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This volume presents a development of the ideas of harmonic analysis with a special emphasis on application-oriented themes. In keeping with the interdisciplinary nature of the subject, theoretical aspects of the subject are complemented by in-depth explorations of related material of an applied nature. Thus, basic material on Fourier series, Hardy spaces and the Fourier transform are interwoven with chapters treating the discrete Fourier transform and fast algorithms, the spectral theory of stationary processes, H-infinity control theory, and wavelet theory.
List of contents
1 Periodic Functions.- 1.1 The Characters.- 1.2 Some Tools of the Trade.- 1.3 Fourier Series: Lp Theory.- 1.4 Fourier Series: L2 Theory.- 1.5 Fourier Analysis of Measures.- 1.6 Smoothness and Decay of Fourier Series.- 1.7 Translation Invariant Operators.- 1.8 Problems.- 2 Hardy Spaces.- 2.1 Hardy Spaces and Invariant Subspaces.- 2.2 Boundary Values of Harmonic Functions.- 2.3 Hardy Spaces and Analytic Functions.- 2.4 The Structure of Inner Functions.- 2.5 The H1 Case.- 2.6 The Szegö-Kolmogorov Theorem.- 2.7 Problems.- 3 Prediction Theory.- 3.1 Introduction to Stationary Random Processes.- 3.2 Examples of Stationary Processes.- 3.3 The Reproducing Kernel.- 3.4 Spectral Estimation and Prediction.- 3.5 Problems.- 4 Discrete Systems and Control Theory.- 4.1 Introduction to System Theory.- 4.2 Translation Invariant Operators.- 4.3 H?Control Theory.- 4.4 The Nehari Problem.- 4.5 Commutant Lifting and Interpolation.- 4.6 Proof of the Commutant Lifting Theorem.- 4.7 Problems.- 5 Harmonic Analysis in Euclidean Space.- 5.1 Function Spaces on Rn.- 5.2 The Fourier Transform on L1.- 5.3 Convolution and Approximation.- 5.4 The Fourier Transform: L2 Theory.- 5.5 Fourier Transform of Measures.- 5.6 Bochner's Theorem.- 5.7 Problems.- 6 Distributions.- 6.1 General Distributions.- 6.2 Two Theorems on Distributions.- 6.3 Schwartz Space.- 6.4 Tempered Distributions.- 6.5 Sobolev Spaces.- 6.6 Problems.- 7 Functions with Restricted Transforms.- 7.1 General Definitions and the Sampling Formula.- 7.2 The Paley-Wiener Theorem.- 7.3 Sampling Band-Limited Functions.- 7.4 Band-Limited Functions and Information.- 7.5 Problems.- 8 Phase Space.- 8.1 The Uncertainty Principle.- 8.2 The Ambiguity Function.- 8.3 Phase Space and Orthonormal Bases.- 8.4 The Zak Transform and the Wilson Basis.- 8.5 AnApproximation Theorem.- 8.6 Problems.- 9 Wavelet Analysis.- 9.1 Multiresolution Approximations.- 9.2 Wavelet Bases.- 9.3 Examples.- 9.4 Compactly Supported Wavelets.- 9.5 Compactly Supported Wavelets II.- 9.6 Problems.- A The Discrete Fourier Transform.- B The Hermite Functions.
Summary
This volume presents a development of the ideas of harmonic analysis with a special emphasis on application-oriented themes.
Report
"The author has written a very careful, complete and readable introduction.... The treatment is mathematically sophisticated and precise.... This distinguishes the text and and makes it a very valuable reference for professionals interested in all relevant applications of Fourier analysis. This book would also be appropriate as a text for a graduate course in mathematics of advanced engineering."
-Mathematical Reviews
