Discrete Fourier Analysis
This textbook presents basic notions and techniques of Fourier analysis in discrete settings. Written in a concise style, it is interlaced with remarks, discussions and motivations from signal analysis. The first part is dedicated to topics related t
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Managing Editor M.W. Wong (York University, Canada)
Editorial Board Luigi Rodino (Università di Torino, Italy) Bert-Wolfgang Schulze (Universität Potsdam, Germany) Johannes Sjöstrand (Université de Bourgogne, Dijon, France) Sundaram Thangavelu (Indian Institute of Science at Bangalore, India) Maciej Zworski (University of California at Berkeley, USA)
Pseudo-Differential Operators: Theory and Applications is a series of moderately priced graduate-level textbooks and monographs appealing to students and experts alike. Pseudo-differential operators are understood in a very broad sense and include such topics as harmonic analysis, PDE, geometry, mathematical physics, microlocal analysis, time-frequency analysis, imaging and computations. Modern trends and novel applications in mathematics, natural sciences, medicine, scientific computing, and engineering are highlighted.
M.W. Wong
Discrete Fourier Analysis
M.W. Wong Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3 Canada [email protected]
2010 Mathematics Subject Classification: 42-01, 47-01, 42A45, 47G30, 65T50, 65T60 ISBN 978-3-0348-0115-7 e-ISBN 978-3-0348-0116-4 DOI 10.1007/978-3-0348-0116-4 Library of Congress Control Number: 2011929860 © Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained.
Cover design: deblik Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com
Contents
Preface 1 The Finite Fourier Transform
vii 1
2 Translation-Invariant Linear Operators
17
3 Circulant Matrices
23
4 Convolution Operators
27
5 Fourier Multipliers
33
6 Eigenvalues and Eigenfunctions
37
7 The Fast Fourier Transform
41
8 Time-Frequency Analysis
45
9 Time-Frequency Localized Bases
53
10 Wavelet Transforms and Filter Banks
61
11 Haar Wavelets
67
12 Daubechies Wavelets
79
13 The Trace
87
14 Hilbert Spaces
95
15 Bounded Linear Operators
107
16 Self-Adjoint Operators
113
17 Compact Operators
117
vi
Contents
18 The Spectral Theorem
121
19 Schatten–von Neumann Classes
125
20 Fourier Series
129
21 Fourier Multipliers on S1
141
22 Pseudo-Differential Operators on S1
151
23 Pseudo-Differential Operators on Z
163
Bibliography
171
Index
175
Preface Fourier analysis is a prototype of beautiful mathematics with many-faceted applications not only in mathematics, but also in science and engineering. Since the work on heat flow of Jean Baptiste Joseph Fourier (March 21, 1768–May 16, 1830) in the treatise entitled “Th´eorie Analytique de la Chaleur”, Fourier series and Fourier transforms have gone from triumph to triumph, permeating mathematics such as partial differential equations, harmo
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