Distributions and Nonlinear Partial Differential Equations

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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pp.

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+/ 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,'>


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,'>




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

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)
AQ(v,s' ,p)

> AQ(v,s ' ,p)
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 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) = {

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)

=

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-~

:

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 - = 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 • 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 ~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

-
~ . 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 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_ 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

:

~ 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




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

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 = -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 ¢=~ P(D)t e

UQS~

the hypothesis.

, hence

.

. But (49) implies the hypothesis.

1 .

=

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)

~ 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_ 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 :

**)

- 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

lCo 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 - 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_-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 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 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_