction.
The question in Remark M, §8, reformulated for the case of filter algebras obtained from
§I0.
I = I, = ~W=nd , remains open.
REGULAR ALGEBRAS
It is worthwhile noticing that the Dirac ideals the locally vanishing ideals
%
IG, p
(see (12)), the ideals
(see §4 and Proposition 6 in §6), WB
(see (42)) as well as the
56
ideals
l,
I@
and
15
used in chapters
5, 6 and 7, are all sgbseffuence, invariant
(chap
§6).
It will trary
be shown i n t h e p r e s e n t
subsequence invariant
ons c o n s t r u c t
An ideal
algebras
I
in
(553
W
un
section
ideals
containing
(see Proposition
Z
in
starting
W , one can u n d e r r a t h e r
with arbi-
general conditi-
the distributions.
is called regular,
(v ° + ~ )
10) t h a t
only if
= o
and V
veln
V
~ ~ N ,
U¢ (56)
V
:
o
N :
x¢
~ > U
:
Rn :
v(~) (x) = o or, shortly, if
(56')
v ~ I n V
then
o
v(v)
does not vanish in
for at most a finite number of
Proposition
v ¢ N
9
Suppose T
Rn
in
I
is a regular
SO
ideal.
Then, there exist compatible v e c t o r subspaces
and vector subspaces
S1
in
SO
satisfying
(6) and
(25).
Proof We shall once more use the method of p r o o f in Propositions
4, 7 and 8.
Assume
E = (I n S o ) /
Then
(e i I i c I) i
i
V x ( Rn i(
is a Hamel base in the vector
e. = s. + I n V
I
. Finally,
Assume
i
assume
vi(V)(x)
Denote by
(58)
s. ¢ I n S a
. Assume
space
~ c C ~ ( R n)
where
(I n V o)
(-i,i)
injective
and
define
vi ( V°
T
= (a(i)) v • ~(x)
the vector
prove that
I
and
subspace in T
,
V
SO
g e n e r a t e d by
are compatible
v ~ N ,
, with
(see
(4),
V° n T = 0 v ¢ V
n T , then
v c T
implies
v =
J c I , J
Z ci(s i + vi) icJ finite and
c i c C 1 . But (59) gives
x ¢ Rn { si (5)).
* vi First
•
~b(x) $ 0 ,
0
: I ~
O
(59)
such that
,by
(57)
shall
O
with
[ i ¢ I } . We the relation
57
(60)
Z
c.
ieJ since
si • I
the
v
-
~
c.
ieJ
v,v i • V °
. Then
(59)
v-
i
. Now will
n
~: I
F
l
o
(60] r e s u l t s
give
v ~ O
in
Z ieJ
, ending
c- e. = 0 ~ E l l
the proof
of
hence
(58).
relation
(61)
c
I n S o
Assume
=
i
and
ci = 0 , V i • J Now,
s.
i
Volt
s • I n S
then o
(62)
~ E
hence
o
s + I n V
=
Z ieJ
o for certain
s + I n V
~
J c I ,
J
c. e. l i
finite
s -
E icJ
c. s. = v 1 1
s
E ieJ
ci
'
and
e I
c.i e C 1 . But
n
V
(62)
can be written
as
o
therefore
since
v,v i • F°
In o r d e r (63)
t c f n T
(64)
t =
J c I ,
(65)
since
to p r o v e
(si+vi)
and
(60)
that
I
+ v
-
Z i~J
ci
vi
c T ~ V _ _ ,,
is p r o v e d . and
T
are
compatible,
it r e m a i n s
, then
Z ci i•J J
V =
t c T
implies
(si+vi)
finite E ieJ
v i ~ Fo
V
and
c. e C 1 . H e n c e l
C. V. = t - Z I i ieJ
and
t,s i ~ I
MeN,
M->~
x~Rn
. But
C. s. E I N V o l i (56)
applied
to
v ~ I n VO
:
:
v[v) (x) : 0 which
together V
with
(65) r e s u l t s
~ ~ N ,
~ -> ~
:
in ~
x e Rn
:
Z C i Vi(~) ) (X) = 0 icJ Now (66)
to show
I n T = 0
Assume
with
=
(57)
and the Z i•J
fact
that
~(X)
ci(a(i)) ~ = 0 ,
# 0 V
,
V x c Rn
v • N
,
~ >-
, will
imply
gives
that
58
The well known property of the Vandermonde V
i e J , hence
t ~ 0
determinants
applied to (66) gives
c. = 0 , 1
due to C64) and the proof of (63] is completed.
Now, we prove the existence of vector subspaces
S1
in
SO
satisfying
(6) and (25).
First, we prove U n (Vo~)T)
(67) Obviously red
= 0
T c V ° + I , hence
S1
(67) follows from (55). Now, the existence of the requi-
results easily from (67)
Vq
Theorem 7 Given a regular ideal
I
in
W , there exist vector subspaces
T
in
S o
tible with
I
(68)
VoQTQs
(69)
U c S1
as well as vector subspaces
I :
in
So
compa-
satisfying
so
Choosing any vector subspace
(V,TQS
S1
V
in
I n V ° , one obtains a regularization
1 ) •
Proof It results from Proposition 9 and Theorem 1 in §3
W?
The existence of regular ideals is granted by: Proposition
i0
A subsequence Therefore,
invariant ideal
a subsequence
I
in
invariant,
W
is proper only if it satisfies
proper ideal
I
in
W
(56).
which satisfies
(55) is regular.
Proof I
It suffices to show that (56) holds whenever v E I n V
o
V
Define variant.
is proper.
Assume it is false and
such that
w E W
~EN
by
:
~
~
oN,
w(p) = v(~p)
But obviously
,
~
:
v(up)
V p c N . Then
~ 0
on
Rn
w c I , since
I
is subsequence
in-
i/w ~ W , therefore
u(1) : w • (l/w) c I • W c contradicting
~P
the fact that
I ~ W T
I ~V
It follows that the ideals mentioned at the beginning of this section are regular
(in
59
WB , the additional condition that
the case of the ideals ter base on
Rn
B
is a strongly dense fil-
is needed].
One can easily notice that the set of regular ideals is chain complete, therefore, due to Zorn's lemma, there exist maximal regular ideals containing any given regular ideal.
Suppose given a regular ideal For any ideal space
V
in
I
in
I n V
W ,
I1
and an ideal
I.
in
W
such that
I z I. , compatible vector subspace
and vector subspace
S1
in
So
T
in
I. c I 1 . S O , vector sub-
satisfying (6) and (7), the alge-
bras
A Q ( v , T C ) S 1 ,P) , where
Q
P ¢ hP
is an admissible property, will be called regular algebras.
The regular algebras will find an important application in chapter 3, §4, where a general solution scheme is established for a wide class of nonlinear partial differential equations.
Remark 3 The condition (55) in the definition of a regular ideal is needed in order to secure the condition (69) in Theorem 7 (see (7) in Theorem i, §3 and (67) in the proof of Proposition 9, as well as (20.2) in chap. I, §7). However, due to Proposition 5 in §6, one can replace (55) by the weaker condition (70)
where
un
(v °
V = I o V
+z)
and T o case the relation holds V° Q T
c u n (V(p)(~)T)
,
v
p ~ i$
was constructed in the proof of Proposition 9, since in that
= V° + Z n S O
C h a p t e r
3
SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL APPLICATION
§I.
EQUATIONS
TO NONLINEAR SHOCK WAVES
INTRODUCTION
It will be proved in §2 of this chapter that, the piece wise smooth weak solutions of nonlinear partial differential efficients,
equations with polynomial nonlinearities
satisfy these equations
tion and derivatives
in the usual algebraic
defined in the Dirac algebras
and smooth co-
sense, ~Jith the multiplica-
containing
D ,(R n)
introduced
in
chapter 2. An application
to the shock wave solutions of nonlinear hyperbolic partial differenti-
al equations will be given in §3. When dealing with partial differential
equations,
one has to consider various nonvoid
open subsets ~ in R n and restrict the functions and distributions is obvious that the construction out in chapters
§2.
(i)
on
~ c Rn ,
are polynomials
(2)
T. u(x) i c.. ij
~ $ ~ , open.
OPERATORS AND SOLUTIONS
operator
T(D)
u E C~(~)
,
is called polyno-
Z Li(D ) T i u(x) l~i~h
,
V
x ~ ~ ,
operators with smooth coefficients,
~hile
of the form =
Z c.. (x)(u(x)) j l~j~k, lj i
,
V
u (C~(~)
,
x ~ ~ ,
smooth.
The polynomial only if
=
are linear partial differential
Li(D )
It
carried
~ ~ only if
T(D)u(x)
where
to such subsets.
containing the distributions,
~ c R n , ~ $ ~ , open, a partial differential
mial nonlinear
with
1 and 2, remains valid for any
POLYNOMIAL NONLINEAR PARTIAL DIFFERENTIAL
Given
T. i
of the algebras
nonlinear partial differential
u(x) = 0 , for
Obviously,
the polynomial
ses of the operators
x c ~ , implies
operator
T(D)u(x)
T(D)
= 0 , for
nonlinear partial differential
is called homogeneous, x ~ ~ .
operators
are particular ca-
in chap. i, §9.
The nonlinear hyperbolic
operators
studied in §3 are examples of homogeneous
polynomial
61
nonlinear partial differential linear wave operators
operators.
The same is the case of several types of non-
studied in recent literature,
well as other nonlinear partial differential A function
u : ~ + C1
(3)
T(D)u(x)
There exists a set the set (see chap.
If
[801.
x ¢
conditions
A
(4),
of mappings
2, §2)
zero Lebesque measure in
(4)
operators,
is called a piece wise smooth weak solution of the equation
= 0 ,
only if the following
[51, [8-111, [911, [1211, [122],
(5),
y
(6) and (7) a r e s a t i s f i e d :
: ~ ÷ R mY , with
FA = ~ x ~ R n ] ~ y E A Rn
Y e O~ ,
my e N , such that
: y(x) = 0 ¢ RmY } is closed, has
and
u e C (~\F A) k = max { k i ] 1sigh } (see (2)) then
(5)
u
k
is locally integrable on
The weak solution property holds
(6)
where
J ( l~i~h Z Ti
L*i
(D)
u(x) L*i
(D)~(x))dx
is the formal adjoint of
v
= 0 ,
~ ~ D(~)
,
L i (D) m
For each
y e A there exists a bounded
and balanced neighbourhood
By
of
Oe
R
such that
(7)
{ y-l(By)
] y ¢ A }
is locally finite in
The nonlinear hyperbolic partial differential
~ .
equations
studied in §3, are known,
[1331, [52], to possess piece wise smooth weak solutions
The main result of the present chapter is presented
in the above sense.
in
Theorem 1 Given a homogeneous
polynomial
nonlinear partial differential
defined on a nonvoid open subset u : ~ ÷ C1
(8)
~ c Rn
operator
T(D)
and a piece wise smooth weak solution
of the equation
T(D)u(x)
= 0 ,
x e ~ ,
there exist regularizations sible property
(V,S')
(see chap.
i, §7) such that for any admis-
Q , one obtains
I)
u ¢ AQ(v,S',p)
,
2)
in the case of derivative the usual algebraic
V
p algebras,
u
satisfies
the equation
sense, with the respective multiplication
vatives within the algebras
AQ(v,s',p)
,
p ~
~n
•
(8) in and deri-
as
62
Proof m ~y : R Y + [0,i] ,
Assume given
~y ~
for each y ~ A
(9.1)
ey = 0
on a certain neighbourhood
(9.2)
~y = 1
on
For u
v • N
and
R*'~ \ By
x • Rn
__VY of
in such a way that
0 e R'Y ,
(see (7))
define a regularization of the piece wise smooth weak solution
by
u(x)- -T-F- c~y((v+i)y (x)) (io)
if
x • fl\FA
],cA
s(v)(x) 0
if
x e FA
We prove that (ii)
s c W(~)
Assume
~ E N
(i2)
given. If
x ~ ~\F A
{ y • A I (~+l)y(x) c By } (~+l)y(x) c By
Indeed, (7) and
the
fact
the product Thus
that
only if Ry
, with
finite.
x • y-I ( ~ T B y ) .
s(v)
(13)
V
of
Now, (13) and (i0) imply that x c FA
then
only
But a finite
~ \ FA
(12) w i i i result
(12)
and
number
(9.2)
imply
of factors
from that i 1 .
is open, one can take a compact
x , V c ~ \ F A . Then, as in (12), one obtaines
{ y ~ A I (v+l)y(V) n By ~ ~ }
such that
Therefore,
y • A , are balanced.
--~v((v+l)y(x)) in (i0) contains y•A " is well defined on ~ \ F A . Since
neighbourhood
If
then
s(~) • C ~
finite. in
x . V
y(x) = 0 for a certain y ~ A . Take 1 Vy (see (9.1)), then
a neighbourhood of
x ,
y(V) c ~
(14)
s(~) = 0
on
V
according to (9.1) and (i0). Therefore,
s(w) e C °O in
x
and the proof of (Ii) is
completed. Define
v e I¢(~)
(15)
v
=
by T(D)s
The sequence of smooth functions by replacing
u
v
is obviously measuring the error in (8) obtained
with its regularization
role in constructing the ideals
s
IQ(v(p),S ')
in the construction of the algebras
given in (I0) and it plays the essential of sequences of smooth functions needed
AQ(v,S',p)
(see (24), chap. I, §7).
63
We p r o v e t h e r e l a t i o n s V
(16)
K c fi k F A ,
%/ V c N ,
s (v)
Indeed, te
on
K
v())) = 0
on
K
AK = { y c A ] y ( K ) n By $ ~ } , t h e n
{ [[y(x)[[y
] y ¢ h K , x e K } *)
K n FA = ~ , K c o m p a c t . sup
Then
(v+l)y(K)
(10),
(15)
An o t h e r
Obviously,
there
I I xy[ I y - < ~ a , m c R Y \ By ,
V
relation
needed,
~c
results
easily
A last
property
(18)
N ,
f r o m (14)
The preliminary
and
<s,'>
from (11),
on
(16) r e s u l t s
easily
T(D) i s h o m o g e n e o u s . s , given in
= u (4) and ( 5 ) ,
since
in the last
relation
one can as-
being trivial.
above lead to the following
essential
property
of the error se-
and
the relation
v c IFA
v c IFA
to prove that
,P ,
,P
,
V
p ¢ ~n
%l p E ~n
results
v e V (~) . Assume o
(see chap. from
(16),
~ c D(~)
and
2, §3) (17). ~) ¢ N
then
imply
I
I v(-~)(x)¢(x)dx
[ = I
S
-<
I
Z
T. s ( , ) ( x )
l_
Li
" (D)¢(x)dxl
1
=
S
= [
Z (Tis(V)(x) l_
E l
J
-Tiu(x))
[ Ti s ( v l ( x )
i
-T i u(x)
(D)0(x)dx
f " [ L*i
[ <
(D)•(X)
] dx
supp I[
from
FA ,
since
(16),
T(D)
results
v ~ %(S)
It only remains
*)
and
~ e N , such that
:
(19)
(6)
is finite
Deno-
~n :
and ( 1 5 ) ,
k z 1 , otherwise
Indeed,
AK
(7).
y~A
o f the regularization
obviously
v
a > 0 , since
due to
given in
pc
s c So(S )
quence
is finite
V y c A , v ¢ N , ~ -> ~ . Now,
DPs(~)) = DPv(~)) = 0
sume
then
exists
AK
and ( 6 ) .
(17)
follows
:
x~ -> II :
= u
denote
a = inf
K compact
I]y
is
a n o r m on
fY
, with
y ¢ A , so that
ycAsup xyeBySUpI lxy[ Iy < ~
(15) and
64 Therefore,
it suffices to prove
(20)
I [ Tis(~)(x)
lim V+oo
- Tiu(x)
I dx = 0 ,
V
l<_i_
K a ~ ,
K compact
K
First,
(2) and (I0) imply for T.s(~)(x) 1
(21)
x E ~ \ FA
- Tiu(x ) =
the relation
Z cij(x)(u(x))J l<j
" (--~T-(~. ((~+l)y(x))) j - I) yea Y ' V
l~i~h ,
V £ N .
And, due to (9) one obtaines
r -7-7-
(22)
(ay((~+l)y(x))) j - i ] ~ I ,
V
j E N ,
~ E n
yea while taking into account also (23)
Now,
FA
in
The above relation
Rn
is zero, will imply
I
of the algebras
the ideal in
v Iv
W(~)
j e N
x E ~ \ FA
v E IFA ,P
chap. 2, §6, implies that
I
Assume now given any Dirac v
AQ(v,S',p) v
the ideals
is based.
then
v
,P ,
V
therefore,
P ~ ~n
(chap. 2, §4)
I v c IFA ,P
and Proposition
6,
is a Dirac ideal.
ideal
I , such that
a I
then, according to Theorem 4, chap. 2, §6, there exists a Dirac class with
IQ(v(p),S ')
is a Dirac ideal (chap. 2, §5) and
Indeed, according to (19),
I
(20), completing the proof of (19).
generated by
I v c /FA
(25)
V
(19) offers the possibility of constructing
upon which the construction
(24)
- i) = 0 ,
(22) and (23) together with (5) and the fact (see (4)) that the Lebesque
measure of
Denote by
'
(7), it follows that
lim (-7-T-(~v((w+l)y(x)))J V+oo yc£ " (21),
x ~ '
Z . Thus, there exists a vector subspace
S1
in
T , compatible
So(Q ) , satisfying the condi-
tions (26)
Vo(~)(~T(~S 1 = So(Q)
(27)
U(~) c S 1
If
(28)
u
is not smooth, then
S1
can be c h o s e n so t h a t
s ~ SI
Indeed, in that case
s *
Vo(~)(~T(~U
since
u = <s,,>
is piece wise smooth and
is a Dirac class. One c a n c h o o s e a v e c t o r (29)
v c V(p)
,
suhspace V
P ~ ~n
V
in
I n Vo(~ )
such that
T
65
Indeed,
(19) implies that Dqv e V 02) ,
q
V
¢
Nn
o
while (16) and (17) and the fact that Dqv ~ IFA
,P ,
S' = T G S
1
V
~ \ FA
q e Nn ,
is open, result in
p e ~n
Denote now
C3o)
then Theorem 4 in chap. 2, §6, will imply that admissible property
(V,S')
is a Q-regularization for any
Q .
The relation u e AQ(v,S',p)
(31)
V
,
p c ~n
results easily from (18) and the fact that It only remains to prove that
u
D'(~) c AQ(v,S',p)
, with
p E ~n .
satisfies (8) in the usual algebraic sense, with the
respective multiplication and derivatives in the algebras
AQ(v,S',p)
•
Due to (28) and (30) one obtains (32)
u = s + IQ(v(p),S ') E AQ(v,s',p)
,
V
p c ~n
therefore, taking into account §9 and Theorems 2, 3, §8, chap. I, as well as (i5) and (29), one obtaines in the case of derivative algebras, the relations (33)
T(D)u = T(D)s + IQ(V(p),S ') = v + IQ(v(pO,S') = 0 E AQ(F,S ',p) ,
The relations (32) and (33) end the proof of Theorem 1
V p E ~n
WV
Remark 1 The regularizations
(V,S')
whose existence is stated in Theorem I, are obtained in a
rather simple, constructive way. Obviously, the algebras
AQ(V,S',p)
obtained are Di-
rae algebras.
§3.
APPLICATION TO NONLINEAR SHOCK WAVES
Consider the nonlinear hyperbolic partial differential equation (34)
ut(x,t) + a(u(x,t)) • Ux(X,t) = O ,
(35)
u(x,0)
where
a : R1 ÷ R1
= Uo(X)
,
x ~
R1
,
t > 0
x ~ R1
is a polynomial.
Obviously, the left part of (34) is a polynomial nonlinear partial differential operator on
~ = R 1 x (0,oo) c R 2
and it is homogeneous.
66
Under rather general conditions, for smooth, [133], [52], or even piece wise smooth, [32], initial data
u ° , the equation (34), (35) possesses shock wave solutions
u : fl ÷ R 1 , with the properties: There exists a finite set A of smooth curves
y : fl ÷ R 1 , such that
(36)
u ¢ ~(fl
(37)
u
(38)
J (u(x,t)~t(x,t) + f(u(x,t)) • ~x(X,t))dx dt = 0 ,
where
f
\ FA)
locally bounded on
is a primitive of
fl V
~ ¢ D(~)
a .
Obviously, such a solution
u
will be a piece wise smooth weak solution, in the sen-
se of the definition in §2. Therefore, Theorem 1 in §2 results in :
Theorem 2 If
u : ~ ÷ R1
is a shock wave solution of (34), (35) which satisfies (36),
[V,S')
(37) and (38), then, there exist regularizations such that for any admissible property ,
V
p ¢
(see chap. I, §7)
Q , one obtains
Nn
i)
u c AQ(v,S',p)
2)
in the case of derivative algebras, u
satisfies (34) in the usual al-
gebraic sense in each of the algebras
AQ(v, S',p) ,
p e ~n , with
the respective multiplication and derivatives.
§4.
GENERAL SOLUTION SCHEME FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
With the help of the regular algebras introduced in chapter 2, §I0, the general result on the piece wise smooth weak solutions of homogeneous polynomial nonlinear partial differential equations, established in Theorem i, §2, can be seen as a particular case of a yet more general solution scheme for a fairly arbitrary class of nonlinear partial differential equations presented next, in Theorem 4. Suppose given a nonlinear partial differential operator (see chap. i, §9) of the general form (39)
T(D)u(x) =
7
ci(x )
1_
~ cR n
-]-]--
is nonvoid and open, and
A distribution
S ~ D' (~)
T(D)u(x) = 0 ,
,
x ¢ ~ ,
ci ~ C~(~) '
Pij ¢ Nn "
is called a (nontrivial) regular weak solution of the non-
linear partial differential equation (40)
DPiJu(x)
K<j_
x ¢ ~ ,
67
only if there exists a weakly convergent
sequence of smooth functions
s
e So(g)
sa-
tisfying the following three conditions (41)
S = <s,'>
and there exists a nonvoid open subset
smooth function
(41.1)
S = u = s(v)
u ~ ~(G)
on
G ,
V
G c g
and a
, such that v ~ N ,
further (42)
v = T ( D ) s c Vo(~ )
and finally, the ideal
Z
generated
v
W(~)
in
by
DPv , with
p e Nn
, satisfies
the
condition
(431
[v
n
(V °
+ U + R)
c
V°
where
•
R = C1 • s
Theorem 3 A piece wise smooth weak solution of a homogeneous differential
polynomial nonlinear partial
equation is a regular weak solution.
