*(English)*Zbl 1038.42001

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.

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.

##### MSC:

42-02 | Research monographs (Fourier analysis) |

42C15 | General harmonic expansions, frames |

33-02 | Research monographs (special functions) |

33D15 | Basic hypergeometric functions of one variable, ${}_{r}{\phi}_{s}$ |

33D50 | Orthogonal polynomials and functions in several variables |