Methods of applied Fourier analysis
A key object of signal processing is the mathematical description of signals, be it for numerical calculation, compression, filtering, or holographic storage. In this context, Fourier analysis has always played a dominant role. Linear expansions are present in Fourier's original work on the inversion of the Fourier transform [\textit{P. Gabriel}, Matrizen, Geometrie, Lineare Algebra. Birkhäuser Advanced Texts, Birkhäuser, Basel (1996; Zbl 0847.51001)], in Haar's construction of the first wavelet, in Gabor's Nobel prize winning work on optical holography, in clinical magnetic resonance tomography [\textit{J. Jin}, Electromagnetic Analysis and Design in Magnetic Resonance Imaging. CRC Press, Boca Raton, London, New York (1999)], as well as in the Kustaanheimo-Stiefel regularization procedure of spinorial celestial mechanics [\textit{K. Stumpff}, Himmelsmechanik, Band III. VEB Deutscher Verlag der Wissenschaften, Berlin (1974; Zbl 0296.70006)]. In present day signal processing, Fourier analysis is of central importance for fast algorithms such as the fast Fourier transform (FFT) as well as in the applications to image processing and video compression. Two of the novel additions to the field of signal processing have been wavelets and their cousins, filter banks or subband encoding [\textit{M. Vetterli} and \textit{J. Kovačević}, Wavelets and Subband Coding. Prentice Hall, Englewood Cliffs, NJ (1995; Zbl 0885.94002)]. The book under review presents a mathematical introduction to a selection of these topics accessible to graduate students in mathematics with a background of analysis. The first chapter includes a treatment of the basic \(L^1\) and \(L^2\) Fourier analysis together with important ancillary topics such as the Fourier analysis of measures. Chapter 2 forms an introduction to the theory of Hardy spaces. The structure of inner and outer functions is presented along with a proof of the Szegő-Kolmogorov theorem. Chapter 3 is devoted to the prediction theory of discrete stationary stochastic processes including a discussion of the spectral theory of stationary processes, whereas the fourth chapter studies the connections of Fourier series and discrete control theory including Nehari's theorem and the commutant lifting theorem. Chapters 5 and 6 are again of foundational nature by dealing with harmonic analysis on the vector space \(\mathbb{R}^n\) and Schwartz distribution theory. Chapter 6 includes a rudimentary treatment of Sobolev spaces, too. The results of Chapter 7 are centered around the Paley-Wiener theorem [\textit{R. M. Young}, An introduction to nonharmonic Fourier series, Academic Press, New York, (1980; Zbl 0493.42001)]. Chapter 8 deals with a study of phase-frequency analysis and the Heisenberg uncertainty principle of quantum physics. The subject of the final chapter is a study of wavelet analysis which includes the basic ideas of multiresolution approximation. Because the author restricts himself to commutative Fourier analysis which excludes a serious application to quantum physics, the book fails to present more advanced topics of signal processing such as the holographic transform. This powerful technique of image processing allows the construction of filter banks as it is used in quantum holography by the clinical scanners of magnetic resonance tomography [\textit{W. J. Schempp}, Magnetic Resonance Imaging: Mathematical Foundations and Applications. Wiley-Liss, New York (1998)]. Moreover, it presents the noncommutative background for the Weil-Brezin transform of the compact Heisenberg nilmanifold, or Zak transform of solid state physics, studied in Chapter 8 without any convincing motivation. This objection points to the main drawback of the book. For many topics the motivation rooted in physics, information theory and computer science is missing and only the experienced reader will be able to appreciate the historical origin and the power of the methods exposed in the text.
discrete stationary stochastic processes
scientific article; zbMATH DE number 1201222