Proof It follows from the relations
(18),
(19) and (24) in the proof of Theorem 1 in §2
WV
Theorem 4 Given a nonlinear partial differential regular weak solution
(44)
T(D1u(x)
S c D'(g)
= 0 ,
x e g
there exist regularizations
operator
T(DI
of the form in (39) and a
of the equation
, (V,S')
such that for any admissible property
Q ,
one obtains i)
S e AQ(v,s',p)
,
V
p e An
2)
in the case of derivative
algebras,
S
satisfies the equation
usual algebraic sense, with the respective multiplication within the algebras
AQ(v,s,
p) ,
(44) in the
and derivatives
P ~ ~n .
Proof Since
S
is a regular weak solution of (44), there exists
s c So(a)
satisfying
(41),
(421 and (43). We notice that the ideal condition
Z
is regular in the sense of chapter 2, §i0. Indeed, the v (43) above implies (55) in chap. 2, §i0. Further, (41.1) and (42)
68
a b o v e imply v(~) = 0
on
G ,
v e N ,
V
the re fo re V
wcI
: v w(V) = 0
(45)
which obviously al in
W(~)
implies
on
G ,
V
(56) in chap.
(46)
I ~ I
(47)
I
Then,
2, §I0. Therefore,
Iv
is indeed a regular ide-
.
Assume now given a regular ideal
So
~) c N
f
in
W(~)
, such that
v
n (V° + U + R) c V°
according
compatible
to Proposition with
I
9 in chap.
as w e l l
2,
as v e c t o r
§10 ,
there
subspaces
exist S1
in
'
Vo(~T(~S 1 : So
(49)
U c S1 S
is not
(50)
smooth,
(47)
(51)
in
as i n t h e p r o o f
of Proposition
S1
c a n be c h o s e n s a t i s f y i n g
in
I n Vo(~ )
9 in chap.
2,
s ~ S1
Now, one can choose (52)
v ~ V(p) v ¢ Z
v
c Z .
= rQs
a vector subspace ,
V
s'
one obtains
a regularization
1
S c AQ(v,S',p) from
(V,S')
,
to show that
multiplication
The relation
according to Theorem 7 in chap.
V
p ~ An
S
D'(~)
satisfies
and derivatives
c AQ(v,S,,p)
,
=
V
(44) in the usual in the algebras
(51) will give S
2, §I0.
Q , the relation
(41) and the fact that
It only remains respective
such that
p ~ An
Given an admissible property (54)
V
Denoting
(53)
(55)
T
s t V + T + U o
V + T c V + I i f one t a k e s T o o ' §10. Then ( 4 8 ) , (49) and (50) i m p l y t h a t
results
subspaces
satisfying
implies
since
since
S o
(48)
In case
vector
s + IQ(v(p),S ') c AQ(v,s ',p)
,
W
p E An
p ~ An . algebraic
AQ(v,S',p)
sense, with the ,
p c ~n
.
69
Now, §9 and Theorems 2 and 3 in §8, chap. 1 as well as (52) above, imply in the case of derivative algebras• the relation (56)
T(D)S = T(D)s + IQ(v(p),S ') = v + IQ(v(p),s') = 0 E AQ(v,s',p)
The relations
(55) and (56) end the proof of Theorem 4
,
V p c An
VW
Remark 2 I)
The condition (43) in the definition of a regular weak solution can be written under the following explicit form:
u(~) = v I + Z w. ~ D i i j where
v I ( V (~) O
w i c W(~)
pij r(D)s ~ ~ = 0 and
Pij
~
on
~ ,
Nn
•
Taking into account Remark 3 in chap. 2, §i0, the condition (43) above can be replaced by the weaker one given in (70), in the mentioned remark. 2)
The algebras
AQ(v,s ' ,p)
obtained in Theorem 4, are obviously regular algebras in
the sense of chapter 2, §I0.
Chapter
4
QUANTUM PARTICLE SCATTERING IN POTENTIALS POSITIVE POWERS OF THE DIRAC g DISTRIBUTION
§i.
INTRODUCTION
Potentials [273,
[283,
considered
with
strong
[i1S],
local
[116],
were those
powers
[140].
have been studied
The s t r o n g e s t
local
[273.
The p o t e n t i a l s
in scattering
singularities
o f m e a s u r e s w h i c h n e e d n o t be a b s o l u t e l y
to the Lebesque measure, sitive
singularities
in this
chapter,
(6) m , 0 < m < ~ , o f t h e D i r a c ~ d i s t r i b u t i o n ,
theory,
[3],
of the potentials
continuous
with respect
given by arbitrary present
obviously
postron-
g e r local singularities. The wave function solutions obtained possess the scattering property of being given by pairs
~_ , ~+
of usual
C~
solutions of the potential free motions, each valid on
the respective side of the potential and satisfying special junction relations on the support of the potentials.
In the case of the potential
~6, i.e.
m = 1 , the only
one treated in literature, [44], the junction relation obtained is identical with the known one.
§2.
WAVE FUNCTIONS, JUNCTION RELATIONS
One and three dimensional motions are considered. The one dimensional wave function ~ is given by
(11
~"(x)
+
(k-U(x))~(x)
= 0
,
x c
R1
(k
E
R1)
with the potential (2)
U(x) = ~(6(x)) m ,
x ~ RI
Ca c R I ,
m ~ (0,~))
The solution of (11, (2) is expected to be of the form
I ~b_(x) (3)
~+(x) where
if
x < 0
if
x > 0
(x)
~_ , t~+ c C~(R I) l~''(x)
are solutions of
+ ktp(x) = 0 ,
x e R1 ,
satisfying certain initial conditions ~-(Xo) = Yo '
~'-(Xo) = Yl
71
~$(xI) = z 1 •
¢ + ( x 1) = z o , where
_.m ~ Xo ~ 0 ~ x I ~ ~
Yl
and
a r e g i v e n and t h e v e c t o r s
Yo ' Yl ' Zo ' Zl c C 1
izo
'
Zl
might be in a certain relation. As known, [44], that is the situation in the case of the junction relation in
x = 0
between
~
and
m = 1
and
x ° = x I = 0 , when
~ + is given by
r,01 01 01
(4)
k*; (o)
~_(o))
1
m e (0,~) , the following three problems
In the case of an arbitrary positive power arise: 1)
tO
define the power
2)
to to
prove that the hypothesis
3)
,
(6(x)) TM
x E
R1
, of the Dirac 8 distribution,
(3) is correct, and
obtain a junction relation generalizing (4).
The first problem is solved in §5, where a special case of the Dirac algebras constructed in chapter 2 will be employed. The solution of the second problem results from Theo rem 4 in §5, and is based on the smooth representation of ~ constructed in §4. The third problem will be the one solved first, using a standard 'weak solution' approach presented in §3. That approach will also suggest the way the first two problems can be solved. The ~unction relations in
(5)
/ kb+ (0)
i,,-(o) ] :
x = 0
ZCm,~)
between
~
and
~+ , will be:
I ~ - (0) 1
where
(5.i)
Z(m,cO
=
{i0} ,
for
m E CO,I) ,
,
for
~ e
c~ER 1 ,
1
(5.2)
Z(l,~)
=
(see [44] and (4))
1 I
(5.3)
Z(2, - (V~) 2)
=
(-1) v
[
0 o
k
(5.4) with
Z(m,c 0 c~ = + 1
and
=
[: 01
- ~ -< K -< ÷~'
,
]
C-l) v
for
arbitrary.
/
for
W = 0,1,2,...
'
m e ( 2 , ~)
,
~
~
(-%o)
72
The interpretation of (5) in the case of one dimentional motions (i) in potentials (2) results as follows: I)
For
2)
If
m = 1 , the known, [44] , motion is obtained. m = 2 , then for the discrete levels of the potential well
(6)
U(x) = -(v~)2(~(x)) 2
x c R1
,
v = 1 , 3,5,7
,
,-..
there is motion through the potential, which causes a sign change of the wave function, namely 3)
If
,+(x) = -,_(x) ,
x c R1 .
m ~ (2,~) , there is motion through the potential (2) in the case of a
potential well only; however, the junction relation (5.4) will not give a unique connection in K
x = 0
between
~_ and
¢+ as the parameters
~
and
involved can be arbitrary.
As known, [44], the problem of the three dimensional spherically symmetric motion with no angul~r momentum, and the radial wave function
R
given by
(r2R'(r)) ' + r2(k-U(r)) • R(r) = 0 ,
r c (0,~)
where the potential concentrated on the sphere of radius U(r)
=
~(6 (r-a)) m
,
r
c
(0,~)
(~
c
a R1
(k ~ R I)
is ,
m
,
a
c
(0,~))
,
can be reduced to the solution of (i), (2). Therefore, the above interpretation for the one dimensional motion will lead to the corresponding interpretation for the three dimensional motion.
53.
WEAK SOLUTION
The solution (3), (5) of (i)~ (2) will be obtained in two steps. First, a convenient nonsmooth representation of ~ will give in Theorem i a weak solution of (i), (2). The second step, in §4, constructs a smooth representation of 6, needed in the algebras containing
D'(R I) . That representation gives the same weak solution, which pro-
ves to be a valid solution of (i), (2) within the mentioned algebras and therefore, in-dependent of the representations used for 6 . The nonsmooth representation of ~ , employed for the sake of simpler computation of the junction relations, is given in (7)
6(x) = lim V(~ v , i / ~ x)-~o
, x) ,
x c RI ,
where (8)
lim ~ v+co
= 0
and
my > 0
with
v c N ,
~ ~ N ,
73 while
(9)
where
v(~,K,x)
~ > 0
and
K
if
0 < x <
0
if
x ~ 0
=
or
X k~O
K e R1 .
Given
m c (0,~),e,k e R 1 , x ° < 0 , Yo 'Yl c C 1 and @v c C (R 1 \{0,~v}) n cl (R I) , the unique solution of (10) with
~''(x) the
initial
(ii)
,~/~v)m,x))~(x)
~'(Xo) = Yl
M(k,Xo) the set of all
(~v I v E N)
(12)
(m,~) c (0,~) × R 1
for which there exists
satisfying (8) and such that
lim ~
Suppose given
=
exists and finite, for any
~C%)
(m,~) C M(k,Xo)
@''(x)
Yo ' Yl c
•
zI
and
Yo ' Yl E C I , one can define
(13)
X ¢ R1
= 0 ,
conditions
$(Xo) = YO '
Denote by
any
+ (k-V(m v
c N , denote by
+ k~(x)
= 0
,
(~ @
I ~ c N)
satisfying (8) and (12), Then for
, ~+ c C~(R I)
x c RI
as the unique solutions of
,
satisfying respectively the initial conditions
(14) where
IYo,l ,
I ~2-(xO) z°
,
z 1 c C1
is
obtained
through
zo 1
IZl 1
(12).
Theorem i given in (3) with
Suppose functions to
~
(~v [ ~ E N)
~
, ~+ from (13) and (14). Then, the sequence of
resulting from (I0) and (ii) is convergent in
D'(R I)
.
Proof Obviously
~
= ~_ on (-°%0], for every
v E N . Thus, it remains to evaluate
on (0, oo) . The relation (12) and the second relation in (14) imply that V
a,~ > 0
:
~ p c N
: V ~ c N , ~ >- p
:
(iS)
Now, from the proof of Theorem 2, below, on can obtain that
~+ - ~ )
74
K>0
:
WreN
:
(16)
I 9\> I , I '7,, I -< ~
on
[0,~]
Indeed, according to (19) in the proof of Theorem 2, it follows that
9~(x)
=
W(k-~/(c0~)m,x)
-
,
L*_' (o)7
~ • N ,
V
which implies the following two evaluations, respectively for Assume
~ > 0 , then for any
o ¢ N
and
x e [0,u0~] .
~ > 0
and
~ < 0 .
x ~ [0,0~] , one obtaines
- *v(m~) I < ( ]exp(xH¢ - e x p L~l + lexp(-xHv) - e x p ( - L w ) ] ) "
] ¢v(x)
(17)
I,_(0)l
(
+ l g ' ( o ) l/s v ) / 2 _< (exp Lv +1) • ( ]9_ (0) 1 + i*_' (0) I/H v ) / 2 •
the last
inequality
resulting
Now, t h e r e l a t i o n s
from the fact
(21) a n d (22)
that
in the proof
0 < xHv -< L~
since
o f T h e o r e m 2, t o g e t h e r
<
0 -< x < ~ with
.
(17) a n d (12)
will imply (16). Assume
~ < 0 , then for any
I *~(x) - % ( % )
(18)
~ ¢ N
and
x ¢ [0,~ w] , one obtaines
I ~ I cos ~
- c o s L~ I "
I ,_(o)
I +
+ I sin xH~ - sin L~ ] " I 9 I ( 0 )
I/~
Now, the relation (23) in the proof of Theorem 2, together with (18) and (12), will again imply (16). The relations (15) and (16) obviously complete the proof
According to Theorem i, the function 9 in (3) with
~YV
9_ , 9+ from (13), (14) is a weak
solution of (i), (2) obtained by the respresentation of 6 in (7), (8), (9), provided the potential M(k,Xo)
(2) is obtained from
(m,~
¢ M(k,Xo)
. The problem of the structure of
is solved new.
Theorem 2 The set
M(k,xo)
does not depend on
M = ((0,1]
x R 1) o ({2}
k ¢ R1
x{-n2,-4~2,-9~
xo < 0
and
and
2 . . . . }) u ( ( 2 , ~ ) x
(-~,0))
u ((o,~) × {o}) Proof If
u c C~(R I)
is the unique solution of
u''(x) + h u(x) = 0 ,
x E R1 ,
( h ~ R 1)
with the initial conditions u(a)
= b ,
u'(a)
= c ,
then
09)
u(X)
u ' (x)
=
li(h,x)
W(h,-a)
,
c
x ~ R
1
,
u
75
where
Assume
(m,~) ~ (0,~) x R 1 . Applying
(19) to the functions
~
, one ohtains
,
V
V eN
It thus remains to make the condition (20) explicit in terms of
m
and
=
L ¢~(cO'O) ) Therefore, (20)
(m,a) ~ 14(k,Xo)
W
(k-c~/(tov)m,~ov) W ( k , - X o ) only if
lim W(k-~/(m~)m,mv) v+oo
= Z(m,~)
e > 0 . Since
First, suppose
Yl
~v ÷ 0
1 W(k-c~/(m'°)m'cav) = 2-
exists and finite.
one can assume
k - ~ / (~)m
e .
< 0 , therefore
exp Cv + exp (-L,o)
~.(exp
Lv - exp (-L,))
ttv (exp Lv - exp (-L~)))
exp I ~ + exp ( - I ~ )
)
with H
= (-k+e~/(~o~)m)I/2 , LV = m H
Obviously
(21)
limH
(22)
lim L,) +
= + ~ if
0 1/2
m = 2
if
m e (2,~)
Therefore,
M(k,Xo)O ( [ 2 , ~ ) x ( 0 , ~ ) )
Assume now
m ¢ (0,2)
( e x p L~ - exp ( - L ) ) 2 term,
m c (0,2)
if
= ~ •
, then the three
terms in
, have got a finite
W(k-~/(~V)
limit
when
m
'~V ) ' e x c e p t
÷ ~ . Concerning
the latter
one o b t a i n s
lim
~-~. (exp ~
Thus, one can conclude that Suppose now
< 0 . Since
- exp (-Lv))
=
if
m c (0,1)
if
m=
+ ~
if
m ~ (1,2)
M(k,x o) n ((1,2) × (0,~)) = @ ~v ÷ 0 , one can assume I cos L)
W(k-~/[co))m,o~))
0 ~
=
-H v sin L)
with H 9 = (k-~l(~))m) I12 , L~ = ~o)H~
and
1
(0,I] x (0, ~) c M(k,Xo)
k - ~ / (~)m
i___ H) sin L)] cos L~)
> 0 , therefore
76
Obviously lim H
(23)
= + oo 0 (-cO I/2
lim Lv =
+oo ASsume now
m e (0,2)
have got a finite
if
m c (0,2)
if
m = 2
if
m ~ (2, °°)
, then the three terms
limit when
~0÷co
if
= -~
$herefore
M ( k , x o) n ( ( 1 , 2 ) x ( - ~ , 0 ) )
Now, assume
limit when
,to)
, except
-H
sin L~) ,
one obtains
m e (0,I) m=
if
m e (1,2)
= ~
m
the latter term,
if
l
and
m = 2 , then again the three
have got a finite
W(k-(z/(~0v)
• Concerning 0
lim (-Hx) sin L)) v+co
in
(0,1] × (-~,0}
terms in
~ ÷ ~ , while
c b1(k,Xo)
.
W(k-a/(~)m,0J ) , except
the latter term tends
-H
sin L
to a limit according
to Lv~J
=
I]
0
lira (-H v s i n v+~ Therefore Finally,
÷~
M(k,x o) n ((2}× (-~,0)) assume
if
m c (2,~)
~ = _(p~)2
~ = 1,2,...
otherwise
= {2}× { _ ( ~ ) 2
, then
with
lim H
] B = 1,2 .... )
= lim L
= +~
thus a necessary
condition
for (20) is (24)
lim sin Lv = 0 v->oo
The condition
(24) will
has to be chosen A,B e (0, ~)
indicate
in order to secure
such that
O : (0,A) ÷ (B, ~) 0
defined by
is strictly
Therefore,
the inverse
on
and
(25) Moreover, (26)
lim o-l(y) y-~
(27)
lira
= o~
on
lira O(~) m+ A 0 -I
(0,A)
satisfying
(8)
k c R 1 , there exist
and the function
= ~(k - ~/ m)i/2
decreasing
has the properties
,
= B . : (B,o) ÷ (O,A)
exists,
is strictly decreasing
= 0
(n v I ~ c N)
n~
given
V ~ E (0,A)
I ~ ~ N)
and
of nonzero
-v,+,o
Indeed,
(~
= 0
lira DO-I(Y) y~
is a sequence
@(~)
function
O -I e CI(B, ~)
ASsume now that
(20)•
k - ~/0~TM > 0 ,
lira 0(00 = ~ c0-~o '
(B ,~)
the way the sequence
is a sequence
real numbers, ,
l i m e,o v+oo
=
0
of positive
integers
and
(ev
such that and
n~
+ e~o >
B ,
~
"o ¢ N
.
I ~ £ N)
77 Define (28)
Then
cov = o - l ( n ~
(m
] ~ ~ N)
(29)
+ e)
,
g
~ e N
satisfies (8), according to (27) and (25). Further, one obtains
cos L
= (-i)n~ cos ev , -H v sin Lv = (-i) n~+l H~ sin e~ ,
V
~ c N
Now, (29) and (27) imply t h a t (30)
lim cos L
exists
provided that (31)
n
~ with
~ c N , have c o n s t a n t p a r i t y .
Therefore, (20) will hold only if (32)
~lim ( - H sin Lv)
But, due t o ( 2 9 ) , (33)
exists and finite
(27) and ( 3 1 ) , t h e p r o p e r t y i n
lime Hv x)-~o
(32) i s e q u i v a l e n t w i t h
exists and finite
It is s i m p l e r t o compute t h e s q u a r e o f t h e l i m i t i n (33) which due t o ( 2 8 ) ,
(27) and
(26) becomes lira ( e H ~ 2 = lira (e~2(k-c~/(@-l(n ~+e )) m) = v~oo v+~o =
since
~ lim (e~ 2 / (@-l(n ~ + e @ ) m = x~+oo
=
- ~%~oolim ( [e [2/m o-l(n ~) + le [l-2/m Do-l(n ~+~ev)) -m =
=
- ~ lim
xk~o
~V ~ (0,i) ,
V
( e ) 2 / (o-l(n
9 c N
g
But, due t o (25) and ( 2 7 ) , t h e l a s t and
~)) TM
a proper choice of
n~
imply that for any
~ E {-i,i}
limit
can assume any v a l u e i n [ 0 , ~ ] ,
d e p e n d i n g on
ev • Therefore, (30) and the second relation in (29) will and
K ~ [-~ , +~] , there exists
(~
I ~ c N)
fying (8) and such that v+~ lim W(k- ~/(m~) m,~v)
Now, obviously
( 2 , ~ ) x (-~',0) c M(k,Xo)
and t h e p r o o f i s c o m p l e t e d
Remark 1 The relations (S.l) - (5.4) result easily from the proof of Theorem 2.
WV
satis
78
§4.
SMOOTH REPRESENTATIONS
FOR 6
In o r d e r t o p r o v e t h a t t h e weak s o l u t i o n s
(3),
(5) o f ( 1 ) ,
lid within the algebras containing the distributions representations lutions
(7),
(8),
(9) used f o r 6, we f i r s t
can b e o b t a i n e d from c e r t a i n
w i l l b e o b t a i n e d by a p p r o p r i a t e l y 'rounding off' ( s e e chap.
