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Zbl 1038.42001
Suslov, S. K.
An introduction to basic Fourier series. (English)
[B] Developments in Mathematics 9. Dordrecht: Kluwer Academic Publishers. xv, 369~p EUR~194.00; \$~190.00; \sterling~122.00 (2003). ISBN 1-4020-1221-7/hbk

This marvellous book, dedicated to Dick Askey with a foreword by Mizan Rahman, gives a very nice introduction to the new branch of classical analysis called Basic Fourier Series. The author is a great expert in this new field and the book is much more than just an introduction to $q$-Fourier Analysis. The book comprises, besides the foreword and a preface, twelve chapters, six appendices, a quite extensive bibliography and an index. Each chapter, except for the last one, concludes with a nice collection of exercises, which makes it suitable for a course on the subject.\par The first two chapters form a quite elementary introduction to basic exponential and trigonometric functions. Chapter~3 deals with addition theorems for these functions and in Chapter~4 the expansions in terms of these functions are discussed. The fifth chapter is an introduction to basic Fourier series, followed by a thorough investigation in Chapter~6. The latter chapter deals with, for instance, asymptotics of zeros, methods of summation and analytic continuation of basic Fourier series. Chapter~7 deals with completeness of basic trigonometric systems in general, illustrated by some important examples. Chapter~8 studies the asymptotics of zeros in more detail, which leads to improved results in comparison with the asymptotics obtained in Chapter~6. Chapter~9 deals with expansions in basic Fourier series; many explicit expansions are given of (basic) elementary special functions. In Chapter~10 the author introduces basic Bernoulli and Euler polynomials and numbers, and also a basic extension of the Riemann zeta function. Chapter~11 deals with a numerical investigation of basic Fourier series and in Chapter~12 the author gives several suggestions for further work in the theory of $q$-Fourier Analysis and related topics. The appendices comprise a selection of basic summation and transformation formulas, some theorems of complex analysis, tables of zeros of basic sine and cosine functions and some numerical examples.
[Roelof Koekoek (Delft)]

MSC 2000:: *42-02 Research monographs (Fourier analysis)
42C15 Series and expansions in general function systems
33-02 Research monographs (special functions)
33D15 Basic hypergeometric functions of one variable
33D50 Orthogonal polynomials and functions in several variables

Keywords: basic Fourier series; $q$-Fourier analysis; basic exponential functions; basic trigonometric functions; asymptotics of zeros; completeness

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