Methods of Applied Fourier Analysis

1998
ISBN: 978-0-8176-3963-1
Verlag: Birkhäuser Boston

Buch, Englisch, 329 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 1450 g

Reihe: Applied and Numerical Harmonic Analysis

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.

Zielgruppe

Research

Autoren/Hrsg.

Ramanathan, Jayakumar

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

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.