(2) o b t a i n e d i n §2, a r e v a -
and t h e r e f o r e ,
independent of the
need t o show t h a t t h e same weak s o -
smooth r e p r e s e n t a t i o n s
o f 6. These r e p r e s e n t a t i o n s
'rounding off the corners'
in (7),
i s a c c o m p l i s h e d w i t h t h e h e l p o f any p a i r o f f u n c t i o n s
(81 and ( 9 ) . The ~ , y • C:(R 1)
1, §8) s a t i s f y i n g : "1
(34)
[3 = 0
**) ***) ****)
(-%-i]
on
0 -< 8 < M
on
~ = I
[i,°°)
on
DPB(0) ~ 0 ,
(-1,1)
V p • N
and *)
(ss)
y = 1
**) ***)
The e x i s t e n c e
on
(-m,-1]
0 ~ y s 1
on
y = 0
[ 1 , m)
on
of the functions
(-1,1)
S
and
y
results
from Lemma 1, a t t h e end o f t h i s
section. (~M [ M • N)
Given now a sequence
(to~ ] M e N) , (L0$ I M • N) ' ' tO" tom M
(36) define
s6 • W
(37)
>
0
'
satisfying
(8) and two o t h e r s e q u e n c e s
such t h a t to'0 ' COi ' t0½
V V
• N
iim
(to,~ + to~') / %
'
,
"'"
are pair wise different
and
= o ,
by
s6(M) (x) = ~(x/to~) ¥ ((x-%)/to~)
V
/ tom '
veN,
xeR
1 ,
then f
supp s6(M) c [-m~ , ~M+ toni
and
]1 - I sd(~) ( x ) ~ I S 2((M+l)toO+to~)/to9 , i
therefore, (38)
V
M~N,
due t o ( 3 6 ) , one o b t a i n s s 6 • S O n W+ and
with the relation
<s 6 , .> = 6
s 6 • W+ ( s e e c h a p . 1, §8) i m p l i e d by (37) and t h e f a c t t h a t
Now, the smooth representation
of 6 obtained in (37) will be the one replacing
and (9). It remains to prove that (37) generates used in solving
again the weak solution
(m,e) • M , k • R 1 , x ° < 0 , (to9 I M • N) satisfying ,, C1 and (tom I ~ • N) satisfying (36), Yl ' Y2 • and
(i), (2). Given
(8) and (12), (to~ I V • M • N , denote by
NI
(7], (8)
[3), C5) when
XV • C~(R I)
the unique solution of
79
(39)
X"(x)
(k-~(s6(~)(x)) m)
+
X
(x)
=
x¢
0 ,
R1
with the initial conditions (40)
X(Xo) = Yo ' X'(x o) = Yl
Theorem 3
is possible to choose (%
It
I ~ ~ N)
such that the sequence of functions is convergent in
D'(R I)
to
~
and (~
(X~ I ~
I ~ ~ N)
E N)
satisfying
(36),
and
resulting from (39) and (40)
given in (3), where
~_ and
~+ are from (13)
and (14).
Proof A Gronwall inequality argument will be used. First, the equations (i0), (ii) are written under the form
F~(x) = A,o(x ) Fv(x) ,
x c R1 •
F,(Xo) = ] Y o [ ()Yl
where F~(x)
5q
=
,
A(x)
[,,,,', C~)/t Similarly,
(39),
(40) can be w r i t t e n
G~(x) = B~(x)
G~(x)
with
IK~(x)1
%(x) = L ×~(x)) Denote
H
= F9
-
H~(x)
G%, ,
II
,
B(x)
GV(x°) =
I
o m
, o,/(~ ) ,x)
1] 0
[,o] Yl
1}
°
-k+~Cs 6CV) (x))m
0
then
X
]]
-k+V(~
=
therefore
Applying the
I
as
x c R1 ,
,
=
X
f
I
X
X
O
RI
O
[l~ vector, respectively matrix norms, denoted for simplicity by
I[ , one obtains
x
x
I[Hv(x) l I _< J IIAv(~)-B (~)ll • IIF~(~)[I d~ + I lIB ( ~ ) [ I . [ [ H ~ ( ~ ) I I d~ , x0 R1 • X0 X ~ Now, the Gronwall inequality implies
80 X
x
X
o
x~
R1
But
~v (x) - Xv (x) I -< I I ~ (x) I I ,
x
e R1
and '
'
~)
~)
x) '
while
m%(~)-%(~)im -< Is
I / (coj) TM
,
~ ¢ [-co~ , oJ~-I u _l-co.o-~o".~ , co +¢o~]
Further IB (n)[l -< max { 1 , [ k ] + ] s [ / (co) m } , n ~ R 1 • Therefore,
one obtains
(41)
[~b~(x)-x~(x)l
< 2f~'~0"] . . ~~"
Is] • ~ .
exp(2(~+m~)(l+[k[+[s[/(~
where
x c R1 %
For given
~
= max { I[F~(~)Jl
v c N
decreasing in and
)m))/(co~) TM
> 0
0~'
F~
I ~ ~ [-co~ , ~x~] u [w .-co~ , co -~.o~'-] }
depends only on
and
co
~0!
and not on
or
~"
Therefore,
m"~ • That fact, together with (41) imply that for given
the function
ximated by the function
~
can be arbitrarily and in a uniform way on
Xv • provided
m'~
and
m~"
RI
~
~ ~ N appro-
are chosen small enough. Taking
into account Theorem i in §3, the proof is completed
~TV
Lemma i There exist functions
1
8,y ~ C+(R )
satisfying
(34) and (35) respectively.
Proof Define
n c C+(R I)
n(x)
Assume
by [
0
if
x -< 0
I exp
(-l/x)
if
x > 0
0 < a , b < 1
and define
~i ' 62 ~ C~(RI)
by
BI(X ) = n(x+l) / (TI(x+I) + n(-x-a)) and ~2(x) = Tl(l-x) / (n(l-x) + n(x-b)) , for x e R 1 . Defining 8 e C°°(RI) by (see Fig. 3) B(x) = (~l(X) exp x-l) ~2(x) + i , for
x c R1 ,
~ will satisfy (34) with
y(x) = ~(l-x) / (Q(l-x) + ~(x+l))
, for
M = e . Defining x ~ RI ,
y e C~(R I)
y will satisfy (35)
by V~7
is
81
rt 1 1
•,
I
-1
-ct
,
02 1
~x
I
-1
b
1
b
1
ft e
T
I
-1
-Q
Fig. 3
X
82
§5.
WAVE FUNCTION SOLUTIONS IN THE ALGEBRAS CONTAINING THE DISTRIBUTIONS
It is shown in this section that, given any
(m,~) ~ M , the weak solution ~ of (i),
(2) obtained in §§3, 4 is a solution of (i), (2) in a usual algebraic sense, considered in certain algebras containing
D'(RI), with the multiplication,
and positive powers defined in the algebras.
derivatives
Therefore, the wave function solution
obtained is independent of the particular representations
used for the Dirac $ di-
stribution.
Theorem 4 Suppose given (i), (2) with
(m,a) ~ M
and let ~ be the weak solution of (i),
(2) constructed in §§3, 4. Suppose ~ is not smooth, that is, ~ or ~t is not continuous in there exist regularizations sible
property
(V,S')
0 c R 1 . Then,
(see chap. i, §7) such that for any admis-
Q , one obtains
1)
~ c AQ(v,s ' , p )
2)
i n the case o f d e r i v a t i v e and p o s i t i v e power algebras (see chap. i , satisfies
,
V
p c §7)
(1) i n the usual a l g e b r a i c sense i n each of the algebras
AQ(v,S',p), p c N , with the respective multiplication,
power and deriva-
tives. Moreover,
there exist
s ~ SO
not depending on
~ = <s,'> = s + IQ(V(p),S ') ~ AQ(v,S',p)
3)
Q ,
or V
p , such that p ~ N •
Proof Since xO se
< 0 (~
(m,~) E I# , there exists and
Yo ' Yl c C 1
I ~ c N)
smooth functions
and
~v
Iv ~ ~
satisfying
then according to Theorem 3 in §4, it is possible to choo
•f~"w I v c N)
(Xv I ~ c ~
satisfying
(36) and so that the sequence of
resulting from (39) and (40) will converge in
to ~ given through (3), (13), (14). Therefore, defining W ~ ~ N , one obtains (42)
s c S
o'
<s,.> =
and, due to (39), (40), the relation
(43)
D2s + (k-~(s6)m)s = u(0) ~ O
and (44)
/
Ds (~) (Xo)
(8) and (12). Assume given
=
lye1 Yl
,
V
v
~ N
.
s ~ W
by
D'(R I)
s(V) = Xv ,
83
The idea of the proof is to show that (43) is valid in the algebras
(V,S')
(45)
satisfies the condition:
s,s~ e V(p) Q S '
Indeed,
s c V (p)QS'
(46)
,
V
, V p e N
p e N
(see (22) in chap. i, §7)
and (42) imply (see (25) in chap. I, §8) that
~ = s + IQ(v(p),S ') c AQ(v,S',p)
In the same time, (47)
, with
(V,S') . In this respect it suffices that the regula-
suitably chosen regularizations rization
AQ(v,S',p)
s~ e V ( p ) Q S '
, U p e ~
,
V
p ~
and (38) result in
~ = s~ + IQ(V(p),S ') ~ AQ(v,S',p)
,
V
p ~ N .
~bw, (46), (47), (43), (38) and Theorems 3, 4 in chap. i, §8, will obviously imply 2). Further, i) and 3) will result from (46). Therefore, it only remains to obtain regularizations
(V,S')
which fulfil (45).
We shall use the method given in Theorem i, chap. 2, §3. Take
J c W
(48)
such that for each supp w(V)
w c J
either is void for {0} c R 1
[49)
w(w)(0) = 0 ,
when
for
w ~ N
and denote by
I1
Denote by
the vector subspace in
T1
, the relation holds
the ideal in
W
v ¢ N big enough or shrinks to
v + big enough
generated by S
J
generated by O
{s~ , Ds~ , D2s~ . . . . We prove that
II
I 1 n S O c VoQT thus
and
TI
are compatible. Obviously
1 . Indeed, assume
t ~ VoQT
} V n T = 0 • Further, o 1 t c I 1 n S O , then (48) implies supp c {0} ,
1 , taking into account (38). Finally, we prove that
Assume indeed
t ~ I1 n T1
(50)
t
=
with
p c N ,
I1 n T1 = 0 .
then
Z k i D1s~ o~i~p
~. ~ C 1 . Now, according to (49) and (37), the above relation (50) imi
plies
0-- t(v)(0)
--
z x i DiB(0) / w ~)(w,) ~) i o_
•
for
w ~ N
b i g enough
which due to (36) and ****) in (34) results in o
= ... = ~
Now, (50) will give follows that (VoQTI)
I1
and
p
=0
t E O . Recalling the conditions T1
are compatible. Obviously,
n U = 0 . Moreover,
according to (42).
.
s ~ VoGTIQU
(4) and (5) in chap. 2, 53, it s ~ VoQT
1
and
, since ¢ is not smooth and
<s,.> =
84
Assume
Z ~ I 1 , with
such that
I
and
vector subspaces Sume finally ply that
V
(V,S')
(45), since
Z
T
ideal in
W
are compatible
S1
in
with
S' =
and
in
TQS 1
s~ ~ T 1
c
T ~ T_~, with
vector subspace
in
So ,
It follows that there exist and
s ~ S 1 , U c S1
. As-
Z n V ° . Now, Theorem 1 in chap. 2, §3 will imis a regularization.
T
T
s ~ VoOTQU__ . VoQTQS 1 = So
and
S o , such that
a vector subspace
s c SI
and
Obviously,
(V,S')
satisfies
WV
Remark 2 i)
The regularizations
(V,S')
rather simple, constructive
whose existence way. Obviously,
is obtained
in Theorem 4 result in a
the algebras
AQ(v,S',p)
are Dirac
algebras. 2)
(sl)
The smooth representation
DPs6Cv)(0) # 0 , which was e s s e n t i a l l y particular
of 6 given by
V ~ c N ,
s6 E S o
p c N
needed i n t h e p r o o f o f Theorem 4. That p r o p e r t y i m p l i e s i n
t h a t n_~osymmetric r e p r e s e n t a t i o n
In chapters 5 and 6, a generalization will be used for defining
in (37) has the property
o f 6 can be u s e d .
of the relation
(Sl) to the n-dimensional
important classes of Dirac algebras.
case
Chapter
S
PRODUCTS WITH DIRAC DISTRIBUTIONS
§I.
INTRODUCTION
A class of relations containing products with Dirac distributions encountered in the theory of distributions is given in (X-xo)r ° Dq6 x
(i)
= 0 ,
V
x ° ¢ Rn ,
q,r ~ ~I ,
r $ q
o where
6
is the Dirac ~ distribution concentrated in x o o The importance of the relations (i) is due to the fact that they give an upper bound x
of the order of singularities the Dirac distributions and their derivatives exhibit. It is worthwhile mentioning the role played by relations of type (i) in the way derivative operators may or may not be defined on the algebras containing the distributions (see chap. i, §8, as well as Remark D in §7). As a first result, Theorem I, §4, will establish relations of type (i), within a wide class of Dirac algebras constructed in the present chapter.
In §4, several other types
of relations involving products with Dirac distributions will be proved valid within the mentioned algebras. A second result is obtained in Theorem 6, §5, where known formulas in Quantum Mechanics, involving irregular products with Dirac and Heisenberg distributions are proved to be valid within the Dirac algebras constructed in the present chapter. These algebras are obtained according to the general procedure given in Theorem i, chap. 2, §3, which presumes the existence of compatible pairs ideal in
W
and
T
is a vector subspace in
Z,T
where
I
is an
S o . The construction of such compatible
pairs is given in §§2 and 3. Here the main problem is the construction in §3 of suitable vector subspaces T , whose existence is based on a rather sophisticated algebraic argument involving generalized Vandermonde determinants. The present form of the respective conjecture in Theorem 8, §7, as well as its proof was offered by R.C. King. A third result is presented in §8. For a subclass of the Dirac algebras constructed in §§2-4, a stronger version of the relations with products given in Theorem i in §4, involving this time infinite sums of Dirac distribution derivatives,
is obtained.
86
§2.
THE DIRAC IDEAL
Denote by x
o
~ Rn
I6
I~
the set of all sequences of smooth functions
w c W
which for any
satisfy the condition
(2)
w(~)(Xo) = 0 ,
for
W c N big enough,
and for a certain neighbourhood (3)
V n supp w(~)
V
of
x o , the condition
either is void for or shrinks to
~ e N big enough
{x }
when
9 ÷ ~
O
Proposition i is a Dirac ideal (see chap. 2, §6)
Proof A direct check of the conditions
(13) and (26) in chap. 2, will end the proof. However,
it will be useful to give a second proof, showing that where
G = FF~
and
§§2,4).
F~
is acutally an ideal Rn
ZG, °
(see chap. 2,
Then, Proposition 6 in chap. 2, §6 will complete the proof.
~bw, the singularity generator with
Z6
is a certain singularity generator on
F6
is chosen as the set of mappings
Yx
x ° ~ Rn , defined by
: Rn ÷ R1 ' o
Yx (x) = ([[X-Xo[[)2
x ¢ Rn
O
where
[[
[[
is the Euclidean norm. Then, obviously
Fyx
= {x o} , for
xo c Rn .
O
Now, the relation tion 1
§3.
18 = IG, o
will follow easily, ending the second proof of Proposi-
WV
COMPATIBLE DIRAC CLASSES
Tx
First, several auxiliary notions. For
m c N denote P(n,m) = { p = (Pl . . . .
and by
g(n,m)
, pn ) c Nn [ tp[ = Pl + "'" + Pn
the number of elements in
P(n,m)
One can see that there exists a linear order Nn = { p ( 1 ) p(1) and
P(n,m)
--[
, p(2)
p(2) = { p(1)
--[
....
}
.
--[ on
Nn
such that
,
.... ,
...
, p(.%(n,m))
} ,
V
m ¢ N .
-<m}
87 It is easy to notice %(l,m)
that
=
m
+
Z(n,m) 1
,
can be obtained
V
m
E
N
from the recursive
relations
,
and Z(n+l,m)
=
Z
i(n,k)
,
V
m 6 N .
o
To any s e q u e n c e o f smooth f u n c t i o n s rix of smooth functions on
will
be associated
Wronskian type infinite
with the help of the above linear
mat-
order --I
Nn "-
W(w)
Denote by
24
(x)
:
DP (1)w(O) (x) . . . . . . . .
DP(P)w (0) (x) . . . . . . . .
DP(1)w(V) (x) ........
DP(V)w(V) (x) ........
the set of all infinite
with a finite number of nonzero An infinite singular,
matrix
vectors
components
of complex numbers
of complex numbers
~'
)
x
~R
n
A = (IN I V ~ N)
Xp .
A = (aop
[ v,p ~ N)
is called column wise non-
only if Y
A ~ M
:
And now,
the definition
functions
representing
of all weakly
AA ~ I'4~ A = 0 of an important
class of weakly convergent
Dirac 6 distributions.
convergent
sequences
Given
sequences
x ~ R n , denote by
of smooth functions
of smooth
Zx
s c S o , satisfying
the set the condi-
tions (4)
<
s
,
"
>
=
(5)
supp s(~))
(6)
W(s) (x)
x
shrinks
to
of sequences
The condition
(6) can be called,
x ° , due to its meaning case
, when
~) ÷ oo •
is column wise nonsingular.
The existence
dimensional
{x}
n = 1
s e Zx
will be proved
[1283,
strong local presence
in the following and
in §7.
~ e D(R I)
particular
case.
such that
~i ~(x)dx
of the sequence
Suppose,
s
in
we are in the one
= 1 . Define
s@ e W
by (7)
s~(9)(x)
Then, s~)
s~
obviously
will satisfy
= (v+l)~(x/(~+l)) satisfies
(6), only if
,
V
~ ~ N ,
(4) and (5), with
x ~ RI
x ° = 0 £ R I . Now, one can see that
88
(8)
DP@(0) $ 0 ,
in particular,
V
p • N ,
@ must be nonsymmetric
(see
(37)
and ****) in
(34)
in chap.
4)
Denote g6 and for
= ~
gx
~ = (s x [ x • R n) ¢ Z~
denote by
T~
the vector subspace in
S
generated o
by
{DPs x [ x e Rn , p E Nn} .
Proposition 2 For each al
I~
Z e Z6 ,
TZ
is a Dirac class which is compatible with the Dirac ide
(see chap. 2, §§3,6).
Proof First, we prove that
21
is a Dirac class, that is, it satisfies
(24) in chap. 2. Indeed, the conditions
(17.1),
(17.2) and
(17.1) and (24) result easily. Assume now
t E T~ , t ~ 0 , then (9)
t
=
Z xeX
Z 1 Dqs q~N n xq x q-
where (10)
X c Rn , X ~
finite, nonvoid,
x ° eX,
qoeN
Px c Nn
n
qo ~ Px xo
(11)
l) -> II
:
V
~) ¢ N ,
:
o
We show that (17.2) is satisfied for ~ ¢ N
and
lxq e C 1 . Moreover 1
Xoqo
~
0
given in (i0). Assume it is false and :
t(~)) (Xo) = 0
then (9) gives (12)
X xeX
Z qeN n
~
Dqsx(V)(x o) = 0
xq
'
V
V ~ N
'
v ~
q~Px But, due to (5) and the fact that
X
is finite, one can take ~ such that (12) imp-
lies (13)
E qeN n q~Px
X
Dqs Xoq
I'
vector IXoP(~+l)
=
0 then, obviously
V
v E N ,
v ->
o
D e f i n e now t h e i n f i n i t e
(14)
(v)__ (o) x = 0 , xo
A c M
of complex numbers if
p(~+l)
_< Px
otherwise
and (13) is equivalent to
A = ()t~ I IJ e N)
o
where
89
W(sx ) C x o )
A ~ M
o
Therefore,
(6) w i l l imply
proof of (17.2)
A = 0
which through
It remains to prove that
TE
and
16
(S) in chap. 2, ~3, are satisfied. rac ideal.
(15)
(14) will contradict
(i0), ending the
i n c h a p . 2.
In order to prove
are compatible,
The relation
that is, the relations
(5) follows easily since
I~
(4) and is a Di-
(4) it suffices to show that
I ~ n TE = O
since
V O n T Z = 0 , as it was noticed
then (9) and (i0) hold again, since thus the reasoning
§4.
above. Assume therefore
t E TZ . But, t E 16
above, contradicting
t E I~ n TZ ,
t % O ,
and (2) will again give (ii)
(I0) will end the proof of (IS)
W?
PRODUCTS WITH DIRAC DISTRIBUTIONS
Based on the compatibility
of the Dirac ideal
~ Z 6 , we shall follow the procedure
I~
with the Dirac classes
in Theorem i, chap.
TE ,
for
2, §3, and construct the Di-
rac algebras used in the present chapter. Suppose given
E e Z~ .
For any ideal
I
if
V
in
W , I = I~
is a vector subspace in
and compatible vector subspace
I n V°
and
SI
T
in
is a vector subspace
in
SO , T D T E , SO
such
that
(16)
Vo Q T Q S
(i7)
u~ VCp)(2)~Qs I
then
(V,TQSI)
perty
Q
= So , v
,
p
is a regularization,
~ ~
,
therefore one can define for any admissible pro-
the Dirac algebras
(18)
AQ(v,T(~)S I _
, p) ,
p
~n
which for the sake of simplicity will be denoted within the present chapter by with
Ap ,
p e ~n .
Properties
of type (i) concerning products with Dirac distributions
are given in:
Theorem 1 In case
I6nV
(19)
o
any q - t h o r d e r x
o
i)
~ Rn
cV (q E Nn)
derivative
has the properties:
Dq6
I X0
0 E A
V P "
p c Nn
of the Dirac delta distribution Dq~ xo
in
90
@ ( x - x o)
2)
for (20)
• Dq6xo = 0 c Ap ,
every
Dry(0)
@ ¢ C~(R n)
= 0 ,
¥
V
P < ~n ,
which satisfies
r ¢ A~ ,
r ~ p
or
r ~ q
in particulax 3)
(X-xo)r
• Dq6xo = 0 ¢ A p
,
V
p ¢ Nn ,
r ¢ Nn ,
r $ p
and
r ~ q
Proof
l)
Assume
E = (s x I x c R n) . According to 3) in Theorem 2, chap. i, §8, the relation
holds Dq6xo therefore,
: Dqs x
, 2C)SI)
+ IQ(V(p)
o
Dq6xo = 0 e ~
, only if
DqSxo E T Z . Thus
Dq6 x° = 0 ~ Ap
and
Dqs
the relation
( 0
X
E Ap
Dqs Xo e IQ(V(p),TQSI)
implies
. But, obviously
DqSxoe T Z n I Q ( v ( p ) , T Q S I )
is absurd due to
c T n I = 0
(4).
