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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Many physical, chemical, biological and even economic phenomena can be modeled by differential or partial differential equations, and the framework of distribution theory is the most efficient way to study these equations. A solid familiarity with the language of distributions has become almost indispensable in order to treat these questions efficiently.
This book presents the theory of distributions in as clear a sense as possible while providing the reader with a background containing the essential and most important results on distributions. Together with a thorough grounding, it also provides a series of exercises and detailed solutions.
The Theory of Distributions is intended for master's students in mathematics and for students preparing for the agrégation certification in mathematics or those studying the physical sciences or engineering.
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Seitenzahl: 298
Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Table of Contents
Title Page
Copyright Page
Preface
Introduction
Chapter 1. Topological Vector Spaces
1.1. Semi-norms
1.2. Topological vector space: definition and properties
1.3. Inductive limit topology
Chapter 2. Spaces of Test Functions
2.1. Multi-index notations
2.2.
C
∞
function with compact support
2.3. Exercises with solutions
Chapter 3. Distributions on an Open Set of ℝ
d
3.1. Definitions
3.2. Examples of distributions
3.3. Convergence of sequences of distributions
3.4. Exercises with solutions
Chapter 4. Operations on Distributions
4.1. Multiplication by a
C
∞
function
4.2. Differentiation of a distribution
4.3. Transformations of distributions
4.4. Exercises with solutions
Chapter 5. Distribution Support
5.1. Distribution restriction and extension
5.2. Distribution support
5.3. Compact support distributions
5.4. Exercises with solutions
Chapter 6. Convolution of Distributions
6.1. Definition and examples
6.2. Properties of convolution
6.3. Exercises with solutions
Chapter 7. Schwartz Spaces and Tempered Distributions
7.1.
S
(ℝ
d
) Schwartz spaces
7.2. Tempered distributions
7.3. Exercises with solutions
Chapter 8. Fourier Transform
8.1. Fourier transform in
L
1
(ℝ
d
)
8.2. Fourier transform in
S
(ℝ
d
)
8.3. Fourier transform in
S
′
(ℝ
d
)
8.4. Exercises with solutions
Chapter 9. Applications to ODEs and PDEs
9.1. Partial Fourier transform
9.2. Tempered solutions of differential equations
9.3. Fundamental solutions of certain PDEs
Appendix
References
Index
Other titles from iSTE in Mathematics and Statistics
End User License Agreement
List of Tables
Chapter 8
Table 8.1. Operations and Fourier transform
Table 8.2. The Fourier transform of a tempered distribution
Guide
Cover
Table of Contents
Title Page
Copyright Page
Preface
Introduction
Begin Reading
Appendix
References
Index
End User License Agreement
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Series EditorNikolaos Limnios
The Theory of Distributions
Introduction
El Mustapha Ait Ben Hassi
First published 2023 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2023The rights of El Mustapha Ait Ben Hassi to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2023938463
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-937-2
Preface
This book is a collection of course notes written in the spirit of presenting distributions in as clear a sense as possible, without overburdening them with even more interesting notions, and not to deviate from the main objective, which is to provide students with relatively simple course supporting material that contains the most essential and important results on distributions. The reader interested in more details on distributions will find a bibliographic list given at the end of the book.
P.1. Summary
This book is organized into nine chapters and an appendix, and almost all of them comprise a theoretical foundation in the form of definitions, propositions and theorems illustrated by examples, and a series of exercises with their answers.
In Chapter 1, the essentials on topological vector spaces are presented, which will allow building topology on the space of test functions.
Chapter 2 is dedicated to the presentation and study of c∞: the space of test functions ends with some density results that will be useful later on.
Chapter 3 introduces the notion of distributions through the presentation of equivalent definitions and various examples.
Chapter 4 addresses some operations on distributions, namely multiplication by a C∞ class function, differentiation of distributions, and classical transformations on distributions.
Chapter 5 is dedicated to the notion of distribution support, its properties allowing a classification of distributions according to their support.
In Chapter 6, the fundamental principle of convolution regularization is presented. More precisely, distribution convolution is defined and its properties are exposed.
In order to define temperate distributions in Chapter 7, Schwartz spaces S(ℝd) are first defined and some properties are given that will be needed in the second part of this chapter, where the space of temperate distributions S′(ℝd) is presented as well as its fundamental properties.
