Asymptotic behaviour and Stieltjes transformation of distributions

204 Pages·1990·10.074 MB·English

by  Stevan Pilipovic

TEUBNER T
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
u^zia Band 116
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
Pilipovic • Stankovic • Takaci TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
Asymptotic
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
Behaviour
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
and Stieltjes TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
Transformation of
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
Distributions TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
TEUBNER TEXT
TEUBNER
zur Mathematik TEUBNER-
TEUBNER-
TEUBNER-
TEUBNER-
TEUBNER-
TEUBNER-
TEUBNER-
TEUBNER-
TEUBNER-
TEUBNER-
TEUBNER-
Stevan Pilipovifc, born in Novi Sad, Yugoslavia, May
24, 1950; finished study of mathematics at.the Fa­
culty of Natural Sciences in Novi Sad 1973, received
his dr. degree at the Institute of Mathematics in
Novi Sad 1979. Presently he is the Full Professor
at the Institute of Mathematics in Novi Sad.
Bogoljub Stankovifc, born in Botos, Yugoslavia, No­
vember 1, 1924; finished study of mathematics at the
Faculty of Natural Sciences in Belgrade; received
his dr. degree at the Serbien Academy of Sciences,
1954; worked as an Assistent Professor at the Uni­
versity of Paris, 1959 - 60. Presently he is the
Full Professor at the Institute of Mathematics Uni­
versity of Novi Sad. He is a member of the Serbian
Academy of Sciences and Arts and Academy of Sciences
and Arts of Vojvodina.
Arpad Takaci, born in Novi Sad, Yugoslavia, Novem­
ber 10, 1951; finished study of mathematics at the
Faculty of Natural Sciences in Novi Sad, received
his dr. degree at the Institute of Mathematics in
Novi Sad in 1982. Presently he is the Associate Pro­
fessor at the Institute of Mathematics in Novi Sad.
Pilipovifc
Asymptoti ons /
Pilipovic r,
1990. - 2
NE: Stank
ISBN 3-32
ISSN 0138
© BSB B.
VLN 294-3
Lektor: D
Printed i
Gesamther
Betrieb d
Bestell-N
02500
TEUBNER-TEXTE zur Mathematik • Band 116
Herausgeber / Editors: Beratende Herausgeber /Advisory Editors:
Herbert Kurke, Berlin Ruben Ambartzumiant Jerevan
Joseph Mecket Jena David E. Edmunds. Brighton
Rüdiger Thiele. Leipzig Alois Kufnert Prag
Hans Triebelt Jena Burkhard Monient Paderborn
Gerd Wechsungt Jena Rolf J. Nessel, Aachen
Claudio Procesit Rom
Kenji Uenot Kyoto
Stevan Pilipovié • Bogoljub Stankovió • Arpad Takaci
Asymptotic Behaviour
and Stieltjes Transformation
of Distributions
The asymptotic behaviour of solutions of mathematical models» classicál
or generalized, using Abelian and Tauberian type theorems, is of great
importance and of practical use. In the last two decades many defini­
tions of the asymptotic behaviour of distributions have been presented,
elaborated and applied to integral transformations of distributions.
The main topic of this book is to give a survey of all such definitions
to elaborate the most important, adding new results and to compare them
to point at their application to different problems, mainly to the Abel
ian and Tauberian type theorems for the Stieltjes transformation of dis
tributions. A theory and application of the equivalence at infinity,
quasiasymptotic and S-asymptotic, has been presented. The second part
of the book is devoted to the Stieltjes transformation, its real and
complex inversion formula and to the Abelian and Tauberian type theo­
rems based on the quasiasymptotic behaviour of distributions at zero
and at infinity.
1
Das asymptotische Verhalten der klassischen und allgemeinen Lösungen
von mathematischen Modellen, welche man mit Hilfe der Abelschen und
Tauberschen Theoreme erhält, ist von großer Bedeutung. In der mathema­
tischen Literatur der letzten 20 Jahre sind mehrere Definitionen des
asymptotischen Verhaltens von Distributionen zu finden. Sie alle wur­
den hauptsächlich für Integraltransformationen von Distributionen aus­
gearbeitet und auf diese angewandt. Das Hauptanliegen des vorliegen­
den Buches ist es, eine Übersicht über diese Definitionen zu geben,
die wichtigsten von ihnen unter Hinzufügung neuer Resultate auszuar­
beiten sowie miteinander zu vergleichen und ihre Anwendungsmöglichkei­
ten auf verschiedene Probleme, insbesondere auf die Abelschen und Tau­
berschen Theoreme für Stieltjes-Transformationen von Distributionen
aufzuzeigen. Besonders beachtet werden die Äquivalenz im Unendlichen,
die Quasiasymptotik und die S-Asymptotik. Der zweite Teil des Buches
ist der Stieltjes-Transformation, deren reeller und komplexer Inver­
sionsformel sowie den Abelschen und Tauberschen Theoremen unter Be­
nutzung des quasi-asymptotischen Verhaltens von Distributionen in Null
und im Unendlichen gewidmet.