O
2)
According again to 3) in Theorem 2, chap. i, §8, the relation holds
(21)
~(X-Xo)
• Dq6 x
: ~(X-Xo)
• Dqsxo + I Q ( v ( p ) , T Q S I) E
o Define
v e W
by
v(~)(x) = ~(X-Xo)
• Dqs x (~)(x)
,
¥
~ ~ N ,
x E Rn ,
O
then (21) becomes (22)
~(X-Xo)
• Dq6 x
= v + IQ(V(p),TQSI)
e Ap
O
We shall (23)
prove that v c IQ(V(p),TQS1)
and then, due to (22), the proof of 2) in Theorem i will be completed. First we notice that
v c S
since
~ ~ C~(R n)
and (4). Actually,
v c V
O
(20). (24)
since
Dry ~ V O
Assume now
in
V
r c Nn
'
r ~ N n , r < p , then (20) and (S) will imply that
Dry e i 6
gether with (24) and (19) will give Dry c 16 n V o c V That relation implies 3)
r < q
O
Thus
Y '
,
v c V(p)
It results from 2) choosing
r c Nn
r~p
and the proof of (23) is completed.
~(x) = x r , V x e R n
VW
which to-
91
Remark 1 The relations
in 3) in Theorem I, describing
ing the distributions are identical
between
with the usual
er, even in that case, For instance,
formulas
the relations
p E N
and
x
(X_xo)P+l . ~x
(I) within
case
(X-xo)q+l"
The nontriviality Theorem
D'(R n)
the algebras
contain-
of the Dirac ~ distribution , except when
in the algebras
r ~ p . Howev-
are also valid
n = ] , the relations
in
D'(Rn).
in 3) in Theorem
I,
e R1 :
o
= (X_xo)P+l ~ D~x o
(26)
the p r o d u c t s w i t h i n
and derivatives
proved
in the one dimensional
imply for any (25)
polynomials
= "'" = .[X-XoJ.p+l " DP-16 x o
Dq6xo = 0 e Ap ,
of the powers
V
= 0 e Ap ' o
q e N ,
of Dirac distribution
q a p
derivatives
is given in:
2
In case
(27)
V c I~ n V the relations
o
hold
(Dq~ x )k ~ 0 e Ap , o
V
p e ~n ,
x ° c Rn ,
q ~ Nn ,
k ~ N ,
k
1
Proof X = (s x I x ¢ R n)
As sume
. According
to 3) in Theorem
2, chap.
I, §8, the relation
holds (Dq~ x )k = (Dqs x )k + I Q ( v ( p ) , T Q S 1 ) o
therefore,
defining
v ~ IQ(v(p),TQS
Assume, i t
v ~ Y
by
is false,
t h e n (27) and (2) imply f o r a c e r t a i n V
which due to the definition
of
Dqs x (V)(Xo) o that relation
same point of Theorem
Rn
is valid,
only if
o
= 0 ,
The nontriviality
v = (Dqs x )k , the theorem
1) .
v(~)(Xo)
However,
c Ap
o
= 0 ,
obviously
of the product
~ ~ N ,
v c N ,
contradicts
derivatives
concentrated
in
v ~ p (6)
o
~7V
of Dirac distribution
can be obtained
the relation
V ~ p
v , will result V
P ~ N
in special
in the
cases:
3
In case
(271 in Theorem 2 is valid,
there exist
Z ~ Z~
such that in the corre-
92
sponding algebras Dq°~ Xo
Ap , p • A n , the relations hold " Dqk6x o # 0 • Ap ,
"'"
V
p e ~n qo
'
, Xo •
"'"
' qk
Rn , k c N , ¢ Nn
Proof Assume
E = (s x I x • Rn)
with
to 3) in Theorem 2, chap. (28)
Dq°8
Define
v • W
(29)
v
given in the proof of Corollary I, §7. According
i, §8, the relation holds • Dqk8
...
x
sx
o
x
= Dqos o
x
...
x
+ IQ(v~),r + s 1) • Ap o
by = Dq°s
x
•
...
Dqks
o
x
o
then, due to (28), the theorem holds only if v c IQ(V(p),TQSI)
, then
v ( v ) ( x o)
Now,
Dqks
o
= 0
v ~ IQ(v(p),TQSI)
(27) and (2) imply for a certain
,
v
v
• N
,
.
~ c N
Assume the relation
v ~
(29) gives Dq°sx (~)(x o) " ... o
which implies the existence of nitely many values of However,
Dqksx (~)(x o) = 0 , o 0 ~ i ~ k
such that
V Dqisx
~ • N .
that contradicts
~ ~
(v) (Xo)
vanishes for infi
o
the fact that
in the proof of Corollary i, §7
V • N ,
DP~(0)
= 1 / K > 0 ,
¥ p c N n , established
VV
An expected property of the product of two Dirac distribution derivatives in different points in
concentrated
R n , is given in:
Theorem 4 In case (19) in Theorem 1 is valid, one obtains the D q6
x
" Dr6
y
= 0 c A
p '
V
p c An
'
x,y • Rn
relations ,
x ~ y ,
q,r c N n
Proof Assume
E = (s x I x c R n) . According to 3) in Theorem 2, chap. i, §8, the relation
holds (30) Denoting
Dq~ x • Dr~y = Dqs x • Drsy + v = Dqs x • Drs Y , the relation
Dhv • I ~ n V o ,
¥
(30) will end the proof
h ~ N n. Therefore ?V?
IQ(v(p),TQSI) c (5) together with v ~ V~)
Ap x # y
c IQ(v(p),TQSI]
implies and the relation
93
In the case of derivative algebras,
the relations
and (26) in Remark 1 - will be supplemented the one dimensional
case
n = 1
in 3) in Theorem 1 - see also (25)
in Theorem 5. For the sake of simplicity,
is considered only.
Theorem 5 In case (19) in Theorem 1 is valid, one obtains the relations (X_Xo) q . (Dq6 x )k = 0 c A , o P
V
p ~ N ,
x
c R1 ,
q ~ N ,
o q e p+l ,
k ~ N ,
k e 2
Proof Assume
E = (s x I x e R I) . According to i) and 4) in Theorem 3, chap.
i, §8, the rela-
tion 1 q+l Dp+l((X_Xo ) . (Dr~x)k) o
(31)
= (q+l)(X_xo)q
• (Dr~x)k o
+
. .q+l • k • (Dr6x )k-i + LX-Xo)
Dr+16x
O
holds in
Ap , for any
(32) Therefore,
q,r,k c N . Now, due to 3) in Theorem i, one obtains •
(X-xo)q+l
= 0
Dr~xo
(31) and (32) imply in
6 AP
Ap
,
V
q,r
c
N ,
q ~ p
,
q ~ r
the relation
1 q+l Dp+l((X-X°) " (Dr~xo)k)
(33)
O
= (q+l(X_Xo) q . (Dr~xo)k V
q,r,k c N ,
But, the product in the left side of (33) is computed in
,
q -> p , Ap+ 1
q >- r ,
k -> 2
and according to 3) in
Theorem i, one obtains (34)
(X-xo)q+l
• Dr6x
= 0 c A o
Taking
r = q , the relations
,
V
q,r c N ,
q e p+l
,
q e r
p+l (33) and (34) will end the proof
WV
An example for the application of Theorem 1 is given in: Proposition
3
The Riccati differential y,
x q+r
y
has in the algebras
equation
(y+l) + Dr+l~(x)
with
q,r c N
q_>l
,
Ap , p ~ r , the general solution
y(x) = 1 / (c exp (-x q+r+l / (q+r+l))-l) provided that
x c RI
(19) is fulfiled.
+ Dr~(x)
,
x c R1 ,
c
~
(-p%0]
,
94
Proof Assume
c • R1
and define
~(x) then
~ ~ C ~ ( R I)
by
= c exp (-xq+r+i/(q+r+l))
1 / ~ • C ~ ( R I) , for
c c (-~,0]
- 1 ,
. Therefore
T = 1 / ~ + Dr6 E D ' ( R I) c Ap , Since
~
, with
p e N • are associative,
1 • C ~ ( R I) , one obtains
x • R1 ,
V
p c N ,
commutative
in these algebras
c ~ (-~,0]
.
and with the unit element
the relation
x q+r • T • (T+I) = x q+r • (Dr6) 2 + 2 x q+r • (Dr6)
• (I/~) +
+ x q+r " ---[i/~)2 + x q+r " --(Drd) + x q+r " (I/~) provided
that
c e (-~,0]
. But, due to 3) in T h e o r e m i, one obtains
in
A
, with P
p ~ r , the relation x q+r • Dr6 = 0 • A P since
q + r > r , as
q ~ 1 . Therefore,
one obtains
in the algebras
Ap , with
p ~ r , the relation x q+r • T • (T+I) = x q+r • (i/~) w h i c h means that
§5.
FORMULAS
T
• (i/~+i)
is a solution of the considered
,
V
c • (-~,0]
Riccati equation
, VVV
IN Q U A N T U M MECHANICS
In the one dimensional
case
n = 1 , the Dirac 6 d i s t r i b u t i o n
and the Heisenberg dist
ributions
6+
=
(~ + ( l / x )
6
=
Ca - (l/x) / ~ i )
satisfy the formulas,
[108],
/ "rri) /
2
/ 2
given in:
Theorem 6 There exist such
Z ~ Z6
that within
and regularizations
the corresponding
(V,TQSI)
algebras
lid: (35)
(6)2 _ (l/x) 2 / 2
= _(i/x 2) / z2
(36)
(6+) 2 = -D6 / 4~i - (I/x 2] / 4z 2
(37)
(6_) 2 =
(38)
6 • (l/x) = -D6 [ 2
D~ / 4~i - (i/x 2) / 4~ 2
(see the beginning
Ap , p e N , the relations
of §4) are va-
95
Proof Assume belong
'~ ~ D ( R I) to
Z
Denoting
with
= i
and satisfying
therefore,
, for
s~
a certain
is a '6 sequence'
Pt
and
o
•
g i v e n b y (7) w i l l
q e N , and assuming that
,
Lq
such that
V
q,~
~
N
,
x
E
R1
,
s o = s~ . Define the sequence of smooth func-
t(~) = s~(w) * (i/x)
, with
w ~ N .
Then, obviously
< t , " > = i/x
the vector subspace in
(Vo®UQT
Z) n h :
SO
generated by
{ Dqt I q ~ N } , then
o
according to Lemmas I and 2, below. Therefore, o
s~
according to [106], [108].
by the convolutions t ~ S
S
then
L > 0 , one obtains
Dqs~ (W) (x) I <- Mq
•
I = (sx I x c R 1) e Z 6 t £ W
Denote by
(8),
o
Mq = s u p { [Dq~(x) l [ x ~ R 1 } , f o r
[ x q+l
tions
~(x)dx
RI
supp ~ c [-L,L]
Assume
f
one can choose a vector subspace
SI
in
which satisfies
(39)
v° Q T ~ Q s
(40)
UQp t c S I
Taking now
T = T Z , the relations
(V,T + $1)
will be a regularization.
Since
s$ ( T l = T
I : so
and
(59) and (40) will imply (16) and (17), therefore
t ~ Pt c S I , one obtains according to 3) in Theorem 2, chap.
I, §8, the relations (41)
6+ = (s~+t/~i)
/ 2 + IQ(v(p),TQS
I) ~ Ap
(42)
6_ = (s~-t/~i)
/ 2 + IQ(v(p),TGS
I) ~ Ap
(43)
6 • (l/x) = s~ • t + I Q ( v ( p ) , T Q S I )
Define the sequences (44)
of smooth functions
t I , t2 , t 3 c W
by
t I = (s~+t/wi) 2 , t 2 = (s~-t/~ri) 2 , t 3 = s~ • t
It was proved in [108] that (45)
< t I
(46)
< t2 , " > =
(47)
< t3
The relations
tI , t2 , t3 ( S o
, " > = -D6 / z i
-
( 1 / x 2)
/
and
72
D6 / ~i - (i/x 2) / ~2
" > = -D6 / 2
(41-43) and (45-47) will give through
38). It only remains to prove algebra
~ Ap
Ap , with
(44), the required relations
(36-
(35). From the definition of 6+ it follows that in each
p E N , the relation holds
(6+) 2 = (@+(i/x)/~i) 2 / 4 = (6) 2 / 4 - (i/x) 2 / 4~ 2 + 6 • (l/x) / 2~i
96
which compared with (36) and (38) will give (35)
And now, the two lemmas concerning distributions
WV
in
D'(R I)
needed in the proof of
Theorem 6. Denote by set of
D~(R I) the set of all distributions
S6
R 1 . Denote by
tiens
s e So
in
D'(R I)
with support a finite sub-
the set of all weakly convergent
which generate distributions
D~(R I)
the set of all distributions
certain
q ~ N , %r ¢ C 1 , hq ~ 0 .
<s,.>
in
T ¢ D'(R I)
sequences of smooth func
D~(R I) . Finally, denote by
such that
E
%rDrT ¢ D~(R I)
for
o~r~q
Lemma i For
t ¢ S°
Pt
denote by
the vector subspace in
S
generated by O
{ Dqt
] q ¢ N } . If
t
~ 0 • then
(UQS~)
n Pt = O ¢=~ ~ C~(R I) + D~(R I)
Proof The implication
~. Assume,
it is false and let
s
( U Q S 6) n Pt
¢
be such that
t i O . Then (48)
)2
s = u(~) + t I =
% Drt r
o_
~ ¢ C°°(RI) , t I c S~ , q ¢ N , h r ~ C 1 , %q $ 0
for certain Denote
P(D) =
(49)
Z X D r . Let r o<_r~_q
X ¢ C~(RI)
t 2 = t - u(x) ¢ S o ,
since
P(D)t 2 = t I , due to
(48), while
with
>=~+
~ e C~(R I)
and
= P(D)~ + <s,'> P(D)t e
UQS~
the hypothesis.
, hence
.
. But (49) implies the hypothesis.
¢ C=(R I) + D~(R I) . "Cnen
>
such that
s = P(D)t I e $6
P(D)t = u(~) + s + v , for certain
Now, if
P(D)t ~ O
then
On the other side, if
= 0 ¢ D'(R I) , hence obtains finally
t I ¢ S~
contradicting
it is false and
I ,.
tI ¢ S o
(UQSd)
P(D)t ~ O
¢ C~(R I) , therefore
(UQS~) n
Pt D t 4 0
u
~
then
v e V
O
t c UQS6
m -> 1 .
, since
=
. It follows that contradicting
P(D) = =
again contradicting
m ¢ N ,
!~erefore,
n Pt D P(D)t ~ O
Lemma 2 (i/x m) ~ C~(R I) + D'(R I) ,
P(D)x = ~ . Then
< t 2 , • > ~ D'(R I)
= X + ~ C°°(R I) + D O'(R I) Now, the implication ~ . Assume,
be such that
7/0 c S~ . One
the hypothesis
WV
97
Proof Assume, it is false. Then
(50)
i/x m = @ + T
for certain (51)
tp ~ C°~(R1) and T t
for certain
T e D' (R1)
Hence
0
=X
RI\ { 0}
X e C°°(RI\{0})
. But, a c c o r d i n g t o the d e f i n i t i o n
of
D ' (RI)
it follows
that
(52)
S
=
~
%tOrT ¢ D~(R 1)
o~r~-q for certain
q c N
S1
(53)
,
~r ~ C1
S ] I RI\{0}
=
% q ~ 0 . Then.
,
¢ D~(RI\{0})
Now, (51-53) imply
(54)
S 1 = P(D)x e C#°(RI\{0})
where
P(D) =
SI = 0
E X D r . As o~r~q r
6¢~ n D~ = {0}
the relations (53)
(54) result in
~
which together with (50-52) gives
P(D)(1/x TM) = P(D)~
on
RI\{0}
Computing the derivative in the left side, one obtains
Z ( - 1 ) r (m+r-l)! ( m - l ~ X x q - r = xm+q P(D)~(x) ! r o~r~q Taking the l i m i t f o r (-1)q
x + 0 , one o b t a i n s
(m+q-l)!
(m-l):
X
q
=
0
contradicting the assumption that
§6.
g x ¢ RI\{0}
l
q
# 0
VVV
A PROPERTY OF THE DERIVATIVE IN THE ALGEBRAS
In the present section, the case of derivative algebras containing
D'(R I)
will be con
sidered. According to the general result in I) in Theorem 3, chap. 1 , §8, the derivative mappings within the algebras Dqp+q : Ap+q ÷ Ap , coincide on
C~(R I)
p ~ N ,
q e N
with the usual derivatives
will be strengthened in Theorem 7.
Dq of smooth functions. That result
98
T Z c T , the same I) in Theorem 3, chap. i,
First, we notice that due to the inclusion
§ 8, implies that the derivative mappings within the algebras coincide on C ~ ( R I ) Q D ~ ( R I)
with the usual distribution derivatives.
Theorem 7 Given any distribution
(V,TQS1) tive
T • D ' ( R 1) \
such that
within
mappings coincide on
rivatives, where
MT
oo 1 (C ( R ) + D ~ ( R 1 ) )
the corresponding
algebras
C~(R I) + D~ (RI) + M T
is the vector subspace in
{ T , DT , D2T . . . .
there
exist
regularizations
Ap , p ~ N , t h e d e r i v a -
with the usual distribution deD' (RI)
generated by
} .
Proof Assume
(UQS~)
T =
for
a certain
t e S
n P t = 0 . But, obviously
. Then, according
o S~ =
VoQT z , for
Z ~ g~ , one can choose a vector subspace
(55)
VoQ~ZQs
(s6)
uGp t c sI
Taking
in
So
§5,
Z ¢ Z~ • Therefore, given
any
such that
I = so
T : T~ , the relations
(V,TQSI)
SI
t o Lemma l ,
(55), (56) will imply (16) and (17), therefore
will be a regularization. Noticing that
HT:{<s,'>
I s ~pt }
and taking into account i) in Theorem 3, chap. i, §8, the proof is completed
§7.
THE EXISTENCE OF THE SEQUENCES IN
In order
to prove that z
x
*
~
(see ,
Z
s
v
o
belonging to
Z°
o
§3) x
o
~R n
,
it is obviously sufficient to show that ces
Z
~ ~ . In this respect, a class of sequeno will be constructed by a proper generalization to n > 1
dimensions of the method in (7) and (8). Suppose
(57)
~ e D(R n)
such that
s~(V) (x) = UI(V)
~n ~(x)dx = I
" ...
" ~n(~)
and define
• ~(l~l(u)x I V
N ~
÷
U(~) = ( U I 0 ) ) , - - . , U n ( U ) )
so • W
.....
by
Un(U)x n)
,
V c N , x = (x I ,..., x n) ~ Rn
where the mapping (58)
~V
c Nn
99
is constructed
in (59-62).
First,
N ~ V ÷ k(~) e N ,
define
(59)
k(0)
Define
also
= 0
N By
and
÷ h(v)
k(v+l)
h(0)
D e f i n e now
N ~ ~ ÷ e(v) E N n , by
(61)
e(V)
Finally,
define
and
(62.2)
{ p(k(v)+l)
= (1,...,1)
= h(v)
+ ~ + 1 ,
~
V
9 E N .
,
V
~ E N .
V e N .
V
c Nn
....
, #(k(V+l))
can be written
,
} = P(n,v+l)
in Fig.
in terms
M4 = { p ( k ( 3 ) + l )
Lemma
+ £(n,v+l)
,h(v)))
(58) is illustrated
M4
= k(v)
(58), by
~(0)
The mapping
h(v+l)
= (h(v) ....
(62.1)
noted by
(see §3):
E N , by
(60)
= 0
by
...
+ e(v+l)
4) in the case of
of (62.2),
, p(k(4))
,
V
v e N .
n = 2 . There,
the set de-
as
} = P(2,4)
+ e(4)
3 s~
satisfies
(4) and (5).
Proof It follows
from the fact that lim pi(~__ = + k~,oo
The basic property Proposition
l_
of the sequences
s~
VVV
defined
in (57-62)
is given in:
4
The following
three conditions
*)
s~ ~ Z °
**)
W(s~)(0)
***)
V
J
DP~(o)
are equivalent:
is column wise nonsingular
~ 0 ,
v
p ~ Nn
(see(8))
Proof Taking into account
the definition
and **) are obviously **) and ***).
equivalent.
First) we compute
Dqs~(v)(0)
of Z in §3 as well as Lemma 3, the conditions *) o It only remains to establish the equivalence between
W(s~) (0) . The relation
= (p(~))q+e
D%(0)
,
V
q ¢ Nn
(57) will give easily j
~ ¢ N
)
100
;
\ \
\ \
i,
P
,
;
/
\\
q
F
iIy
•V
•
10
Fig./.,.
15
M4
101
where
e = (1,...,1)
~ Nn
W(s~) (0)
(63)
. Therefore
: A • B
where A
while
B
=
(
(1J(~>))P(CY)+e I x;,cY c N )
is a diagonal
(64)
DP(V)~b(O)
matrix with the diagonal
,
V
~
N
Now, according to Theorem 8, below, ma 4, below, choose
A
it suffices to show that
m ~ N , such that Vo = k ( m )
+
Then, the conditions
elements
is columnwise nonsingular. A
a = £(n,m+l)
satisfies - 1 ~ q
and
gular, only if
E N . We
..... ~q = k ( m + l )
1
*) and **) in (65) are obviously
(63),
~
k(m) ~ ~ . Now, we choose
orem 8, will directly imply ***) in (65). Therefore Now, the relations
Indeed, due to Lem-
(65). Assume given
(641 and Lemma 4, imply that
DP(~)~(0)
~ 0 ,
V v c N
satisfied,
while
(62.2) and The-
A is column wise nonsingular. W(s~)(0)
is column wise nonsin-
WV
And now, the main result of the present section Corollary
1
Zx
~
0
,
x o ~ Rn
V
o
Proof According to Proposition f ~(x)dx = 1 Rn Define
and
a : Rn ÷ R 1
such that
B ~ 0 K
Defining
by
and
=
4, it suffices to show the existence of
DP~(0)
~ 0 ,
~(x I , ... ,xn) = exp (x I +...+ x n) ~ = 1
in a certain neighbourhood
[~ ~(x)S(x)dx
of
such that
and assume
S c D(R n)
0 ¢ Rn . Then
> 0
~ = ~ • S / K , one obtains the required function,
V p ¢ Nn
~ ¢ D(R n)
V p ¢ Nn •
since
DP~(0)
= I/K > 0
WV
In case arbitrary positive powers of the Dirac ~ distributions in the algebras constructed
in the present chapter
chap. 4) one needs the result given in: Corollary
2 n W+
Zx O
~
~
,
V
x ° ~ Rn
are to be defined with-
(see Theorem 4, chap.
i, g8 and
102
Proof Choosing fore
8 c C~(R n)
in the proof of Corollary i, one obtains
oo n @ ~ C+(R )
and there-
VVV
st~ E Z ° n W+
Lemma 4 The
infinite matrix of c o m p l e x
numbers
A = (auo
[ V 0
E N)
is column wise non-
singular, only if V
~,#cN: o c N , Vo ,..., Vo ~ N :
**)
-
6
***)
(65)
%
o
<
o
< v
...