Given the importance of the Fourier transform, Chapter 8 is dedicated to this notion in different functional spaces, namely the space L1(ℝd), the Schwartz space S(ℝd) and finally the dual space S′(ℝd) of temperate distributions.
Chapter 9 presents an opportunity to apply all this theory exposed for the solution of some differential equations.
An appendix concludes the book with a series of synthesis exercises that can be used for self-evaluation.
P.2. Targeted audience
This book is intended for master’s students in mathematics as well as for students of the physical sciences. It is also intended for students preparing for post-graduate degrees for the teaching of mathematics. It equips them with the most elementary definitions and notions that will help them familiarize with distributional calculus such as the notions of derivation, limit of a sequence or series of distributions as well as other operations on distributions. Through a range of well-chosen examples, they will be able to grasp the difference between what is classical for ordinary functions and what is related to distributions when considering what is usually referred to as “generalized functions”, which are themselves distributions.
Finally, I would like to thank all the people who helped me, in one way or another, in the preparation of this book. I would especially like to thank my students and their contribution to preparing these notes. I will be grateful to those of my readers who will send me their comments about this first edition.
April 2023
Introduction
The emergence of the theory of distributions can be traced back to 1945–1950 and is due to Laurent Schwartz. This theory provides a general setting with a pleasant formalism for studying functional spaces and partial differential equations. Since then, distributions are the natural framework that many analysts use basing themselves on Schwartz’s notations and ideas. A good familiarity with the language of distributions has become almost essential to any analyst. Schwartz himself was well aware that the main merit of his approach was not in the introduction of new tools, but in a clear and accessible synthesis of multiple recipes that had already been employed in various contexts.
There are several reasons for introducing the concept of distribution. Some of them are purely physical (even experimental) reasons, while others are more mathematical ones; these latter, on the one hand, consist of giving meaning to some objects manipulated in physics, as for example the mysterious Dirac function introduced by Dirac (1929) which equals 0 anywhere, except at 0 where it equals + ∞, and whose integral is equal to 1 (contrarily to all of the rules of Lebesgue’s theory of integration). Not only did Dirac use this function for formal computation purposes, but he would also derive it at will, simply observing that the successive derivatives were increasingly singular. On the other hand, they consist of exposing the operations, especially differentiation operations, carried out within the setting of partial differential equations.
In order to motivate the generalization of the point aspect of functions and make it possible to move from the notion of function to the notion of functional, a classical physical example is given here related to the measurement of the temperature of a straight wire “at a given point”. Understandably and for obvious reasons, such a measurement is never perfectly feasible. Any thermometer, regardless of the physical principle used for the measurement, presents a spatial extension that cannot possibly be reduced to that of a point: what should be achieved to be able to measure the temperature at a point x0. It can nonetheless be admitted that, in the case of a realistic temperature measurement, the thermometer takes into account all the temperatures in a “neighborhood”, of the point x0 according to a sensitivity function φ0 so that for a distribution function T (x) of the temperature along the bar (a function about which it is a priori not known what it is really equal to at a specific point of the bar), it can be said that the measured temperature T will be in fact If the measurement would be taken at another point x1, it would yield
It can be seen that the measured temperature T , assuming that the functions T (x), φ0, φ1, etc., are sufficiently regular, appears as a linear expression in the sensitivity function φ. This expression can generously be expressed in the form:
When the product x → T (x)φ(x) is Lebesgue integrable, this notation is perfectly justified. However, in many practical cases, the physical quantity being considered (which here would be T (x)) proves to be too singular for the written integral to have any meaning with a realistic choice of the function φ. The set of functions φ is called the set of test functions or trial functions. The measure of a physical quantity T is then represented by the “bracket”: 〈T, φ〉 independently or not of an integral form for this expression. The set of measurable quantities T by the test functions is then generically called a distribution.
What should be the required minimum for the objects thus considered?
1) The set of test functions constitutes a vector space of functions. That is, any linear combination with complex coefficients of test functions is still a test function.
2) The distribution set is the set of continuous linear forms on the vector space of the test functions.
3) The result of the measure of
T
by
φ
is then the real or complex number 〈
T, φ
〉.
One of the other main interests of the theory of distributions is to allow the construction of a differential calculus that extends the ordinary differential calculus and for which any distribution is indefinitely differentiable. This theory has become an essential tool, especially in the study of partial differential equations. It has also allowed for a mathematical modernization for many physical phenomena.