Le comportement asymptotique des solutions, classiques ou générales,
obtenu en utilisant des théorèmes abêliens et taubériens, est très im­
portant et très utile. Pendant les vingt dernières années, dans la
littérature mathématique plusieurs définitions du comportement asym­
ptotique des distributions ,sont apparu. Elles sont toutes élaborées et
appliquées, tout d’abord, aux transformations intégrales des distribu­
tions. Le but essentiel de ce livre est de donner un l’aperçu de ces
definitions, d’élaborer les plus importantes d’entre elles, en dé­
veloppant, des résultats nouveaux et en les comparant mutuellement;
de montrer les possibilités de leur application aux divers problèmes,
surtout aux théorèmes abêliens et taubériens pour la transformation
de Stieltjes des distributions. On a prêté une attention particulière
à l’équivalence à l’infini, quasi-asymptotiques et S-asymptotiques.
La deuxième partie du livre est consacrée à la transformation de
Stieltjes, à la formule, réelle et complexe, de l'inversion, ainsi
qu’aux théorèmes abêliens et taubériens en utilisant le comportement
quasi-asymptotique des distributions à zéro et à l’infini.
Большое значение и практическую пользу имеет асимптотическое поведение
классических или обобщённых решений, полученных при помощи теорем типа
Абеля и Таубера. За последние двадцать лет в математической литературе
было представлено, изучено и применено к интегральным преобразованиям
обобщённых функций большое количество определений асимптотического
поведения обобщённых функций. Основной целью настоящей книги является
рассмотрение всех таких определений, изучение наиболее важных из них,
добавление новых результатов и сравнение их, применение этих результа­
тов к разным проблемам, в частности к теоремам типа Абеля и Таубера
для преобразований СтилТьеса обобщённых функций. Представлена теория и
приложения эквивалентности в бесконечности, квазиасимптотики и S-асим­
птотики. Вторая часть книги посвящена преобразованию Стилтьеса, его ве­
щественной и комплексной формуле обращения и теоремам типа Абеля и
Таубера, основанным на кваэиасимптотическом поведении обобщённых функ­
ций в нуле и бесконечности.
2
PREFACE
In the last two decades many definitions of the asymptotic behaviour of
distributions have been presented, elaborated and applied to integral transformations
of distributions. The main topic of this book is to give survey on all such
definitions, to elaborate the most important, adding new results and to compare
them; to point at their application to different problems mainly to the Abelian and
Tauberian type theorems for the Stieltjes transformation of distributions.
Chapter I gives some basic notions which are used throughout the book,
namely regularly varying functions, cones, and the Fourier, Laplace and Stieltjes
transformations of functions and distributions.
Chapter II treats, firstly, the simplest asymptotic behaviour of distributions
introduced by Lighthill [ 15 ] which can be applied only for distribution having
a continuous function as its restriction on a neighbourhood of infinity. A refined
version of this definition, called equivalence at infinity, was introduced by J.
Lavoine and O.P. Misra [12 ]. The analysis of this notion is given using
regularly varying functions [11] . The quasiasymptotic behaviour of distributions
at infinity and at zero is the content of the rest of the Chapter II. It was
introduced and analysed by a group of Soviet mathematicians with V.S.
Vladimirov, all in connection with investigations in the quantum field theory [ 83] ,
[ 76 ] . The mentioned authors have given general Abelian and Tauberian theorems
for the distributional Laplace transformation using the quasiasymptotic behaviour
of distributions. Some contributions to these results have been obtained by the
authors of the book. In the last years a theory of the S-asymptotic (shift
asymptotic) of distributions has been developed. It has an origin in the books of
L. Schwartz [48]. A theory of the S-asymptotic and its applications are presented
in Chapter III. A nice property of the S-asymptotic is that it preserves many
important operations with distributions, therefore it is easy to apply this notion.
The last chapter is devoted to the Stieltjes transformation. One can follow the
definition of this transformation, given previously by J. Lavoine and O. P. Misra
[ 12 ], [13], its generalizations and to compare other approaches to this
transformation. Using the concept of the quasiasymptotic, Abelian type theorems
have been proved for the behaviour of the transformation at O+ and at infinity.