............
o
%
o
o
,
a M o 0
............
o
a.ooG
Proof I t follows easily from the definition in §3
VVV
And now, the theorem on generalized Vandermonde determinants
(for notations,
see §3),
whose present form, as well as proof was offered by R.C. King. Theorem 8 Suppose given Then, for each
nc
N,
n>
a c Nn
1
.
, a > e :
(1 .....
1)
~ Nn
and
£ ~ N , .% > 1
, the
rela-
tion holds (a+p(1))
p(1)
...............
(a+p(1))
p(~)
=--~-l~i~n (a+p ( £ ) ) pC1)
where
p(j)
...............
= (pl(j),...,pn(j))
(a+p(~)) p(£)
,
for
1 < j
< .% .
--~--~ (pi(j))! > 0 l~j~i
103
Remark The Value
on
of
the
depends only on
determinant
n,R,p(l),...,p(R)
and does not depend
a .
Proof Let
US
consider
the determinant
A, = det ((a+p(o))P(')) For
1 S T S R , the r-th
, where
a = (al
For
,...,
u , T < R
column in Al is (a 11+P (1))
if
15
P,(T)
x . . . x (a,+p,Ul
I
p,(r)
an) .
Is-cQ, , consider
the column p(l)p(r)
I where
$&)P(r)
0' = 1 whenever it
We obtain
occurs.
then
() P(T)
(66)
C,(T)
= Cl CT) + i
(-a)p(T)-p(x)
C (p(X)) 1
P(X)
where the sum i Introducting
is taken
for
follows
We shall
1 2 X 5 R such that
ip(x)I
y
l
.
the determinant A2 = det (pi)
it
all
3
from
, where
(66) that
A2 = Al , since
A2 with
now simplify
1 F(h,k)
if
C,(T)
is the -r-th
the help of the function k=O
= h(h-l)...(h-k+l)
which obviously
1 2 o,~ 5 9. ,
satisfies
the conditions
if
kxl
column in
A2 .
F : N xN + N defined
by
104
(67)
F(h,k)
= 0 ~ = ~ h - k + 1 _< 0 * = * h < k
(68)
F(h,h)
= h !
Now, for
1 -< T <- ~ , we define the column (Pl(1)
c30)
.... × .F(Pn(1) , pn(T))
, PIG))
x...
=
F(Pl(~) Then,
• pl(T))
it f o l l o w s
× F(pn(~,) , p n U ) )
that p ( 1 ) q (~)
(69)
C3(T ) = C2('r ) + Z ( - W ~ - - ( - j ) )
,
V
1
<£
jcJ (£)q(~) where (69o!)
the
sum
~
Ji
c
with
(69.2)
q(T) the
2]~e relation (70)
where
is
taken
for
all
{1,2,...,pi(~)-l}
= p(Z)
-
(IJ1E,...,iJn])
number of elements
in
J.
1
u ...
o Jn @ ~
for
1 <_ i
-< n ,
, with
IJi[
denoting
.
(69) can obviously be written under the form C3(T ) = C2(T ) + Z ( - - [ ~ - - ( - j ) ) jeJ
(69.1)
J = J1
•
and (69.2)
are
still
C2 ( X ( T , J 1 , . . . ,
valid
and
V
Jn) ) ,
X(T,J 1 ,...,
Jn)
e N
1 -< T - <
~
,
is uniquely defined
by p(X(z,J 1 ,...,
therefore
1 <- X ( T , J 1 . . . .
Denoting by implies
A3
A3
, Jn)
and
(EJll
.....
IJn[)
/X(T,J 1 ,...,
Jn) l < lp(z)l
C3(I ),...,C3(£ ) , the relation
=
A1
(67) gives for any
--I [-- F(Pi(O) l_
1 -< ~,T <- £
the equivalences
, pi(T)) -- 0 ~=* (3 l-
taking into account (68) and the form of the colturms
the relation
(70)
and thus
Now, the relation
]herefore,
-< ~
-
the determinant with the columns
A3 = A2
(71)
Jn) ) = p(z)
(71) will end the proof
VW
C3(T ) , with
I-
105 §8.
STRONGER RELATIONS CONTAINING
The stronger version of
PRODUCTS WITH DIRAC DISTRIBUTIONS
t he relations
in 2) and 3) in Theorem i, §4, obtained in the
present section is of the following type: Given a locally finite subset family of Dirac distribution mily
derivatives
(Ca I a ~ X) of functions
high order vanish in in§§2-4,
(72)
Z ~a(X-a) aeX
qa c N~ , and a faup to a sufficiently
the relations: • Dqa6a(X)
Dirac algebras
dure in §§2-4.
whose derivatives
0 ¢ R n , one obtains within a subclass of the Dirac algebras
constructed
~le mentioned
(Dqa~ a I a c X) , with
Ca c C~(R n)
X c Rn , a
= 0 ,
x e Rn .
are constructed
through a particularization
Nkmely, the family of Dirac classes
Z Z , with
will be replaced by a smaller family which possesses First, we restrict the representatians
of the proce-
Z c Z 6 , defined in §3,
stronger properties.
of the Dirac 6 distributions
given by
Z~
in
§3. Denote by
g~ V
(73)
the set of all X c Rn ,
(73.1)
X
V
x
V V
o
E
Rn
I x c X)
the condition
is locally finite,
V
v c N ,
of
x° ,
~ ~ N
:
~ ~ p :
Vn l _ _ l xcX\{xo} The analog of Corollaries Proposition
satisfying
:
neighbourhood
~ c N ,
I x c R n) e Z~
locally finite:
(supp Sx(V)
(73.2)
Z = (sx
supp
Sx(~)
i and 2 in
=
7 is obtained in:
S
g~ ~ ~
and there exist
Z = (s x I x ~ R n) c g ~
such that
VxcR
sx c ~/+
n
Proof It results from the proof of Corollary
Now, for
Z
=
(s x I x c R n) e
Z~
2, §7
, denote by
WV
TE
the vector subspace
in
SO
genera-
ted by all the sums Z x~X
Z X Dqs qcN n xq x q4P x
where
X c Rn ,
X
locally finite,
Px c N n
and
~
c C I . One can notice that due xq
106
to (73.1), the definition of
TE
is correct.
And now, the analog of Proposition 2 in §3: Proposition 6 For each ideal
E • Zg , TZ
is a Dirac class which is compatible with the Dirac
I~
Proof First, we prove that
~
is a Dirac class, that is, it satisfies (17.1), (17.2) and
(24) in chap. 2. Indeed, the conditions (17.1) and (24) result easily. Assume now t • TZ , t % O , then (74)
t =
E x•X
E ~ Dqs qcN n xq x q<-Px
where
X c Rn ,
(75)
X
locally finite, nonvoid,
x° • X '
~
qo ~ Nn '
We show that (17.2) is satisfied for (76)
p c N : V
~
Px ¢ Nn
and
lxq ~ CI" Moreover
qo -< Px o : ~ Xoqo :~ 0 x
o
given in (75). Assume, it is false and
~ • N , ~ -> P : t(~)(x o) = 0
then (74) gives (77)
E x¢ X
qEENn %xqDqsx(U)(Xo)
= 0 ,
¥
~ e N , v >-
q-
E h Dqs (U)(Xo) = 0 , q•N n Xoq xo
P
%; ~ • N ,
such that (77) will result in u >-
q-
TE
and
Z~
TE
is a Dirac class. It remains to
are compatible. Since, obviously
TE c TE
and
16 , T E
are
compatible according to Proposition 2, §3, it suffices to prove that (79)
I~ n TE = O .
Assume therefore But
t • I~
t • I ~ n T 7' , t % 0 , then (74) and [75] hold again, since
t • TZ .
and [2) will again give (76), thus the reasoning above, contradicting (75),
will end the proof of (79)
VVV
Based on the above result we proceed to construct the Dirac algebras in which relations of type (72) are valid.
107
Suppose given
E ~ Z~ .
For a n y ideal
I
vector subspace
W , I ~ 16 , compatible vector subspace
in V
in
T
in
SO , T ~ TE
I n V
and vector subspace S 1 in S satisfying (16) and o o (17), one obtains a regularization ( V , T Q S I ) . Then, for any admissible property Q,
one can define Dirac algebras according to (18). The analog of Theorem 1 in §4, stating the validity of (72) within the algebras defined above is obtained in Theorem 9. First, we shall specify within the algebras the meaning of expressions more general, (80)
as in (72), or
of the form
E ~a(X-a) a~X
• ( E h qcN n aq Dq~a)
q~Pa where
X c Rn ,
Suppose
X
locally
H = ( h a [ a e X)
(81)
Y
a c X :
]
Va c R n ' ha = 1
(82)
V
finite, is
Va
of functions
neighbourhood
of
a
and
Xaq ¢ C 1
h a ¢ D(R n)
such that
:
a $ b :
h a o supp h b =
The existence of such families
H
results from the fact that
X
is locally finite.
one can define ~(x)
and then
a family
' Pa c Nn
on V a
a,b c X , supp
Obviously,
~ a ¢ C~(Rn)
=
E ha(X ) • ~a(X-a) aeX
~ e C~(R n)
(83)
T =
Z aCX
. Define
xE
,
W ¢ D,(R n)
R1
by
Z l Dq~ qCN n aq a q<Pa
Lemma 5 Within the algebras provided
that
(19)
Ap , p ~ ~n , the product
~-T
does not depend on
H ,
is valid.
Proof Assume
H' = (h a I a ¢ X)
~' ¢ c~(R n)
is an other family satisfying
by
~ ' (X) =
Y: h a ( X ) • ~Pa(X-a) x£X
,
x
R1
(81) and (82) and define
108
We shall prOve that (84)
~b' • T = @ • T
holds within the algebras relation
%
• P ¢ ~n . I n d e e d ,
(83) above, the inclusion T=
E
Z
a¢X
qCN n
X
TE c T
Dqs
aq
a
assume
E = (s x ] x ¢ Rn)
a n d 3) i n T h e o r e m 2, c h a p .
, then the
1, §8, g i v e s
IQ(v(p),T(~SI)e %
+
q~Pa since
t =
E acX
E q¢N n
Dqs
aq
c 7E
a
and
T =
q-<Pa Therefore,
to • T = u(to) 4''
• t + IQ(v(p),TQS1)¢
T = u(@')"
Ap
IQ(v(p),TQS1)E%
t +
hence (85)
IQ(v(p),T(~SI)
tO" T - ~ • T = u(to'-to) • t +
But, due to (73.2), ¥
x
t
satisfies
c
Ap
the condition
ERn\X: O
(86)
~
V
V
x) ~ N ,
neighbourhood ~) _> p
of
xo ,
~ c N :
:
V n supp t(~) = while, due to (81),
(87)
u(to'-to).t
V
XoCX:
~
U
satisfies
neighbourhood
of
x
the condition
: O
(l~'-to) • t(v) = 0 Now, the relations
U ,
V
~) c N
(86) and (87) will imply
Dr(u(to'-to)-t) therefore,
on
¢ f 6 o V° ,
(19) and (85) will give
V
r ¢ h~
(84)
VVV
Lemma 6 Within the algebras ( E
%
ha(X)'toa(X-a))
aCY
for any
, P ¢ ~n , t h e r e l a t i o n s • ( E
aeY
y c X ,
Y
E
finite, provided
Proof Assume
a,b
¢ Y ,
a + b
and
X
qcNn q~Pa
q ¢ Nn , t h e n
aq
that
hold
Dq6a) =
E
toa(X-a)( E
acY
(19) is valid.
qeNn q~Pa
~ a q Dq6a)
109
(88)
ha(X)
Indeed,
" ~a(X-a)
• DC~b = 0 e A P
assume
~ = (s =< I x c Rn ) , t h e n , and the inclusion Z c T , one obtains (89)
ha(X)
But,
(81)
• ~0a(X-a)
and (82)
(90) taking
into
account
will imply (88)
• Dq6 b = U ( h a )
imply that
Dr(u(ha)
t o 3) i n T h e o r e m 2, c h a p .
• U(~a('-a))
DqSb) • ira n V° ,
1
§8,
IQ(V(p),TQ)S1) • Ap
Dqs b +
b $ supp h a , therefore,
• u(tPa(--a)) that
according
one obtains ¥
s b e Zb . Now, t h e r e l a t i o n s
r • Nn (19)
a n d (90)
together
with
(89)
777
The Lemmas 5 and 6 suggest within the algebras
Ap • p c ~n
the following definition
of the expressions in (80) Z ~a(X-a) • ( Z ~ Dq~a) = ( 2 -~a(X-a)) • ( Z a•X qcN n aq aeX h a ( X ) a•X
(91)
q~Pa where
H = (h a ] a • X)
Z X Dq~a) q•N n aq q~Pa
is any family
of functions
h a ¢ D(R n)
satisfying
(81)
and
(82).
Theorem 9
In case (19) is valid, the following relations hold in the algebras p e ~n : I)
2
~a(X-a) • Dqa~a(X) = 0 • A
a•X for
Ap , with
P ' each
X c Rn ,
X
locally
finite,
qa ~ Nn
and
~ a e ~ ( R n)
satisfying the condition Dr~a(0)
(92)
=
In particular• if 2)
Z
%/ a e X •
0 •
p • Nn
r • Nn
,
r
<
p
or
r
-< q a
"
then
(x-a) r a . Dqa~ (x) = 0 • A
a•X
a
for eack
X c Rn
p "
X
locally finite•
ra • qa • Nn •
ra ~ p
and
ra ~ qa "
Proof i)
Assume h
(93)
a
2 = (s x I x e Rn)
• D(R n) Z aeX
and
H = (h a I a c X)
is a family of functions
satisfying (81) and (82). Then, according to (91) ~a(X_a).
Dqa~a(X ) = ( Z a•X
ha(X)-~a(X_a)).
( Z aeX
Dqa~a(X))
110
We shall p r o v e that the right side of the above r e l a t i o n , d e n o t e d b y in
A
. Indeed,
S ~ D' (Rn)
S , vanishes
, thus, taking into account 3) in T h e o r e m 2, chap.
P §8, and d e n o t i n g v = u ( Z h • ~a(--a)) a£ X a
•
Z Dqas ae X a
one obtains
(94)
S = v + IQ(V(p),TC)Sl) since
But
Z Dqasa • ~ a•X v • V
E Ap ,
c T .
due to the fact that
(92) holds for
r • ~
, r -< _ qa
" Therefore
o
(95)
Dry • V° , A s s u m e now ther w i t h
r ~ N~ .
r ~ Nn , r < p , then
(90) and (5) imply that
Dry ~ /o
w h i c h toge-
(95) and (19) will give
Dry
2)
V
That
relation
shes
in
¢ /
n Vo c V ,
implies
¥
v • V(p)
r
• Nn
, hence
,
r
due to
. (94)
the
expression
A P
It follows f r o m I) taking
T ~ . ~barX~ = x a ,
%/
a • X ,
x • Rn
WV
in
(93)
vani-
I,
C h a p t e r
6
LINEAR INDEP£NDENT FAMILIES OF DIRAC DISTRIBUTIONS
§i.
INTRODUCTION
The representations of the Dirac ~ distribution used in chapters 4 and 5, were given by weakly convergent sequences of smooth functions satisfying a condition of strong local presence (see (6) in chap. 5, §3). A first consequence of that condition was the nonsymmetry of these representations• implying that the Dirac distribution derivatives
D°~
of any order
q e N n , are not invariant within the algebras under the
transformation of coordinates Rn9 {see pct.
x
+
a • xc
Rn ,
2 i n Remark 2, c h a p .
5,
a = -1
,
§5).
In the present chapter the following stronger result is proved within the algebras containing the distributions in
D' (Rn)
: Applying to any given derivative
D q~ ,
q ~ N n, of the Dirac distribution the transformations of coordinates Rn B x one o b t a i n e s
÷
for pair
a " x c Rn ,
ac
wise different
a°
Dq6 ( a o x ) . . . .
, Dq6 ( a m X )
.
In §4,
result
extended to
that
is
include
R1 \
{0}
,
• . . . , am c R I \ { 0 }
also
generalized
, linear
independent
Dirac elements
of the form
lim n~ (ax) a->Co The a b o v e p r o b l e m s arbitrary
§2.
are approached within
coordinate
transformations
a general
within
framework established
the algebras
are
i n §2, w h e r e
studied.
COMPATIBLE ALGEBRAS AND TRANSFORMATIONS
Suppose given G i v e n an i d e a l a vector
Z ~ Z6 I
in
subspace
V
{see chap.
§§2-4).
W , I = I ~ , a compatible in
(i)
VoQTQS
(2)
U c V(p) Q T Q S
it follows that
S,
I
n Y
and a vector
1 ,
V
o
vector
subspace
T
subspace S1
in
S
o
in
So , T = T Z ,
such that
1 = SO
(V,TQSI)
p c ~n •
is a regularization, therefore, one can define for any ad
112
missible p r o p e r t y (3)
Q
the Dirac algebras
AQ(v, T Q S
1 , p)
,
~n
p
denoted for the sake of simplicity b y
A P
It will be assumed throughout (4)
§§2 and 3 that
v = f6 n v 0
Given a m a p p i n g
e : R n ÷ Rn • a e C ~
called transformation,
we define
~ : W ÷ W
by (~)(~)(x)
= s(~)(~(x))
obtaining thus a homomorphism An algebra
A
,
v
s
of the algebra
and the transformation
~
w,
~
~N
•
x
~ R1
,
W .
~ are called compatible,
only if
P
IQ(v(p),T+(~SI )A and AQ(v(p),T+(~SI)~ are In that case,
invariant
of
one can define the algebra h o m o m o r p h i s m
~ : W ÷ W . ~ : A
÷ A P
~(s+IQ(v(p),TGSI)) = A transformation -I
Proposition
~ : Rn ÷ R n : Rn ÷ R n
~(s) +
a e C~
exists and
IQ(v(p),TGSI) , V
is called invertible, -I
given by P s
~
AQ(v(p),TQSI)
only if
E C~
i
An algebra
and an invertible
A
transformation
is an invariant of
AQ (V ( p ) , T Q S ~ )
a
~ are compatible•
only if
: W ÷ W .
Proof The
necessity
is obvious.
Now, the sufficiency.
IQ(v(p),T~SI)~
is an invariant
AQ(v(p),TGSI)
generated by
ant of
a
: W ÷ W
of
V(p)
~
One needs only to show that
IQ(V(p),TGSI)
is the ideal in
due to Lemma 1 below,
it is an invari
: W ÷ W . But,
, therefore,
WV
Lemma 1 If ~ is an invertible of
~
transformation,
: W ÷ W .
Proof Since ~ is invertible•
(s) The relation
~ ( v o) c V°
one obtains
easily
then
V(p)
, with
p E ~n
, are invariant
113
(6)
a(I ~) c I ~
is also valid. satisfies Then
w
V e N But
Indeed,
in
x°
the conditions
xI
satisfy
and
c R n given. We have to show that aw o (2) and (3) in chap. S, §2. Denote x I = a(Xo) .
(2) in chap.
w
and
e N
xI
also satisfy
V
5, §2, thus
(3) in chap.
is a neighbourhood
big enough.
for
v c N
of
(aw)(~)(x o) = w(~)(Xl)
= 0 , for
to
{x o}
and (6) will obviously sy to see that
The result Suppose
a(V) c V 1 . Now,
then also
assume that of
~-i
V n supp
imply
~(V(p))
~(V) c V , since
c V(p)
,
V p ~ An
1 above,
has the property
A
= ~
shrinks
for to
{x I} ,
it will follow that V n supp(~w)(v) The relations
(4) was assumed valid,
the following
transformations.
PM ' only if
of
lhen,
(5)
it is ea-
WV
justifies
is a set of invertible
V1
if
(~w)(~)
V 1 n supp w(v)
v ÷ ~ , and the proof of (6) is completed.
in Proposition
M
such that
big enough,
On the other side,
, when
S, §2, for a certain neighbourhood
x°
~ ÷ ~ . Now, due to the continuity
shrinks
W
x
big enough.
V 1 n supp w(~) = ~
in
w c I~
and
x I . Assume
when
assume
definitions.
We shall
is an invariant
say that a subalgebra
of each
~
A
: W ÷ W , with
~ ~ M . Obviously,
PM
The algebras
is an admissible
property
(see chap.
(3) will be called M-transform
I, §6).
algebras,
only if
Q
is stronger
than
PM "
Corollary
1
An M-transform
algebra
A
and a transformation
c M
are compatible.
P
Proof The subalgebra
AQ(v(p),T
+ SI)
is invariant
of
~ : W ÷ W
since
A
is an N-transP
form algebra and
§3.
LINEAR
Denote by defined by
Theorem
e c M
WV
INDEPENDENT
M
FAMILIES
OF DIRAC DISTRIBUTIONS
the set of invertible aa(X)
= ax
,
transformations
a
: R n ÷ R n , with
a
~
RI\{0}
V x e Rn .
1
The Dirac distribution
derivative
q ~ N n , are linear independent
transforms
within
the
Dq~(aoX),... ,DqG (amX) Mo-transform
algebras
with given Ap , p ~ A n ,
,
114
provided
that
m <- Ipl
and
a ° ,..., a m ¢ RI\{0}
are pair wise different.
Proof Assume•
k ° ,.. ., k m c C I
it is false and
(7)
koDq6(aoX)
(8)
]
are such that
+ ... + kmDq@(amX)
= 0 ~ A P
0 ~ i ~ m : k. $ 0 i
Assume
E = (s x
TZ c T
give
I x ¢ R n)
then 3) in Theorem
Dq6 = Dqs ° + I Q ( v ( p ) , T Q S I )
2, chap.
I, §8, and the inclusion
( Ap '
therefore kiDq@ (aix) = ki~ai Dqs ° + I Q ( V ( p ) , T G S I )
(9)
since
A
p
is an
(i0)
then
v =
(7) and
v =
with
vj ¢ V(p)
chap.
