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Sampling theory in Fourier and signal analysis: foundations. (English) Zbl 0872.94010

Oxford: Clarendon Press. xiii, 222 p. (1996).
The book under review is devoted to one of the most interesting and exciting subjects of signal and systems theory: sampling theory and its connections with Fourier and signal analysis. It is hard to imagine present (and future) communications without sampling which provides the possibility of using discrete and quantized signals instead of analogous ones when transmitting information through various channels.
First, we give the chapter headings: (1) An introduction to sampling theory, (2) Background in Fourier analysis, (3) Hilbert spaces, bases and frames, (4) Finite sampling, (5) From finite to infinite sampling series, (6) Sampling for Bernstein and Paley-Wiener spaces, (7) More about Paley-Wiener spaces, (8) Kramer’s lemma, (9) Contour integral methods, (10) Irregular sampling, (11) Errors and aliasing, (12) Single channel and multi-channel sampling, (13) Multi-band sampling, (14) Multi-dimensional sampling, (15) Sampling and eigenvalues problems, modelling, uncertainty stable sampling, Appendix A: Fourier transforms, Appendix B: Hilbert transforms, references and an index.
Before discussing the developed subjects, we remark two special features of the book: a) there is no figure in the text, it is written in an entirely analytical language, b) the text is written by use of LATEX and its beautiful graphics reveal the power and the advantages of this system.
The main merit of the book consists in presenting relevant mathematical facts permitting a deep understanding of sampling. This has been possible as the author is an active and outstanding worker in signal theory (for more than a quarter of the century). It is worth pointing out that from the beginning he presents the foundations of Hilbert spaces and especially, unusual to standard expositions, objects such as Riesz spaces, frames and reproducing kernels which allow to grasp the refinement of sampling theory. The kernel of the book is the study of Bernstein and Paley-Wiener spaces and entire functions of exponential growth. This approach clearly explains to the reader that these spaces are exactly those for which an interpolation problem leads to the whole reconstruction of the initial function. An important aspect related to many applications is what is meant by stable and irregular sampling (Chapters 10, 11). The sampling of band-pass signals or multiband ones, an operation of high importance in practice, is discussed in Chapters 12, 13 and 14. The last chapter is devoted to a very acute problem related to sampling (perhaps the most important one from the physical point of view). It is suggestively expressed as: modelling (of real signals), uncertainty (the product band-width \(X\) duration depasses always a constant depending on the family of signals taken into consideration) and stable sampling. Although of few pages, it provides the fundamental result of sampling theory, namely, the Landau theorem giving the value of the minimal sampling rate. The work contained in the last chapter leads to the important concept of time and frequency concentration. The contributions on this subject (here they are only stated) were obtained by Landau, Pollak and Slepian in a series of celebrated papers devoted to prolate spheroidal wave functions. This point of view has been proved to be very productive leading to time-frequency representations of signals and to wavelet theory.
The book is written in a classical style (Hilbert and \(L_2\) spaces) and the distributional point of view is quoted only. Finally, we think this volume provides a splendid and original exposition of the past, present and the future of sampling theory in terms of classical analysis. Applied mathematicians and research engineers will have at their disposal a basic and reference book in the area of signal analysis. We are awaiting the author’s forthcoming book devoted to advanced topics in Fourier and signal analysis.

MathOverflow Questions:

Origin of the term ”sinc” function

MSC:

94A12 Signal theory (characterization, reconstruction, filtering, etc.)
94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
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