The real and complex inversion formula for the distributional Stieltjes transformation
has been given. Tauberian type theorems for the Stieltjes transformation bring
this chapter to an end.
We would like to express our gratitude to " Teubner-Texte zur Mathematik” for
publishing this book.
Novi Sad, June 1989. S. Pilipovic, B. Stankovic and A. Takaci
3
C ОКТ T Eisr T S
CHAPTER I. SOME BASIC NOTIONS ....................................... 7
1. Regularly varying functions ....................................... 7
2. Cones ............................................................... 9
3. Fourier transformation and the convolution of tempered
distributions ....................................................... 10
4. Classical and generalized Laplace transformation ............... 12
5. The Stieltjes transformation ...................................... 14
CHAPTER II. EQUIVALENCE AT INFINITY AND QUASI ASYMPTOTIC OF
DISTRIBUTIONS ............................................. 15
1. LighthilltS definition ............................................. 15
2. Sebastiao e SilvatS order of growth of distributions .......... 17
3. Equivalence at infinity ........................................... 21
4. Quasiasymptotic behaviour of distributions at infinity ........ 29
4.1. The one dimensional case ....................................... 29
4.2. Quasiasymptotic behaviour at infinity of tempered
distributions with supports in a cone ......................... 37
5. A modification of quasiasymptotic at infinity .................. 43
6. Quasiasymptotic at O+ .............................................. 45
7. Quasiasymptotic expansion at ® and at O+ ....................... 50
8. Quasiasymptotic of Schwartz distributions at ±® ................. 53
8.1. Fundamental theorem ..........................................
8.2. Quasiasymptotic at ±® .........................................
8.3. Fourier transformation, convolution and the quasiasymptotic at±® 52
9. Quasiasymptotic at 0 ............................................... 64
9.1. Basic definition and properties .............................. 64
9.2. The structural theorem for the quasiasymptotic at 0 ....... 67
CHAPTER III. S-ASYMPTOTIC OF A DISTRIBUTION ........................ 74
1. Introduction .................................’...................... 74
2. Results of Yu. A. Brychkov........................................ 74
3. Definition and main properties of the S-asymptotic ............ 76
4. Characterization of the numerical function c(h) and the limit
distribution U ......................................................
5. S-asymptotic of some special distributions ...................... g6
6. Relation of the S-asymptotic with asymptotic, quasiasymptotic
and equivalence at infinity ....................................... 88
4
6.1. S-asymptotic and asymptotic behaviour of a function at
infinity .......................................................... 88
6.2. Relation between quasiasymptotic and S-asymptotic ........ 90
6.3. Equivalence at infinity of a distribution and the S-asym­
ptotic ............................................................ 95
7. S-asymptotic and mappings of some subset ofV 1 into P' ...... 97
8. S-asymptotic and the support of a distribution.............. 98
9. Characterization of some subspaces of V1 by the S-asymptotic 103
10. S-asymptotic in the subspaces of V1 ........................... 106
11. S-asymptotic and the Fourier transform ....................... щ
12. Application of the S-asymptotic ................................... 112
12.1. Application to partial differential equations ................. 112
12.2. Abelian and Tauberian type theorems for the Weierstrass
transform...................................................... 115
13. S-asymptotic expansion .......................................... 117
13.1. Definition of the S-asymptotic expansion ................ 118
13.2. Properties of the S-asymptotic expansion................. 119
13.3. Application of the S-asymptotic expansion to partial
differential equations ...................................... 124
14. Generalized S-asymptotic ........................................ 125
14.1. Definition and properties .................................. 125
14.2. Comparison of the generalized S-asymptotic and
S-asymptotic .................................................. 126
15. Structural theorems for the distributions having S-asymptotic 129
CHAPTER IV. STIELTJES TRANSFORMATION OF GENERALIZED FUNCTIONS . 131
1. Introduction ..................................................... 131
2. The Stieltjes transformation .................................... 132
2.1. Spaces J’(r) and I*(r) .......................................... 132
2.2. Definitions of the Stieltjes transformations Sj. and Sj. .... 134
2.3. Existence of the !^-transformation and its connection with
the Sr-transformation ........................................... 135
2.4. The Stieltjes transformation as the iterated Laplace
transformation ................................................... 139
3. Erdélyi's approach ............................................... 143
3.1. Testing function spaces ...................................... 143
3.2. The Stieltjes transformation on M1-type spaces ............. 145
4. Abelian theorems ................................................. 148
4.1. Final value Abelian theorems ...... 148
4.2. Initial value Abelian theorems ............................... 151
4.3. On Abelian type results at « at O+ .......................... 157
4.4. Asymptotic expansion of the Sj.-transformat ion................ 163
5
5. Inversion formulae ............................................... 168
5.1. Real inversion formulae ....................................... 168
5.2. Complex inversion formula ................................... 177
6. Tauberian-type results for the Sf-transformation ............ 183
6.1. A Tauberian-type theorem with the Keldysh condition......... 184
6.2. Tauberian-type results related to the quasiasymptotic
behaviour ....................................................... 185
6.3. Tauberian results for non-negative distributions ........... 190
REFERENCES ........................................................... 195
INDEX....................................... .......................... 200
6
I. SOME BASIC NOTIONS
The aim of this chapter is to introduce some notations and no­
tions (like regularly varying functions, cones and integral transforma­
tions) and state several facts on them. The content of Chapter I is de­
termined by the following ones, so it should help the reader in their
understanding. A more complete exposition on regularly varying func­
tions is given in [49], on cones in [74] and [75], and on generalized
integral transformations for instance in [48], [69], [74], or in the
recent monograph [76].