E o~i<_m
and
wj ~ A Q ( v ( p ) , T G S I ) (4) above,
re
Nn ,
ueN
(12) V
~¢
0 _< i _< m ,
hence
v. • w. J J
S, §2, one obtains V
¥
Denote
v c IQ(v(p),TQSI)
~ o~j~h
into account
algebra.
,
k.~ Dqs • ai o
(9) imply
(ii)
Taking
M -transform o
¢ %
.
as well as condition
(2) in the definition
of
16
in
from (II) the relation r
:
~)>U
:
: N,
Drv(~) (0) = 0 Since
m-<
Ipl
, the relations
IL o
(13)
(i0) and (12) will give
ki(ai )Irl]
D q+r So(V)(0)
= 0 ,
V
r e Nn , Irl _< m
for a suitable But
(14)
so
r < p , •
~ e N ,
U' e N .
Z o ) therefore V
re
@
V ¢ N ,
Nn ,
Drs
(V)(0)
oe
N:
V >- g
:
~ 0
O
since the matrix Now, the relations
W(s o) 60] is column wise nonsingular (13) and (14) result
in
(see chap.
5, §3).
~ >- U'
)
115
ki(ai )£ = 0 ,
V
Z c N ,
Z ~ ]pl
o~i~m Since
m ~ Ipl
and
a° ,..., am
are pair wise different,
Vandermonde determinant will imply
the known property of the
k ° = ... = k m = 0 , contradicting
(8)
W?
Corollary 2
§4.
The family
(Dq6 (ax)
with given
q e N n , is linear independent within the
GENERALIZED
Within
I a c RI\{0})
derivative
Mo-transform
transforms algebras
DIRAC ELEMENTS
D' (Rn) , the operation
(15)
lim an~ S a_+Oo a
has a meaning for certain distributions K = D/n f(x)dx thus
of Dirac distribution
, then
(iS) gives
(15) will give
an ~
lira a+oo
algebras
S = f c LI(R n)
S = ~ , one obtains
due to Corollary
A class of algebras,
anea S = S
,
x c Rn ,
2 in §3.
similar to the ones used in §§2 and 3 will be constructed
section and it will be shown in Theorem 2 that within those algebras, (16) exists and it is different of The mentioned
I~
w(~)
a e RI\{0}
the limit in
. I~
defined in chap. S
condition
vanishes outside of a bounded subset of
~) e N
R n , provided
that
is big enough.
It is easy to see that
PropOsition
, with
in this
, consisting of all the sequences of smooth functions
which satisfy the additional
(17)
an6(ax)
algebras are constructed by replacing the ideal
§2, with the smaller one w c I~
and
Ap , p c ~n , Ipl = ~ , the problem of the limit
= lira an~(ax) a_>=o
a
becomes nontrivial
K~ . In case
if
S = ~ .
Within the M - t r a n s f o r m (16)
S . For instance,
16
is indeed an ideal in
W , actually a Dirac ideal.
2
For each
E e Z6 , the Dirac class
TE
and the Dirac ideal
I~
are compatible.
Proof It follows from the inclusion
I~ c 16
and Proposition
2 in chap. 5, §3
VW
116 Suppose now given
Z ~ Z6 .
Given an ideal
in
I
a vector subspace
W , I = 16
V
in
I n V°
(IS)
VoQTGs 1 :
(19)
U c V(p)QTQS
it follows
ble property
Q
vector subspace
and a vector subspace
S1
in
T
in
So
So , T = TE ,
such that
SO
1
(V,TQSI)
that
, a compatible
V
p c An
is a regularization,
thus,
one can define for any admissi
the Dirac algebras
AQ(v,TQS 1 ,p] , p c ~n ,
(20) denoted
for simplicity by
A P
We shall assume in the sequel that V = I~ n V °
(21)
Within the above algebras
~
, p c ~n , the limit in (16) will be obtained
as given
b y a relation (22) with b
lim t c W
an6(ax)
IQ(v(p),T(~SI)~ c PA
= t +
, t(v) = ( b ) n s(v) ( b x)
> 0 , limb
V w ~ N , x ~ R n , where
,
The entities of type
(22) will be called generalized
Using a p r o o f similar to the ones in Proposition Proposition
A
and an invertible
AQ(v (p) ,T Q S ~ )
given
homomorphism
is an invariant
b = (b v I v c N)
transformation of
, with
a
: W + W
b w c RI\{0}
Ibvl n w(v)(b x)
We shall only be interested
~ are compatible,
only if
•
• Then,
one can define the algebra
,
V
w ~ W ,
~ ~ N ,
x ~ Rn .
in the case when
lim b~ = +
Since in the definition a : Rn ÷ R n
definition
of compatibility
between
given in §2, the transformation
ra h o m o m o r p h i s m ned b e t w e e n
besides
Dirac elements.
1 and Lemma 1 in §2, one obtains:
~b : W + W , where (c~w) (v) (x) =
(23)
and
3
An algebra
Suppose
s c go
=
~ : W ÷ W , it follows
the algebras
A
of M - t r a n s f o r m
transformations
an algebra A and a transformation P appears only through the generated algeb-
that the compatibility
and algebra homomorphisms
P algebras
given in §2, will
also algebra homomorphisms
of
of
has actually been defi-
W . In the same way, the
still be correct W .
if
M
contains
117 Proposition 4 The algebra
c~
and the algebra homomorphism
A
AQ ( Y ( p ) , T Q S 1
is an invariant of
p
are
compatible, only if
0% .
Proof S e e the proof of Proposition i, §2 and the following:
Lemma 2 V(p) , with
Nn
p E
, are invariant of
~b "
Proof Assume
(24)
lim by = + = 12+oo
First we prove
(25)
% 4 % ) c f6
Indeed, assume
w E I~
and
quence of smooth functions
x ° E Rn ~b w
given. First, we notice that due to (24), the se
will obviously satisfy (17), since
that condition. We shall now show that (3) in chap. S, §2. Indeed, in case (24), while for
x° = 0
~b w
satisfies in
xo
w E f~
satisfies
the conditions (2) and
x ° ~ 0 , the two conditions result easily from
they are obvious.
Now, we prove that (26)
~b4V)
Assume,
indeed
show that
(27)
c V
v E V , then
~b v E V°
lira [
, which
v ~ I6 is
n Vo
therefore,
equivalent
due to
425),
it
will
suffice
to
to proving
v4~)4x)~(x/bv)dx = 0 ,
V
~ E D(R n)
First, we notice that 428) for such
supp v(~) suitable that
K c Rn X = 1
429)
on
lira [ %~+0o
But, for any
c K , , K
v E N ,
bounded
K . Due t o
and (28),
V >-
~ E N , since the
relation
v(v)(x)x(x)~(x/b )dx = 0 ,
~ ¢ D(R n) , the sequence
V x E Rn , is convergent in
vEY=f~nVo
V
c yo
D(R n)
to
(~
427)
. A s s u m e now
will
hold
only
X ~ D(Rn)
if
~ E D(R n)
I v E N)
~(0)
hence, the sequence
V
v ¢ I~
with
~(x)
X • Therefore,
(v(~)) [ ~ ¢ N)
= X(X) ° ~4x/bv)
converges in
Thus, the relation (26) is proved. It follows then easily that
,
(29) holds, since N'(R n) to
0
118
~b(V(p))c V(p) The c a s e
when
lira b
= -~
,
V
p ( ~n .
is similar
WV
k>+¢o
Suppose given
b = (b~ ] ~ c N) , with
= M o u {~b }
where
Mo
by c RI\{0}
(23) and denote
was defined in §3.
As noticed above, one can define Mb-transform to Propositions
, satisfying
algebras
A p • with p c ~n . According 3 and 4, these algebras will be compatible with any B ~ M b •
And now, the result concerning generalized Dirac elements. Theorem 2 The Dirac distribution derivative
transforms
neralized Dirac element derivative
Dq~b6(X)
Dq6(aoX),...,Dq6(a~nX) , with given
and the ge
q E Nn • are linear
independent within the Mb-transform algebras Ap , p c ~n , provided that R 1 \ {0} are pair wise different. and a° ,..., a m c
m ~ Ip]
Proof Assume,
k ° ,. • ., k m ,
it is false and
(30)
koDq~(aoX)
(31)
k = 0
~
k c CI
+ -.- + kmDq~(amX) ~
Z = (s x l x c R n)
TZ c T
imply
(32)
is an ~b-transform kiDqS(aix)
= 0 ~ Ap
then 3) in Theorem 2, chap. I, §8, and the inclusion
~ Ap
Dq8 = Dqs ° + I Q ( v ( p ) , T Q S I ) Ap
+ k~bDq6(x)
0 ~ i ~ m : ki $ 0
Assume
Since
are such that
algebra,
it follows that
= ki~a. Dqs ° + I Q ( v ( p ) , T Q S I )
c Ap ,
V
i
(33)
kDq~bs(X)
= k • t + I Q ( v ( p ) , T Q S I) ~ Ap
where (34)
t(~)(x) = Lb~l n Dqso(~)(b~x)
,
V
~ ~ N ,
Denote
(35)
v =
then (30),
Z k.~ Dqs + k • t 1 a. o o_
(32) and (33) imply
(36)
v =
with
vj. ~ V(p)
v c I Q ( v ( p ) , T Q S I)
Z v. • w. o<j
w.j c A Q ( v ( p ) , T Q S I )
•
thus
x ~ Rn •
0 ~ i ~ m ,
119
Taking into account chap.
(21) above as well as condition
5, §2, the relation
(2)
in the definition
of
I~
in
(36) will result in
V
r~
Nn ,
r
:
V
~e
N
~>-~
:
(37) Dry(v) (0) = 0 But
m ~ Ip[
, therefore
(35) and (37) will give
Irl
Z ki(ai) o~i~m
(38)
+
klbv[ n + l r l )
V r e # for a suitable Now,
that
Z ki(ai )~ + klb~ o~i~m
must v a n i s h s i n c e
a ° ,..., am
terminants
Corollary
£ ~ N ,
~ =
0 ,
are constant
(31)
~o
~)c
N,
O->~
' ,
V
£ e N ,
k ° = ... = k m = 0
~ < m _
(38) results
in
% -< m
''''' ~E e
N ,
in (39). Therefore,
above can be a r b i t r a r y .
are pair wise different,
will imply
will contradict
V
o c N , one can find suitable
Z ki(ai )~ o~i~m Since
Irl ~ m ,
r ~ p ,
In+£ = 0 ,
k ° ,..., k m , k , a ° ,..., a m
k
0
(14) in the proof of T h e o r e m 1 in §3, the relation
where for any given But
=
~' ~ N .
according to
(39)
,
Dq+rso (V) (O)
Then,
~o ''''' ~
~ o
(23) will imply
(39) becomes
.
the known property of the Vandermonde
which together with
k = 0
obtained
de-
above,
VVV
3
The family
(Dq6(ax)
I a e RI\{0})
together with the generalized
Dirac element derivative
q e N n , are linear independent p e N n , ]pL = ~
of Dirac distribution
within the M b - t r a n s f o r m
derivative
~b Dq
(x)
algebras
transforms
with given Alp ,
C h a p t e r SUPPORT,
§i.
7
LOCAL PROPERTIES
INTRODUCTION
An important property of the distribution [II], [61],
[66-69], V
multiplication,
[783, which can be formulated
S,S',T,T' •)
¢ D'(R n)
S • T ,
• *)
S = S'
,
E c Rn •
S' • T' ,
T = T'
on
will be proved valid for the multiplication butions.
E # ~ ,
exist in
namely
D'CRn)
E
open ]
: ~
S • T = S' • T'
as well as several
on
E
J
within the algebras
An extension of the notion of support of a distribution
ing the above,
its local character,
as follows
other local properties
containing
the distri-
is used in establish-
of the elements
in the algeb-
ras.
§2.
THE EXTENDED NOTION OF SUPPORT
Suppose
P
is an admissible property,
sible property, Given
such that
S c AQ(v,S ',p)
Q -< P
and
*) **)
for a certain
~eN S
S
vanishes
on
E , only if
S = s + IQ(v(p),S ') s(x)) = 0
on
E , with
v ~ N ,
~) >-I/
,
. will be the closed subset
of
(2)
supp S = R n \ {x c R n
Proposition
! in
C°~(Rn )
I S
vanishes
on a n e i g h b o u r h o o d
Proof from (20.2)
of
x}
the above notion of support is identical
usual one.
It results
Q is an admis-
:
The support
For functions
is a P-regularization,
p ~ NP .
E c R n , we say that
s c AQ(v(p),S') (I)
and
(V,S')
in §7 and Theorem 2 in §8, chap.
i
WV
.
with the
121
The local character of the multiplication the distributions
is established
and addition within the algebras containing
in the following two theorems.
Theorem 1 If
S,T c AQ(v,S',p)
, then
I)
supp (S + T) c supp S u supp T
2)
supp
(S • T) c supp S n supp T
Proof It follows from a direct verification
Given
S,S' c AQ(v,S',p)
vanishes
on
and
of the definitions
E c Rn , we say that
WV
S = S'
on
E , only if
S - S'
E .
Theorem 2 Suppose on
S,S',T,T'
E AQ(v,S',p)
and
Ec
R n . If
S = S'
on
E
and
T = T
E , then
I)
S + T = S' + T'
on
E
2)
S • T = S' • T'
on
E
Proof It follows directly form the definitions
An important
WV
feature of the above notion of support is pointed out in the case of the
algebras where the Dirac ~ function is represented smooth functions
satisfying
by weakly convergent
the condition of strong local presence
sequences of
(see chap. 5, §3
and also chap. 4, §6). Theorem 3 In the case of the algebras constructed any order
q ~ N n of the Dirac ~ distribution,
l)
s u p p Dq~ = (0}
2)
Dq~
v a n i s h e s on any
3)
Dq6
d o e s n o t v a n i s h on
c h a p . 5,
*)
cl
E
E c Rn
such that
possess the properties:
0 + cl
E
*)
Rn\(o} , provided t h a t the c o n d i t i o n
§4 i s v a l i d .
is the topological
closure of
Dq6
in chapter 5, the derivatives
E c Rn
(19) i n
of
122
4)
Dq6
does not vanish on
chap.
{0} , provided
that the condition
(27) in
5, §4 is valid.
Proof Assume
Z = (s x [ x ~ R n)
TZ c T
gives
(3)
for
Dq6
3)
We notice
C4) be
I, §8, and the inclusion
I) •
A Q ( v , 2 ~ ) S 1 ,Pl •
(5) in chap.
p c ~n
5, §3.
that in any representation
Dq 6 = t will
easily from
2, chap.
representation
Dq6 = Dqs o + I Q ( v ( P ) , T Q S
Then i) and 2) result
t
, then 3) in Theorem
the following
IQ(v(p),TQS1) c AQ(v,T(~S 1
+
a sequence
of
smooth
functions.
,p)
Therefore,
p •
,
if
Dq6
y~n , vanishes
on
Rn\{0}
,
Thus,
due
then
(5)
t(V)
for certain
= 0
on
Rn
V
~) c N
U • N . But (5) obviously
V >
implies
Drt e 16 n V
V r e Nn O
to (191 in chap.
5, §4, one obtains
Dq6 = 0 • A Q ( v , T Q S 4)
Assume,
(6)
I)
t(~)(o)
for certain
V p • ~n
,
V p ~ ~n
, contradicting
'
. Now,
(4) will
11 in Theorem
a representation
imply
i, chap.
5, §4.
(4) with
such that =
o
,
v
~
c N
,
v
~'
>
U' e N . Denote
(7)
v = t - Dqs
O
(3) and (4) imply v =
with
,
it is false and there exists
t • AQ(v(p),TQS
then
1 ,p)
t • Y(p)
Z o_<j<_m
v •
IQ(v(p),TQS I)
therefore
v. • w. J 3
vi • V(p I , w i • A Q ( v ( p ) , T Q S I )
. Now, the condition
(27) in chap.
5, §4, will
give v. c V(p) c V c 16 ] which together
(2) in chap.
vj(~)(0)
(8)
for certain Now,
with
(6),
dition
=
0
,
V
V
0 _< j -< m
5, §2 results 0 -< j -< m
,
~
in •
N
,
v > U"
U" ~ N .
(81 and (71 imply
(91 with
,
11 ~ N
Dqso(~)(0)
= 0 ,
V
suitably
chosen.
But,
(61 in chap.
5, §3
WV
9 c N ,
~ >-
s o • Z o ' while
C9) ohviously
contradicts
the con
123 Remark 1 The p r o p e r t y
i n 4 ) , Theorem 3, t h a t t h e D i r a c d i s t r i b u t i o n
x ° c R n , q ¢ N n , do not vanish on strong local presence The
and it is proper
'delta s e q u e n c e s '
generally used,
137], [ 1 6 2 ] , do n o t n e c e s s a r i l y
A characterization
{x o}
sequences
Dq~
[4],
[35-41],
[53],
prevent the vanishing of
[68-69], Dq6
X
in algebras
of smooth functions
, with
X
of the con__tion°o_g~#
for the algebras used in chapters
of the support of the elements
of the representing
derivatives
is a consequence
is presented
4, 5 and 6.
[105-110],
on
[136-
{x } . O
O
in terms of the supports in:
Theorem 4 Suppose
S ~ AQ(v,S',p)
supp S = n cl
lim supp s(v)
where the intersection
(10)
then
is taken over all the representations
S = s + I Q ( V ( p ) , S ')
AQ(F,S ' , p )
with
s e A Q [ v ( p ) , S ')
Proof The inclusion c
. Assume
s
V n supp s(V) = ~ , for a certain neighbourhood Conversely, such that
assume S
V
V of
~ c N , x
and
vanishes
on
. Then
v ~ ~ ,
~ ~ N . Now, obviously
x e Rn\supp S , then there exists
V n lim supp s(v) = ~ v+oo
x % supp S .
an open n e i g h b o u r h o o d
V . Hence, one can obtain a representation
V
of
in chap.
6, §4, an additional
result on the
can be obtained.
Theorem 5 Suppose given S = sI +
S e
AQ(v,TQS 1
IQ(v(p),TQS I)
Then, the subsets
in
lim supp Sl(~)
,p)
and two representations
= s2 + IQ(v(p),TQS
~
Proof sI - s2 e
,P)
•
supp s2(~)
differ in at most a finite number of points,
Obviously
I) e AQ(v,TQS 1
Rn ,
IQ(F(p),TQS I) , hence
provided
that
x ,
(i0), such that
WV
In the case of the algebras constructed support
x e R n \ cl lim supp s(V) v+oo
given in (i0) and
V c 16 n V ° .
124
(8)
SI - S2 =
with
vj c V(p)
~ V. " w. o~j~m ~ J
, wj ~ A Q ( v ( p ) , T Q S I )
vj ~ V(p) c V c I~ therefore Then,
vj
, with
due to (8),
Corollary
,
. Now,
V
in chap.
S, §4 implies
0 ~ j ~ m
0 ~ j ~ m , satisfy
Sl - ~2
(27)
(3) in chap.
will also satisfy
5, §2 and (17) in chap.
those two conditions
6, §4.
VVV
I
Under the conditions Sin= 0 for certain
in Theorem
5, if
AQ(v , T Q S 1 , P) m e N , m e i , then
supp S
is a finite
subset of points
in
Rn .
Proof Assume
given
a representation
S = s + IQ(V(p),T~SI)~_~
, with
s c AQ(V(p),T~SI),~
then S m = s m + IQ(v(p),TQSI) s m ~ IQ(v(p),TQSI)
therefore
Using an argument s
m
~ I~
similam
therefore
.
to the one in the proof of Theorem
lim supp sm(w) v+oo
supp s(w) = supp sm(~)
,
= 0 c A Q ( v , T Q S 1 , p)
V v ~ N
is a finite
5, it follows
subset of points
in
Rn
that Finally,
VVV
Remark 2 In the case of the algebras
constructed
lary 1 will still be valid, provided
§3.
5, the results is replaced
in Theorem
by locally
5 and Corol-
finite.
LOCALIZATION
Given S
in chap.
that finite
S ~ AQ(v,S',p)
vanishes
on
denote by
ES
the set of all open subsets
E c Rn
such that
E .
The relation ~-J
E
=
R n \ supp S
EcE S is obvious.
In case
stributions
are used,
S e D'(R n)
and the usual notions
the corresponding
set
E$
of vanishing
and support
has the known property
for di-
that any union
125
of sets in
ES
is again a set in
ES .
In particular R n \ supp S = ~ J E e ES E•E S A first p r o b l e m approached ty to the algebras the structure
of
in the present
containing ES
section is the extension of the above proper
the distributions.
In this respect,
several results on
will be given.
Theorem 6 Suppose
and
S • AQ(v,S',p)
EI , E~ • E S . I f
d ( E l k E 2 , E2\E 1) > 0
*)
then
E 1 u E2 • E S .
