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Generalized functions. Theory and technique. (English) Zbl 0538.46022

Mathematics in Science and Engineering, Vol. 171. New York-London etc.: Academic Press, a Subsidiary of Harcourt Brace Jovanovich, Publishers. XIII, 428 p. $ 58.00 (1983).
[The complete version of this abrigded review is available on demand.]
The purpose of this book is to present the basic concepts of the theory of distributions of L. Schwartz and then to use this analysis in applications. First let us survey the content of the book. To motivate the concept of distribution, the Dirac delta ”function” and delta sequences are introduced in Chapter 1. Various sequences \(\{f_ m(x)\}\) are considered whose limit is the delta ”function” in the sense that \[ \lim_{m\to \infty}\int^{\infty}_{-\infty}f_ m(x)f(x)dx=f(0) \] for functions f(x) which are ”sufficiently smooth” in \(-\infty<x<\infty\). In Chapters 2 and 3 the space of test functions \({\mathcal D}\) and the space of distributions \({\mathcal D}'\) of L. Schwartz are introduced over \(R^ n\); the usual distributional operations and definitions, such as derivative and support, are given; and additional facts concerning distributions, including transformation properties of the delta distribution, convergence of sequences and series of distributions, and distributional convergence of Fourier series, are presented. We note that no topological vector space analysis is explicitly used in this book; neither does the author assume that the reader has knowledge of this topic. Chapter 4 contains a number of examples and calculations concerning distributions defined by divergent integrals through the Cauchy principal value and Hadamard finite part of divergent integrals techniques. The vector analysis of functions with jump discontinuities across surfaces and boundaries is developed in Chapter 5; the analysis is used later in the book to solve problems in the potential, scattering, and wave propagation theories. Distributional derivatives of functions at jump discontinuities of the function are studied in Chapter 5. The functions of rapid decrease \({\mathcal S}\) and the tempered distributions \({\mathcal S}'\) of L. Schwartz are defined in Chapter 6; the culmination of this chapter is the definition of the Fourier and inverse Fourier transforms of \({\mathcal S}'\) elements with properties and examples of these transforms being given. In Chapter 7 the tensor product (direct product) and convolution of distributions are defined and studied; the use of convolution to ”regularize” distributions is indicated as is the Fourier transform property of mapping the convolution into the product of the Fourier transform. The Laplace transform of distributions in one dimension is defined in two ways in Chapter 8. The original definition given by L. Schwartz of the Laplace transform of elements in \({\mathcal D}'\) with support bounded on the left (i.e. \({\mathcal D}'\!_ R)\) is indicated. Here the existence of a real number c such that \((e^{-ct}U_ t)\in {\mathcal S}'\) for \(U\in {\mathcal D}'\!_ R\) is assumed, and the Laplace transform of U is defined by \[ <U_ t,e^{-st}>=<e^{-ct}U_ t,\lambda(t)e^{-(s-c)t}> \] where \(\lambda\) (t) is the usual \(C^{\infty}\) function which is 1 over the support of U and which has support in a neighbourhood of the support of U. In addition the author introduces ”distributions of exponential growth” which are the continuous linear functionals on a test space of \(C^{\infty}\) functions whose derivatives have exponential growth of order b; for these distributions V, the Laplace transform is simply defined to be \(<V_ t,e^{-st}>\) for real s satisfying \(b<s<\infty\), a direct extension of the classical Laplace transform.
Chapters 1-8 contain the properties and analysis of the distributions; the remainder of the book contains the applications. In Chapters 9 and 10 the author studies classical and generalized (i.e. distributional) solutions of ordinary and partial differential equations; the Laplace, heat, wave, and other fundamental operators are studied in relation to distributional solutions of partial differential equations. The analysis of partial differential equations is applied to boundary value problems in Chapter 11, and this analysis together with the vector analysis of Chapter 5 yield applications to wave propagation in Chapter 12. In Chapter 13, which extends the analysis of Chapter 5, functions which have infinite singularities at an interface are studied; this situation arises in the theories of electromagnetism and magnetohydrodynamics. Chapter 14 contains calculations concerning linear systems; and the concluding Chapter 15 contains various topics associated with distributions such as distributional weight functions for orthogonal polynomials and applications of generalized functions to probability, statistics, and economics.
The emphasis in the book is on the applications which are many and varied, and the analysis presented in some of the applications was originally due to the author and his collaborators. In the preface to the book the author states ”I have attempted to furnish a wealth of applications from various physical and mathematical fields of current interest and have tried to make the presentation direct yet informal.” As for mathematical rigor, the author states his intention for the book in the preface: ”definitions and theorems are stated precisely, but rigor is minimized in favor of comprehension of techniques.” The author has done an excellent job in presenting examples and in displaying the calculational techniques associated with distributions and the applications. Throughout the book there are a wealth of examples concerning the distributional topics and calculations introduced and concerning the applications, and the examples are presented in detail. The author has provided exercises at the end of Chapters 1-4 and 6-10 and states in the preface that these chapters are suitable for a one semester course; the remaining chapters do not have exercises. The book contains a list of references, a list of papers and books of additional reading, and has an index at the end.
Reviewer: R.D.Carmichael

MSC:

46F10 Operations with distributions and generalized functions
46F12 Integral transforms in distribution spaces
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
35Dxx Generalized solutions to partial differential equations
34Axx General theory for ordinary differential equations
34Bxx Boundary value problems for ordinary differential equations