Throughout this,chapter and also the book,]N stands for the set
of natural numbers, INq = M U (0),2 is the set of integers, Ж the set
of real and (П the set of complex numbers. Further on, V (resp. S) is the
space of infinitely differentiable functions with compact support (resp.
with rapid decrease),V1 (resp. S’) its dual, namely the space of distribu­
tions (resp. tempered distributions). denotes the space of tempered
distributions with supports in the first octant Ж^ = {(x^,...,xn) e
e Жп|х^ £ 0, i = l,...,n), and, at last, £' denots the space of dis­
tributions with compact support.
1. Regularly varying functions
Regularly varying functions were defined by J. Karamata in the
early thirties as a natural generalization of power functions. The
first paper with this notion and important implications within theorems
of the Abelian and Tauberian type for the Laplace and Stieltjes trans­
formation was [11]. We shall give some elements of the theory of regu­
larly varying functions which are sufficient for this book; the reader
interested in the complete theory can consult the book [49] and the
extensive bibliography cited there. Let us start with regularly vary­
ing functions at infinity.
DEFINITION 1.1. A function p:(a,°°) Ж, a e Ж, is called *egu-
ZajJiyt varying at tn^LYUty if it is positive, measurable and there exists a
real number a such that for each x > 0
P (kx)
(1.1) Iim Xa.
к -*» p(k)
The number a is called the index of p. Specially, if a = 0, then p is
called àiowly varying at bxJ^ixdty and for such a function the letter mLm will
be used. In fact, we have
PROPOSITION 1.1. A positive and measurable function p:(a,«) -*■ Ж
7
is regularly varying at infinity if and only if it can be written as
(1.2) p(X) = xaL(x), X > a,
for some real number a and some slowly varying function L at infinity.
One can prove that the convergence in (1.1) is uniform on every
fixed interval [a',bf], a < a’ <b’<®,and that p is necessarily bounded
(hence integrable) on it. Let us state a few properties of slowly vary­
ing functions at infinity. Their proofs can be found in Seneta's book
[49].
PROPOSITION 1.2. Let L be a slowly varying function at infinity.
Then, for each e > 0
(i) there exist constants C1, > 0 and X > a such that
(1.3) C1x”E £ L(x) < C2Xe for X ¿ X;
(ii) we have Iim xeL(x) = +®, Iim x eL(x) = 0;
X-Xo X-XO
(iii) the function L1(X): = x e{sup zeL(z) : X ¿ z ¿ x}, X > a,
satisfies L(x) ^ L1(X) as x -► ®.
The first two statements together with the representation (1.2)
explain the relation of the regularly varying.functions to power func­
tions, while the third shows that such functions with a positive index
are asymptotically equal at infinity to monotone ones.
A sufficient condition for regular variation gives
PROPOSITION 1.3. If the function p is continuously differenti­
able on [a,®), then it is regularly varying if
XPt(X)
Iim a,
X-XO p(x)
the real number a being the index of p.
The assumption of differentiability of p is not at all limiting.
Namely, by [49], p. 17, for a given slowly varying function L1 there ex­
ists another infinitely differentiable slowly varying function L such that
L1(X) ^L(x) as rf-»® and L^n) = L(n) for all integers n sufficiently large.
In Chapter II we define the quasiasymptotic behaviour at zero re­
lated to a regularly varying function at zero. Now, it is obtained by
an obvious change of variable:
DEFINITION 1.2. A function p:(0,a) -► Ж, a > 0 is fiogulasiZy varying
at zafio from the right if the function P1(X) = p(l/x) is regularly vary­
ing at infinity.
In Chapter IV we need a more refined property of slowly varying
functions on the asymptotic behaviour of certain improper integrals.
8