Proof Assume
such that
s I • s 2 • AQ(V(p),S ')
(9)
S = s i + IQ(v(p),S ') • AQ(v,s',p)
(i0)
si(v ) = 0
for certain
(n)
on
Ei ,
> • N . Denote
V
1 < i < 2 ,
one obtains
(12)
for
v(V) (x) =
According
~ • N ,
v ~ N ,
~ _> ~ ,
~ = 1/2
on
~ ->
if
x • Rn \ ( ~
s 1 (v) (x)
if
x • E2 \ E1
- s 2 ( V ) (x)
if
x • E1 \ E2
0
if
x • E1 n E2
to Lemma 1 below,
thus denoting
- s2(, )(x)
e C = ( R n)
there exists
~ / E 2 . Denote
s = (Sl+S2)/2
+ w
w = u(~)
S = s + IQ(v(p),S '3 • AQ(v,S',p)
=
0
on
(14)
s(~)
Now,
(13) and (14) will
E1 u E2 , imply
, such that
• v , then
, the relation
But, due to (12), it follows obviously
*)
1 < i -< 2 ,
v = s I - s 2 , then (9) implies
Sl(X; ) ( x )
(13)
V
v • IQ(v(p),S ')
Further,
and
,
(ii) implies
(9) will give
,
s ~ AQ(v,S',p)
that V
~
E1 u E2 • E S
~ N
,
~
~
VVV
d( , ) is the Euclidian distance on R n and Rn d(E,F) = inf {d(x,y) [ x • E , y 6 F) for E,F c
u E 2)
= -1/2
on
E2\E 1
w c IQ(v(p),S ') ,
126 Lemma i Suppose
F c G c Rn
such that
d(F,Rn~G) > 0 ,
then there exists
t~ e U°°(Rn)
with the properties I)
0 ~ ~ ~ 1
on
Rn
2)
~ = 1
on
F
3)
~ = 0
on
Rn\G
4)
@ ~ D(R n)
if
F
bounded
Proof Define
X : Rn
(0,~) ÷ R I
x
K ×(x,s)
by " exp (e2/(iixll2-e2))
0 where
if
llxl]
< e
if
llxll
~ ¢
=
Ks = 1 /
exp
(c2/(llx[12-e2))dx
llxl <~ Assume
0 < E < d(F,Rn\G)/2 ~(x)
=
J
and define
~ : Rn + R I
by
X(X-y,e)dy
F(~) where
F(¢)
= {y ¢ Rn [ d ( y , F )
It can easily
Corollary
be seen that
~ e}
~ is the required
function
WV
2
Suppose
S c AQ(v,S',p)
i)
cl E n cl F =
2)
F
then
E u F c ES
and
E,F E E S . If
bounded
PrOof The
sets
Theorem
E
and
F
satisfy
the conditions
in Theorem
6
VW
7
Suppose then
S ~
AQ(V,S',p)
E[ u E 2 ~ E S .
and
E 1 , E 2 E E S . If
E~ c E 1
and
d(Ei
, Rn\E1 ) > 0
127
Proof We shall use the notations According
in the proof of Theorem 6.
to Lemma I, there exists
= 1/2
on
(15) therefore,
Corollary
it can be seen that the relation
E i . Then, s(~) = 0
such that
~ e Cm[R n]
on
the relation
Ei u E2 , (13) will
V
~ ~ N ,
imply
~ = -1/2
on
E2/E 1
and
(14) becomes
~ a p ,
E i u E2 e E S
VW
3
Suppose
S c AQ(v,S',p)
i)
cl E n supp S =
2)
E
then
E ~ ES
and
E c Rn \ s u p p
S ,
E
open.
If
bounded
Proof Assume
K c R n \ supp S ,
x ° .... , x m ¢ K
(16) If
K c m = 0
Assume
[ J o~i~m
K = E . It follows
then
pair wise different, B(xi
from
(2) that
*)
such that
, ¢x./2) i
E c K c B(Xo
, gx /2) o
and the proof is completed.
m = 1 . Denote E1 = B(Xl
then
compact,
x ~ K : ~ ex > 0 : B ( x , ex ) c E S
V Assume
K
E1 , E2
and
' gXl ) '
E2 = B(x°
fulfil
the conditions
E~
and due to (16) the proof Assume
' gXo ) ,
E i = B(x 1 , eXl / 2 )
in Theorem
7, therefore
~
o E2 e ES
is again completed.
m = 2 . Denote E 1 = B ( x 2 , ex2 )
,
E2 = B(x °
, eXo ) u B(x 1 , eXl / 2 )
,
E~ = B ( x 2 , ¢x2 / 2 ) then
S
fulfil
vanishes
on
the conditions
is completed
B(x,e)
and as seen
in Theorem
above,
7, therefore
also
on
E2 . Moreover,
Ei u E2 ~ ES
again.
The a b o v e p r o c e d u r e
*)
E1
= { y ~ Rn
can be used
for
I ][y-xl[
< e}
any
for
m ¢ N , m > 3
x
~ Rn
,
VW
e > 0
E 1 , E2
and
E~
and due to (16) the proof
128 Corollary 4 Suppose
(17)
S ~ AQ(_v,s ' ,p]
V
Kc
~
E £ ES :
Rn\
H ,
and K
Hc
Rn
then
supp S c H
only if
compact:
KcE
Proof Assume (17) and x % supp S
x c Rn\H
since
E
into account that
then,
x c E
for a certain
E c E S . Therefore
is open. The converse results directly from Corollary 3 taking
supp S
is closed
W7
Corollary 5 Suppose V
S c AQ(v,S',p) E c Rn ,
E
, then
supp S = $ , only if
open, bounded:
E e ES
Proof It follows directly from Corollary 3
VW
Theorem 8 Suppose
S c AQ(v,s ' ,p)
,[ V
and
F c Rn ,
$ E D(Rn\F) n:
F
closed. Then
] supp S c F
$.
o ~ A~(V,S ' , p )
s:
In the case of sectional algebras (see chap. i, §6) the converse implication is also valid.
Proof Assume
x ( Rn\F • Since
and a neighbourhood
V
$ • S = 0 (AQ(v,S',p) (18)
F of
is closed, it follows that there exists x
such that
@ = 1
on
, therefore, given any representation
S = s + IQ(v(p),S
')
(AQ(v,s',p)
• with
one obtains
(193
u($) • s c IQ(v(p),S ')
But~ (18) and (19) imply (203
S = 11(1-$) • s + IQ(v(p),S') c AQ(v,S',p)
$ (D(RnXF)
V . Then, due to the hypothesis
s c AQ(v(p),S ')
129
Denoting imply
t = u(1-~)
x ~ supp S
• s , it follows that
t(v) = 0
on
V ,
and the first part of T h e o r e m 8 is proved.
Assume now, in the case of sectional algebras the inclusion • D(Rn\F)
W ~ ¢ N . Then (20) will
. Since
supp ~
is compact,
supp S c F
and
Corollary 4 implies the existence of
E ¢ ES
such that (21)
supp ~ c E
Due to the fact that
S
vanishes on
E , one can assume that
s
in (18) satisfies the
condition (22)
s(w) = 0
for c e r t a i n me
~ = 0
imply
on
E ,
p e N . Now, the p r e s e n c e of sectional in (22). Then,
Two d e c o m p o s i t i o n r u l e s supports
algebras makes" it p o s s i b l e to assu-
(21) and (22) will result in
~ • S = 0 • AQ(v,S',p)
of their
~ m P ,
w ¢ N ,
V
u(~)
• s c 0
hence
(18) will
VW
for the elements of the algebras,
corresponding
to components
a r e g i v e n now.
Theorem 9 In t h e c a s e o f s e c t i o n a l with
F
closed,
K
algebras
suppose
compact and
S e AQ(v,S',p)
. If
supp S = F u K
F n K = ~ , then the decomposition holds
S = SF + SK for certain
S F , S K ¢ AQ(v,S',p)
I)
supp S F n supp S K =
2)
K n supp S F =
3)
F n s u p p SK = ~
and
satisfying the c o n d i t i o n s
s u p p SK
compact
Proof Assume
G1 , G2 , G3 , G4 c R n
cl G 3 c G4 • K1
cl G 4 n F = ~
is compact and
such that
such that
and
G4
K c GI ,
is bounded.
cl G 1 c G 2 •
Denote
cl G 2 a G 3 ,
K 1 = (cl G4) \ G 1 , then
K 1 n supp S = ¢ . A c c o r d i n g to C o r o l l a r y 4, there exists
E eE S
K1 c E . Then, for a certain r e p r e s e n t a t i o n
S = s + IQ(V(p),S ')
e AQ(v,S',p)
s(v) = 0
V
,
with
s e AQ(v(p),S ')
,
one obtains
w i t h suitably chosen
on
E ,
on
Rn\G4 )
~ >- p
~ ¢ N . But the case of sectional algebras allows t h e choice of
H = 0 . Now, Lemma 1 g r a n t s ~F -- 1
~ ¢ N ,
~F = 0
the existence on
c l G3 ,
of
~F ~ C~(Rn)
~K = 1
on
c l G1
and and
~K ~ D(Rn) OK = 0
such that on
Rn\G2 •
130
Then, obviously S K : U(~K)
u(~ F + ~ K ) • s = s . Defining
• s + IQ[v(p),S')
and 3) will be satisfied
In the one dimensional First,
S = SF + S K
, one obtains
• s + IQ(v(p),S ')
and the properties
and i), 2)
VVV
case
n = 1 , a stronger decomposition
a notion of sepamation
for pairs of subsets
are called finitely separated, vering
S F = U(@F)
in
can be obtained.
R 1 . Two subsets
F , L c R1
only if there exists a finite number of intervals
co-
F u L (-~
, CO )
such that no interval
•
(C O
,
C1
contains
.....
)
elements
(c m , ~
F
of both
vals do not contain elements of the same set
)
F
,
and
or
me L
N, while successive
inter-
L .
T h e o r e m 10 In the case of sectional with
F , L
disjoint,
algebras
S ~ AQ(v,S',p)
suppose
closed and finitely
separated,
. If
supp S : F u L
then the decomposition
holds S = SF + SL for certain
satisfying
S F , S L ~ AQ(v,S',p)
I)
supp S F n supp S L =
2)
L n supp S F = F n supp S L =
the conditions
Proof F , L
Since
are closed, F u L c
there exists
(-~,Co-e)
u
g > 0
(Co+e,Cl-e)
such that
u ... u (Cm+e,~)
Denote K = ~ J [ci-~,ci+e] o_<"
K
compact
such that
and
K c R 1 \ supp S . According to Corollary
4, there exists
E E ES
K c E . Assume a representation S = s + IQ(v(p),S ') ~ AQ(v,s',p)
,
with
s e AQ(v(p),S ')
such that s(V) = 0 for certain gebras.
on
E ,
V
~ c N ,
D e N . But, one can assume
Further,
~ ->
D = 0 , due to the presence
Lemma 1 implies the existence of
~o = 1
on (-oO,Co-e/21 ,
~i = 1 ....
~o
''" "' ~m+l c C°°(RI)
on [ c i + ~ / 2 , c 2 - e / 2 1 , ~el
: 1
of sectional
on
[c + e / 2 , °~]
...
al-
such that
131
and
~o
= 0
on [c o-~/3,+ m)
'
~i = on (-~, Cl+E/3] o [c2-~/3 , m) ....
~m+l = 0
on
(-~,
, ...
Cm+~/3]
Denote J
o = (-~'
Co]
'
J1 = [Co
'Cl]
.....
Jm+l
= [Cm ' ~ )
and define ~F =
It
can
easily
Z ~i ogi~m+l FnJ.= ~ I
be
seen
that
and
S L = U(~L)
§4.
THE EQUIVALENCE
If
S = 0 c AQ(v,S',p)
~L =
'
Z ~i l~i~n+l LnJ.= i
u(~ F + ~L)S = s • Defining
S F = U(~F)
• s + IQ(v(p),S ') , the required properties
BETWEEN
S = 0
supp S = ~ . A result on the converse
then, obviously
WV
will follow easily
=~
supp s
AND
• s + IQ(v(p),S ')
impli-
cation is given in: Cor011ary 6 In the case of sectional i)
supp S =
2)
S
then
S = 0 e AQ(v,S',p)
vanishes
algebras
S e AQ(v,s ',p)
suppose
outside of a bounded
subset of
. If
Rn ,
.
Proof Assume
B a Rn ,
B
bounded,
such that
vanishes
S
on
R n \ B . Then,
there exists a
representation S = s + IQ(v(p),S t] c AQ(v,s ',p)
,
with
s ~ AQ(v(p),S ') ,
such that s(~) for a certain Assume (23) But
= 0
on
Rn \
B ,
V
~
~
N ,
v e ~
p c N . Due to the presence of sectional
~ ~ D ( R n)
such that
~ = 1
on
, algebras,
B , then obviously
u(~)
one can assume P = 0 • s = s , that is
~ • S = S E AQ(v,S',p) supp ~ n supp S = ~
and
• S = 0 e A Q • The relation
supp S
is closed,
(23) completes
therefore
the p r o o f
WV
Theorem 8 will imply
Chapter
8
NECESSARY STRUCTURE OF THE DISTRIBUTION MULTIPLICATIONS
§i.
INTRODUCTION
The algebras containing completions A
the distributions
of the smooth functions
is a subalgebra in
That construction
on
in
Rn
W = N ÷ C~{R n) , while
functions,
noti
their connection with certain zero-filters I
A
, where
.
itself is a subalgebra in
an essential
feature of the ideals
generated by subsets of
specific connections
in the case of the quotient
the distributions.
between
N
×R n
I
is
on which
ideals and zero-filters
constructions
will be
giving the algebras containing
In that way, an information on the necessary
stribution multiplications
§2.
is an ideal in
as sequential A/I
vanish.
In the present chapter, established
I
W
C ° ( N × R n , C I) . As known in that context,
in
were constructed
can naturally be placed within the framework of the theory of alge-
bras of continuous
the functions
D'(R n)
and resulted as quotients
structure of the di-
will be obtained.
ZERO SETS AND FAMILIES
For a sequence of smooth functions Z(w) = { (~,x) £ N and call it the zero set of
w 6 W
denote
x R n I w(~)(x)
w .
For a set of sequences of smooth functions Z(H) = { Z
] t ~H
and call it the zero family of
A standard argument
= 0 }
H
H c W
denote
} .
in algebras of functions
gives:
Theorem 1 Suppose
(V,S')
ted by NxR
n
V . Then .
is a regularization Z(1)
and denote by
and in particular,
Z(V)
I
the ideal in
W
genera-
are filter generators on
133
Proof First we p r o v e the r e l a t i o n (1)
Z(v)
Assume it w ¢ W which
is
by
~ ~ ,
false,
then
w(w)(x)
results
in
v
v ~ z v(v)(x)
= 1/v(w)(x) I(V,W)
= I(V#) $ 0 ,
,
V v ¢ N , x ~ Rn
g v ~ N , x ~ Rn
= W . Then
(20.3)
and
, therefore,
. Thus
(20.1)
in
u(1) chap.
one can define
= v • w ¢ 1,
will
I(V,W)
contradict
each other. Now, we can p r o v e (2)
Z(Vo)
Indeed, define jugate of (3) But,
n ...
w ¢ W
n Z(Vh)
by
~ ~ ,
w = v
o
. v
V
v ° ,..., v h ( I ( V , W )
* o + ... + v h • v h
where
V*i
is the complex con-
v i . Then, o b v i o u s l y Z(w) = Z(Vo)
w c I(V,W)
, since
n
...
n Z(Vh)
v ° ,..., v h ( I ( V , W )
. Therefore,
(3) and (i) will imply
(2)
VVV
The f o l l o w i n g three theorems r e f o r m u l a t e in terms of zero sets the results in Theorems 6, 7 and 8 in chap. i, §i0. Theorem 2 S u p p o s e given a local type r e g u l a r i z a t i o n
(V,S') , with
VQS'
sectional in-
variant. T h e n Z(v) for any
n ({~,U+I,...}
v ( V ,
U c N
×G) and
is infinite G c Rn ,
G $ ~ , open.
Theorem 3 Suppose given a r e g u l a r i z a t i o n ant. Then, the zero set Z(v)
n [ J ((v} I/_<~)
Z(v)
(V,S')
VQS'
such that
of each
v c V
x supp t(w)) infinite,
is sectional
invari-
has the property:
V
t ~ V(~S'
,
t~V,
U(N
Theorem 4 Suppose
(V,S')
is a r e g u l a r i z a t i o n .
Then, the zero set
Z(v)
of each
has the property: Z(v)
n U
~(N
({x~}
× supp
t(,))
~ ~ ,
W
t ~ VQS'
,
t ~ V
v ~ V
134
§3
ZERO SETS AND FAMILIES AT A POINT
The results
in Theorems
I, 2, 3, and 4 in §2, will be strengthened
in the present
sec-
tion. Given
w ~ W
,
H c W
and
x c R n , call
Zx(W ) = { , c N [ w ( , ) ( X ) the zero set
of
w
Zx~)
at
x
and c a l l
= { Zx(t)
the zero family of
H
at
I t ¢H
, the zero families
First,
several
Suppose
E
}
x .
It will be shown in Corollaries
(V,S')
= 0 }
i, 2 and 3 that for a large class of P-regularizations
Zx( ~
of
V
at any
x c Rn
are filter generators
on
N
notations.
is a set of subsets
WE={W{W
in
Rn
and denote
%/ xE c~EE : :
t
f Call
E
hereditary, %/
only if
x E E c E :
E \ {x} [ ~ Denote by sets in
Ef
and
E
c
~
E \ {x}
c E
the set of all nonvoid
Rn . O b v i o u s l y
Ef
and
Ec
and finite,
respectively
countable
are hereditary.
Theorem 5 Suppose given a vector
subspace
V
in
W , satisfying
the condition
vc~ Then, f o r any
vo
%/ E ~ E
:
xcE
:
Zx(Vo) n . . . If
E
is hereditary, %/
E ~E:
@
E'c E :
vh e V
, . . . ,
the property holds
n Zx(V h) # then the stronger
I)
car E' = car E
2)
Zx(Vo) n . . .
n Zx(Vh) ~ ~ ,
property
V
results
x c E'
for a certain
E c Ec
sub-
135 Proof It follows
Corollary
WV
from Len~aa 1 b e l o w
1
Suppose (4)
@ ~
Zx(V)
v
v E v
Then, the zero f a m i l y
Zx(Y )
on
(V,S')
the P-regularization ,
,
x
of
c
satisfies
the condition
Rn
g
at any
x c Rn
is a filter generator
N .
Proof It is easy to notice that
Corollary
(4] is equivalent with
V c W~ f
VVV
2
If under the conditions
in Corollary i,
zero family
V
Zx(V )
subsets of
of
at any
Y
is sectional
x c R n , generates
invariant,
a filter of infinite
N .
Proof Assume it is false and
(s)
v'o ,..., v
(6)
~ W
v:
¢ V , since
P c N
such that
by 1
if
v -
vi(v) (y)
if
~_>p+l
v. ~ V
and
V
is sectional
i
Zx(Vo) n . . . n Zx(V~) ~ 0
due t o C o r o l l a r y But,
and
v~(~] (y) =
i
(7)
v ° ,..., v h ~ Y
Zx(V o) n ... n Zx(V h) ~ {0 ..... ~}
Define
Then
x ~ Rn ,
1.
(5) implies that ¥
v ~ N ,
~ -> p + i :
i~ ¢ {0,...,h}
:
vi (~))(x) ~ 0 w h i c h together with
Call a subset
M
{~+I
for a certain
(6) is contradicting
of
N
, ~+2
p ~ N .
sectional, ,
...}
c
M
(7)
only if
VVV
then the
invariant.
Therefore
136
Corollary 3 If u n d e r the conditions in C o r o l l a r y i,
V
is subsequence invariant
i, §6), then the zero family
V
at any
ter of sectional subsets of
Zx(V )
of
(see chap.
x e R n , generates a fil-
N .
Proof It suffices to show that the sets nal in
Zx(V ) , with
N . Assume it is false and
tional in
v • V
and
x ~ Rn
N . Then, there exists an infinite subset
(8)
and
v c V
x • Rn
such that M
of
are all sectioZx(V )
is not sec-
N , such that
Zx(V) n M =
Assume
M = {Po ' Pl " ' ' }
(9)
v'(v)(y)
Then
v' ~ V , since
(10)
Zx(V' ) #
and d e f i n e
= v(~ v) (y)
g
•
v • V and
v ' c W by v c N
,
y e Rn
V i s subsequence i n v a r i a n t .
Therefore
due to C o r o l l a r y I. But,
(8) implies that V
~ c N
:
v(pv) (x) * 0 which t o g e t h e r w i t h (9) i s c o n t r a d i c t i n g
(10)
VW
And now, a lemma of a general interest, used in the'proof of T h e o r e m 5, is presented. Given a n o n v o i d set X , denote by For
w c W
and
x c X
W
the set of all functions
w
: N x X + C1
denote
Zx(W ) = { v • N [ w(v,x) = 0 } For a set
Y
of n o n v o i d and countable subsets in
Wy = { w ~ W
g~
yY •cY Y :
X , denote
t
Zy(W) ~
Lemma 1 Suppose
V
is a v e c t o r subspace in
v ° ,..., v h ~ V , the relation holds
(II)
V
Y~y
~
y c Y : Zy(Vo]
:
n ... n Zy(Vh) #
W
and
V c ~
. Then, for any
/37
If Y
is hereditary, V
Y~Y
:
y'c
(12)
then (ii) obtains the stronger form
y :
(12.1)
car
Y'
= car
Y
(12.2)
Zy(Vo)
n ...
n Zy(Vh)
$ ~ ,
V
y ~ Y'
Proof First, we prove
Y c Y
(ii). Assume it is false and
Zy(Vo) n ... n Zy(Vh) = ~ ,
V
such that
ycY
Then V
~c
N,
ye
Y :
i~,y c {O,...,h}
(13)
v.
1
(~,y)
:
~ o
~,y
Define Rh+l ~ X = (~o ''''' Xh)
+
v% = ~oVo + ... + XhV h E V
and N x y ~ It
is
easy
to
notice
(~,y)
+
that
Av,y
kv,y
= { X c Rh + l
, with
(v,y)
I vX(V,Y)
= 0 }
c N x y , are
vector
subspaces
i n Rh + l .
Moreover A v , y ~ Rh + l
(14) Indeed,
denoting
X = (0 . . . .
tion, one obtains Now,
,
X ~ k ,y
V
(V,y)
e N x y
,0,1,0,...,0)
e Rh + l
, with
1
in the
iv,y+l -th posi-
due to (13).
(14) and the Baire category argument will give
U
a , y ~ ah+l
>cN ycY since
Y
is
countable.
Assume therefore
% c R h+l \ [__J A ~6N v'Y y~Y then vx(V,y)
(is) But
v~ c V ¢ ~
Assume
Y
such that
$ 0 ,
therefore
V
V c N ,
y ~ Y
(15) is contradicted
is hereditary and take
Y c Y
and (ii) is proved.
• Then (ii) implies the existence of
Zy(Vo) n ... n Zy(vh) ¢ ~ . Denote
Y1 = Y \ {y} " If
Y1 = ~
y c Y
then taking
138
Y' = Y , the proof is completed. If
Y1 ~ ~
hereditary. Now, (ii) can be applied to
then
Y] , etc.
Y1 EY VW
due to the fact that
Y
is
139
REFERENCE
i.
Acker A., Walter W.: The quenching problem for nonlinear parabolic differential equations. Springer Lecture Notes in Mathematics, ~oi. 564, 1-12
2.
Alexandrov A.D.: Dirichlet's problem for the equation Vestinik Leningrad Univ. Ser. Mat. Mek. Astr. 13,1,1958
3.
