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Fourier analysis and its applications. (English) Zbl 0786.42001

The Wadsworth & Brooks/Cole Mathematics Series. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software. x, 433 p. (1992).
The book is a detailed and very readable treatise on the theory and practice of series expansions and transforms. The primary audience consists of advanced undergraduates in mathematics, physics, and engineering, but the book is also a useful reference for more advanced workers. The underlying motivation throughout is the study of differential equations, especially the basic equations of physics (heat, Laplace, wave), and Sturm-Liouville problems. The presentation consistently exhibits applications as soon as a new mathematical object is introduced. This feature certainly contributes to its strength as a text.
Here is a more detailed discussion of the chapters. Chapter 1, the “Overture”, describes the wave, heat, and Laplace equations and introduces Fourier series by means of separation of variables. Chapter 2 contains a systematic treatment of Fourier series embellished with helpful graphs, a table of twenty expansions and a partial sum convergence theorem for piecewise smooth functions. This is followed by applications to heat flow. There is a theoretical discussion of inner- product spaces, completeness and related inequalities in Chapter 3. The theory is illustrated by the regular Sturm-Liouville problem. Chapter 4 is a serious treatment of boundary value problems for rectangles, sectors, and disks. Here one can find the Dirichlet kernel, and vibrations of a rectangular membrane. Chapter 5 covers the Bessel functions (type \(J\), \(Y\)) from their differential equations through asymptotics and zeros to their use in vibrations of circular membranes. Orthogonal polynomials of classical type (Legendre, Hermite, and Laguerre) are studied in Chapter 6 by means of Rodrigues’ relations and generating functions. The emphasis is on the use of these polynomials in the solution of partial differential equations. However, there is no mention of the unifying role of the hypergeometric series here. There is a concise introduction to Haar and Walsh functions and wavelets. Chapter 7 on Fourier transforms is a methodical introduction, with a minor dependence on the concept of Lebesgue integrability, but with mostly a practical emphasis on how convolution acts as a smoothing operator (with the usual helpful graphics) and how the Fourier transform is related to signal analysis. This chapter also treats the heat equation and the sampling theorem. Chapter 8 defines the Laplace transform and shows how to solve a variety of ordinary and partial differential equations. The last two chapters concern distributions, weak solutions of differential equations and Green’s functions for a large collection of problems.
The strength of the book comes from its careful presentation of theory followed by detailed applications, with good illustrations, and finally a generous collection of exercises (with answers). The prose is smooth and gives understandable discussions of technical difficulties. The references are mostly to books, and there is an adequate subject index.
This text can surely be recommended for use in a one or two semester course, or as a reference for graduate students or other persons who want to see what sort of problems Fourier analysis was invented to solve.

MSC:

42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
33-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35L05 Wave equation
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
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