Methods of applied Fourier analysis
type: Item
instance of: scholarly article
title: Methods of applied Fourier analysis
zbMATH Open document ID: 0926.94004
author: Jayakumar Ramanathan
published in: Applied and Numerical Harmonic Analysis
publication date: 1998-09-21T00:00:00Z
review text: 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 di…
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 42-01
Mathematics Subject Classification ID: 94-01
Mathematics Subject Classification ID: 65-01
Mathematics Subject Classification ID: 93-01
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 65T50
Mathematics Subject Classification ID: 60G25
Mathematics Subject Classification ID: 60G35
zbMATH DE Number: 1201222
zbMATH Keywords: discrete control theory
zbMATH Keywords: signal processing
zbMATH Keywords: Fourier analysis
zbMATH Keywords: fast Fourier transform
zbMATH Keywords: Hardy spaces
zbMATH Keywords: inner and outer functions
zbMATH Keywords: Szegő-Kolmogorov theorem
zbMATH Keywords: prediction
zbMATH Keywords: discrete stationary stochastic processes
zbMATH Keywords: spectral theory
zbMATH Keywords: Fourier series
zbMATH Keywords: Nehari's theorem
zbMATH Keywords: commutant lifting theorem
zbMATH Keywords: harmonic analysis
zbMATH Keywords: Schwartz distribution
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: Paley-Wiener theorem
zbMATH Keywords: phase-frequency analysis
zbMATH Keywords: Heisenberg uncertainty principle
zbMATH Keywords: wavelet
zbMATH Keywords: multiresolution approximation
zbMATH Keywords: commutative Fourier analysis
reviewed by: Walter Schempp
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1201222