Amrein W.O., Georgescu ?.~ Strong aaymptotik completeness of wave operators for highly singular potentials. Helv. Phys. Acta, 47,1974, 517-533
4.
Antosik P., Mikusinski J., Sikorski R.:
5.
Baumgarten D., Braunss G., Wagner 0.: An extension of the Gel'fand-Shilov regularization method and its application to the construction of causal solutions for nonlinear wave equations. Preprint, 1975
6.
Bogoliubov ~N., Parasiuk 0.S.: Ueber die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder. Acta Math., 97,1957, 227-266
7.
Bellini C.G., Giambiagi J.J., Gonzales Dominguez A.: Analytic regularization and the divergencies of quantum field theories. Nuovo Cim., 31,3,1964, 550-561
8.
Braunss G.: 70-80
9.
Braunss G.: Causal functions of nonlinear wave equations. 1971, 86-92
det(zi.) = ~(z I .... ,zn) J
Theory of distributions. Elsevier, 1973
On the regularization of functional equations.
Math. Ann., 186,1970
J. Diff. 5q., 9,1,
i0.
Braunss G.: Weak solutions and junction conditions of a certain class of nonlinear partial differential equations. Preprint 1971
II.
Braunss G., Liese R.: Canonical products of distributions and causal solutions of nonlinear wave equations. J. Diff. Fxl., 16,1974, 399-412
12.
Bredimas A.: Sur une theorie electromagnetique non lineaire. 6,14,1973, 569-572
13.
Bredimas A.: Generalized convolution product and canonical product between distributions. Some simple applications to physics. Lett. Nuovo Cim., 13,16, 1975, 601-604
14.
Bredimas A.: La differentiation d'ordre complexe, le produit de convolution generalise et le produit canonique pour les distributions. C.R. Acad. Sc. Paris, 282,1976, A, 37-40
15.
Bredimas A.: Applications en Physique du theoreme d'inversion de la transformation spherique de Radon dans Rn . C.R. Acad. Sc. Paris, 282,1976, A, 1175-1178
16.
Bredimas A.: Extensions, proprietes complementaires et applications des operateurs de differentiation a gauche e t a droite d'ordre complexe. C.R. Acad. Sc. Paris, 283,1976, A, 3-6
17.
Bredimas A.: Applications a certaines equations differentielles des premier et second ordre a coefficients polynomiaux des operateurs de differentiation d'ordre complexe a gauche e t a droite. C.R. Acad. Sc. Paris, 283,1976, A, 337-340
Lett. Nuovo Cim.,
140
18.
Bredimas A.: La differentiation d'ordre complexe et les produits canoniques et la convolution generalise. C.R. Acad. Sc. Paris, 283,1976, A, 1095-1098
19.
Bremermann H.J., Durand L.: On analytic continuation, multiplication and Fourier transform of Schwartz distributions. J. Math. Phys., 2,2,1961, 240 258
20.
Bremermann H.J.: On finite renormalization constants and the multiplication of causal functions in perturbation theory. Preprint, 1962
21.
Bremermann H.J.: Some remarks on analytic representations and products of distributions. SIAM J. Appl. Math., 15,4,1967, 929-943
22.
Burrill C.W.:
23.
Carmignani R., Schrader K.: Subfunctions and distributional inequalities. SIAM J. Math. Anal., 8,1,1977, 52-68
24.
Conley C.C., Smoller J.A.: Shock waves as limits of progressive wave solutions of higher order equations. Comm. Pure Appl. Math., 24,1971, 459-472
25.
Conway E.D.: The formation and decay of shocks for a conservation law in several dimensions. Arch. Rat. Mech. Anal., 64,1,1977, 47-57
26.
Dafermos C.M.: Quasilinear hyperbolic systems that result from conservation laws. Leibovich S., Seebass A.R. (ed.): Nonlinear Waves, Cornell Univ. Press, 1974, 82-102
27.
Davies E.B.: Eigenfunetion expansions for singular Schroedinger operators. Arch. Rat. Mech. Anal., 63,3,1977, 261-272
28.
Delft P., Simon B.: On the decoupling of finite singularities from the question of asymptotic completeness in two body quantum systems. Preprint, 1975
29.
De Jager E.M.: Divergent convolution integrals in electrodynamics. Rep. Dept. Math. Univ. Calif. Berkeley, 1963
30.
De Mottoni P., Texi A.: On distribution solutions for nonlinear differential equations: nontriviality conditions. J. Diff. Fq., 24,1977, 355-364
31.
Di Perna R.J.: Singularities of solutions of nonlinear hyperbolic systems of conservation laws. Arch. Rat. Mech. Anal., 60,1976, 75-100
32.
Douglis A.: An ordering principle and generalized solutions of certain quasi linear partial differential equations. Comm. Pure Appl. Math., 12,1959, 87-112
33.
~wards R.E.:
34.
Ehrenpreis L.: Solutions of some problems of division. 1954, 883-903
35.
Fisher B.: The product of distributions. 291-298
36.
Fisher B.: The product of the distributions Camb. Phil. Soc. 71,1972, 123-130
x+-r-I/2
37.
Fisher B.: The product of the distributions Camb. Phil. Soc. 72,1972, 201-204
x
Foundations of Real Nhmbers.
Functional Analysis.
Mc Graw Hill, 1967
ONR, Tech.
Holt, 1965 Amer. J. Math., 76,
Quart. J. Math. Oxford, 22,2,1971,
-r
and
and
~
x -r-I/2 _
(r-l)
(x) .
Proc.
Proc.
141
38.
Fisher B.: Some results on the product of distributions. 73,1973, 317-325
39.
Fisher B.:
Singular products of distributions.
Proc. Camb. Phil. Soc
Math. Ann., 203,1973, 103-116
-r x+ 6(r-l)(x) . Stud. Sci. Math.
40.
Fisher B.: The neutrix distribution product Hung., 9,1974, 439-441
41.
Fisher B.: Distributions and the change of variable. Math. Rom., 19,1975, 11-20
42.
Fleishman B.A., Mahar T.J.: Boundary value problems for a nonlinear differential equation with discontinuous nonlinearities. Math. Balkanica, 3,1973, 98-108
43.
Fleishman B.A., Mahar T.J.: A perturbation method for free boundary problems of elliptic type. Preprint, 1976
44.
Flugge S.: Practical quantum mechanics I. Die Grundlehren der Mathematischen Wissenschaften. Band 177, Springer, 1971
45.
Fountain L., Jackson L.: A generalized solution of the boundary value problem. Pacif. J. Math., 12,1962, 1251-1272
46.
Fuchssteiner B.: Eine assoziative Algebra ueber einem Unterraum der Distributionen. Math. Ann., 178,1968, 302-314
47.
Galambos J.: Representation of real numbers by infinite series. ture Notes in Mathematics, vol. 502, 1976
48.
Garabedian P.R.:
49.
Garnir H.G.: Solovay's axiom and functional analysis. in Mathematics, vol. 399, 1974, 189-204
50.
Gel'fand J.M.: Some problems in the theory of quasilinear equations. Mat. N~uk, 14,1959, 87-158
51.
Glimm J., Lax P.D.: Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs of AMS, nr. 101, 1970
52.
Golubitsky M., Schaeffer D.G.: Stability of shock waves for a single conservation law. Adv. Math., 15,1975, 65-71
53.
Gonzales Dominguez A., Scarfiello R.: Nora sobre la formula Rev. Union Mat. Argen., I, 53-67, 1956
54.
Guettinger W.: Quantum field theory in the light of distribution analysis. Phys. Rev., 89,5,1953, 1004-1019
55.
Guettinger W.: Products of improper operators and the renormalization problem of quantum field theory. Progr. Theor. Phys., 13,6,1955, 612-.626
56.
Guettinger W.: Generalized functions and dispersion relations in physics. Fortschr. Phys., 14,1966, 483-602
57.
Guettinger W.: Generalized functions in elementary particle physics and passive system theory: recent trends and problems. SIAM J. Appl. Math., 15,4, 1967, 964-1000
58.
Guckenheimer J.:
An unsolvable equation.
Bull. Math. Soc. Sci.
Springer Lec-
Proc. AMS, 25,1970, 207-208 Springer Lecture Notes
Uspehi
v.p. 1 . 6 = 1 6' ~ -~
Solving a single conservation law. Preprint, 1973
142 59.
Harten A., Hyman J.M., Lax P.D., Keyfitz B.: ons and entropy conditions for shocks• 297-322
60.
Hepp K.: Proof of the Rogoliubov-Parasiuk theorem on renormalization. Math. Phys. 2,1966, 301-326
61.
Hirata y., Ogata H.: On the exchange formula for distributions. Univ., 22,1958, 147-152
62.
Hopf E.: The partial differential equation Math., 3,1950, 201-230
63.
Horvat J.: An introduction to distributions. 227-240
64.
Hoermander L.:
65.
Hoermander L.: An introduction to complex analysis in several variables. Van Nostrand, 1966
66.
Itano M.: On the multiplicative product of distributions. 29,1965, 51-74
67.
Itano M.: On the multiplicative products of 29,1965, 225-241
68.
Itano M.: On the theory of the multiplicative products of distributions. J. Sc. Hirosh. Univ., 30,1966, 151-181
69.
Itano M.: Remarks on the multiplicative product of distributions. Math. j., 6,1976, 365-375
70.
Jeffrey A.: The propagation of weak discontinuities in quasilinear symmetric hyperbolic systems. J. Appl. Math. Phys., 14,1963, 301-314
71.
Jeffrey A.:
72.
Jelinek J.: Sur le produit simple de deux distributions. rol., 5,4,1964, 209-214
73.
Kang H., Richards J.: Preprint, 1975
74.
Kang H.: A general definition of convolution for several distributions. Preprint, 1975
75.
Keisler H.J.:
76.
On finite difference approximatiComm. Pure Appl. Math., 29,1976,
u t + uu x = ~Uxx .
Comm.
J. Sc. Hirosh.
Comm. Pure Appl.
Amer. Math. Month., March 1970,
Linear Partial Differential Operators.
x~
and
Quasilinear hyperbolic systems and waves.
Springer, 1963
J. Sc. Hirosh. Univ.
B • J. Sc. Hirosh. Univ. x+
Hirosh.
Pitman, 1976 Comm. Math. Univ• Ca-
A general definition of convolution for distributions.
Element~ry Calculus.
Prindle,
1976
4 Kelemen P.J., Robinson A.: The nonstandard ~ : ~2(x] : model. I. The technique of nonstandard analysis in theoretical physics. II. The standard model from a nonstandard point of view. J. Math. Phys., 13,12,1972, 1870-1878
77.
Kelemen P.J.: Quantum mechanics, quantum field theory and hyper quantum mechanics. Springer Lecture Notes in Mathematics, vol. 369, 1974, 116-121
78.
Keller K.: Irregular operations in quantum field theory I, II, III, IV, V Preprint, 1977
79.
Keyfitz Quinn B.: Solutions with shocks: an example of an migroup. Comm. Pure Appl. Math., 24,1971, 125-132
L 1 contractive se-
143
80.
Knops R.J.: Comments on nonlinear elasticity and stability. Notes in Mathematics, vol. 564, 1976, 271-290
81.
Koenig H.:
82.
Koenig H.: Multiplikation und Variablentransformation in der Theorie der Distributionen. Arch. Math., 6,1955, 391-396
83.
Koenig H.: Multiplikationstheorie der verallgemeinerten Distributionen. Bayer. Akad. Wiss. Math. Nat. KI. Abh. (N.F.), 82,1957
84.
Larra Carrero L.: Stability of shock waves. matics, vol. 564, 1976, 316-328
85.
Laugwitz D.: Eine Einfuehrung der 6 Funktionen. turw. KI. 1959, 41-59
86.
Multiplikation yon Distributionen.
Springer Lecture
Math. Ann., 128,1955,
420-452
Springer Lecture Notes in Mathe-
Bayer. Akad. Wiss. Math. Na-
Laugwitz D.: Anwendungen unendlich kleiner Zahlen. I. Zur Theorie der Distributionen. II. Ein Zugang zur Operatorenrechnung von Mikusinski. J.f.d. reine und angew. Math., 207,1961, 53-60, 208,1961, 22-34
87.
Laugwitz D.: Bemerkungen zu Bolzano Groessenlehren. 1965, 398-409
88.
Laugwitz D.: Eine nichtarchimedische Erweiterung angeordneter Koerper. Nacht., 37,1968, 225-236
89.
Lax P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm. Pure Appl. Math., 7,1954, 159-193
90.
Lax P.D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM Regional Conference Series in Appl. Math., nr. II, 1973
91.
Leibovich S., Seebass A.R.
92.
Levine H.A., Payne L.E.: Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations. J. Math. Anal. Appl., 55,1976, 329-334
93.
Lewy H.: An example of a smooth linear partial differential equation without solution. Ann. Math., 66,2,1957, 155-158
94.
Ligeza J.: On generalized solutions of some differential nonlinear equations of order n . Ann. Pol. Math., 31,1975, 115-120
95.
Lightstone A.H.:
96.
Lightstone A.H., Kam Wong: Dirac delta functions via nonstandard analysis. Canad. Math. Bull., 81,5,1975, 759-762
97.
Lightstone A.H., Robinson A.: Non Archimedean fields and asymptotic expansions. North Holland, 1975
98.
Liverman T.P.G.: Physically motivated definition of distributions. Appl. Math., IS,1967, 1048-1076
99.
Lojasiewicz S.: Sur la valeur d'une distribution dans un point. Polon. Sc., 4,1956, 239-242
I00.
(ed.): Nonlinear Waves.
Infinitesimals.
Arch. Hist. Ex. Sc., 2,
Math.
Corneil Univ. Press, 1974
Amer. Math. Month., March 1972, 242-251
SIAM J.
Bull. Acad.
Lojasiewicz S.: Sur la valeur et la limite d'une distribution dans un point. Stud. Math., 16,1957, 1-36
144
i01.
Lojasiewicz S.: Sur la fixation des variables dans une distribution. Math., 17,1958, 1-64
Stud.
102.
Luxemburg W.A.J.: What is Nonstandard Analysis? part II, 38-67
103.
Malgrange B.: Existence et approximation des solutions des equations aux derivees partielles et des equations de convolution. Ann. Inst. Fourier Grenoble, 6,1955-56, p.271-375
104.
Meisters G.H.: Translation invariant linear forms and a formula for the Dirac measure. J. Funct. Anal., 8,1971, 173-188
i05.
Mikusinski J.: Sur la methode de generalisation de Laurent Schwartz et sur la convergence faible. Fund. Math., 35,1948, 235-239
106.
Mikusinski J.: 163-169
107.
Mikusinski J.: Criteria of the existence and of the associativity of the product of distributions. Stud. Math., 21,1962, 253-259
108.
Mikusinski J.: On the square of the Dirac delta distribution. Pol. S~., 14,9,1966, 511-513
109.
Mikusinski J.: 315-325
110.
Mikusinski J., Sikorski R.: The elementary theory of distributions I, II. Rozprawy Matematyczne, 12,1957 and 25,1961
Iii.
Moss W.F.: A remark on convergence of test functions. 20, A, 1975, 73-76
112.
Nokanishi N.: Complex dimensional invariant delta functions and lightcone singularities. Comm. Math. Phys., 48,1976, 97-118
113.
01einik O.A.: On the uniqueness of the generalized solution of the Cauchy problem for a nonlinear system of equations occuring in mechanics. Uspehi Mat. Nauk., 78,1957, 169-176
114.
Palamodov V.P.: Linear Differential Operators with Constant Coefficients. Springer, 1970
115.
Pearson D.B.: An example in potential scattering illustrating the breakdown in asymptotic completeness. Comm. Math. Phys., 40,1975, 125-146
116.
Pearson D.B.: General theory of potential scattering with absorption at local singularities. Preprint, 1975
117.
Plesset M.S., Prosperetti A.: Bubble dynamics and cavitation. Fluid Mech., 9,1977, 145-185
118.
Pogorelov A.V.: 1964
119.
Pogorelov A.V.: On a regular solution of the n dimensional Minkowski problem. Soviet Math. Dokl., 12,1971, 1192-1196
120.
Pogorelov A.V. : The Dirichlet problem for the n dimensional analogue of the Monge Ampere equation. Soviet Math. Dokl., 12,1971, 1727-1731
Amer. Math. Month., 80,6,
Irregular operations on distributions.
Germs and their operational calculus.
Stud. Math., 20,1961,
Bull. Acad.
Stud. Math., 26,1966,
Monge Ampere equation of elliptic type.
J. Austral. Math. Soc.
Ann. Rev.
Noordhoff, Groningen,
145
121.
Portnoy S.L.: On solutions to Appl., 55,1976, 291-294
u t = &u + u 2
122.
Reed M.C.: Abstract Nonlinear Wave Equations. matics, vol. 507, 1976
123.
Richtmyer R.D.: On the structure of some distributions discovered by Meisters. J. Funct. Anal., 9,1972, 336-348
124.
Robinson A.:
125.
Rosinger E.E.: Embedding the D'(R n) distributions into pseudotopological algebras. Stud. Cerc. Mat., 18,5,1966, 687-729
126.
Rosinger E.E.: Pseudotopological spaces. Embedding the D'(R n) into algebras. Stud. Cerc. Mat., 20,4,1968, 553-582
127.
Rosinger E.E.:
A distribution multiplication theory.
128.
Rosinger E.E.:
Division of distributions.
129.
Rosinger E.E.: An associative, commutative distribution multiplication I, II. Preprint, 1976
130.
Rosinger LE.: Nonsymmetric Dirac distributions in scattering theory. ger Lecture Notes in Mathematics, VOlo 564, 1976, 391-399
131.
Rosinger ~.E.: Nonlinear shock waves and distrihution multiplication. Preprint, 1977
132.
Sabharwal C.L.: Multiplication of singularity functions by discontinuous functions in Schwartz distribution theory. SIAM J. Appl. Math., 18,2,1970, 503-510
133.
Schaeffer D.G.: A regularity theorem for conservation laws. 1973, 368-386
134.
Schmieden C., Laugwitz D.: Eine Erweiterung der Infinitesimalrechnung. Zeitschr., 69,1958, 1-39
135.
Schwartz L.: Sur l'impossibilite de la multiplication des distributions. C.R. Aead. Sc. Paris, 239,1954, 847-848
136.
Shiraishi R., Itano M.: On the multiplicative products of distributions. J. Sc. Hirosh. Univ., 28,1964, 223-235
137.
Shiraishi R.: On the value of distributions at a point and the multiplicative products. J. Sc. Hirosh. Univ., 31,1967, 89-104
138.
Shiu Yuen Cheng, Shing Tung Yau: On the regularity of the Monge Ampere equation. Comm. Pure Appl. Math., 30,1977, 41-68
139.
Sikorsky R.: A definition of the notion of distribution. 2,5,1954, 209-211
140.
Simon B.: Quantum mechanics for Hamiltonian defined as quadratic forms. Princeton Univ. Press, 1971
141.
Slowikowski W.: A generalization of the theory of distributions. Pol. Sc., 3,1,1955, 3-6
Nonstandard Analysis.
in two dimensions.
J. Math. Anal.
Springer Lecture Notes in Mathe-
North Holland, 1966
distributions
Preprint, 1974
Pacif. J. Math., 66,1,1976, 257-263
Sprin-
Adv. Math., 11,3,
Math.
Bull. Acad. Pol. Sc.
Bull. Acad.
146
142.
Slowikowski W.: On the theory of operator systems. 1955, 137-142
143.
Speer LR.:
144.
Stroyan K.D., Luxemburg W.A.J.: Acad. Press, 1976
14S.
Struble R.A.: An algebraic view of distributions and operators. 37,1971, 103-109
146.
Stuart C.A.: Differential equations with discontinuous nonlinearities. Rat. Mech. Anal., 63,1,1976, 59-75
147.
Stuart C.A.: Boundary value problems with discontinuous nonlinearities. Springer Lecture Notes in Mathematics, vol. 564, 1976, 472-484
148.
Szaz A.: Convolution multipliers and distributions. 267-275
149.
Tavera G., Burnage H.: Sur les ondes de choc dans les ecoulements instationnaires. C.R. Acad. Sc. Paris, 284,1977, A, 571-573
150.
Temple G.: Theories and applications of generalized functions. Soc., 28,1953, 134-148
151.
Thurber J.K., Katz J.: Applications of fractional powers of delta functions. Springer Lecture Notes in Mathematics, vol. 369, 1974, 272-302
152.
Tillmann H.G.: Darstellung der Schwartz'schen Distributionen durch analytische Funktionen. Math. Z., 77,1961, 106-124
153.
Treves F.: Locally Convex Spaces and Linear Partial Differential Equations. Springer, 1967
154.
Treves F.: Applications of distributions to PDE theory. March 1970, 241-248
155.
Treves F.:
Linear Partial Differential Equations.
156.
Treves F.:
Basic Linear Partial Differential Equations.
157.
Van Osdol D.H.: rigorous.
158.
Van Rootselaar B.: Bolzano's t~eory of real numbers. 2,1964, 168-180
159.
Vol'pert A.J.: On differentiation and quasilinear operators in the space of functions whose generalized derivatives are measures. Soviet Math. Dokl. 7,6,1966, 1586-1589
160.
Vol'pert A.J.: The spaces BV and quasilinear equations. vol. 2,2,1967, 225-267
161.
Zel'dovich Ja. B.: Hydrodynamics of the universe. 1977, 215-228
162.
Walter H.F.: Ueber die Multiplikation von Distributionen in eine~ Folgenmodell Math. Ann., 189,1970, 211-221
163.
Whitham G.B.:
Analytic renormalization.
Bull. Acad. Pol. Sc., 3,3,
J. Math. Phys., 9,9,1968,
1404-1410
Introduction to the theory of infinitesimals.
Pacif. J
Stud. Math.,
Arch.
Math., 60,2,1975
J. London Math.
Amer. Math. Month.,
Gordon, Breach, 1970 Acad. Press, 1975
Truth with respect to an ultrafilter or how to make intuition Amer. Math. Month., April 1972, 355-363
Linear and Nonlinear Waves.
Arch. Hist. Exact Sc.,
Math. USSR Sbornik,
Ann. Rev. Fluid Mech., 9,
Wiley, 1974
