A FIRST COURSE
IN WAVELETS WITH
FOURIER ANALYSIS
Second Edition
ALBERT BOGGESS
Texas A&M University
College Station.
TX
Department of Mathematics
FRANCIS .J. NARCOWICH
Texas A&M University
College Station.
TX
Department of Mathematics
ffiWILEY
A JOHN WILEY
&
SONS, INC., PUBLICATION
Copyright
©
2009 by John Wiley
&
Sons, Inc. All rights reserved
Published by John Wiley
&
Sons, Inc. Hoboken, New Jersey
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any
form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise,
except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, with
out either the prior written permission of the Publisher, or authorization through payment of the
appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,
MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests
to the Publisher for permission should be addressed to the Permissions Department, John Wiley
&
Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at
http://www.wiley.com/go/permission.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts
in preparing this book, they make no representations or warranties with respect to the accuracy
or completeness of the contents of this book and specifically disclaim any implied warranties of
merchantability or fitness for a particular purpose. No warranty may be created or extended by sales
representatives or written sales materials. The advice and strategies contained herein may not be
suitable for your situation. You should consult with a professional where appropriate. Neither the
publisher nor author shall be liable for any loss of profit or any other commercial damages, including
but not limited to special, incidental, consequential,
or
other damages.
For general information on our other products and services or for technical support, please contact
our Customer Care Department within the United States at (800) 762-2974, outside the United States
at (317) 572-3993 or fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print
may not be available in electronic format. For more information about Wiley products, visit our web
site at www.wiley.com.
Library of Congress Cataloging-in-Publication Data:
Boggess, Albert.
p. cm.
A first course in wavelets with fourier analysis
I
Albert Boggess, Francis J. Narcowich. - 2nd ed.
Includes bibliographical references and index.
ISBN 978-0-470-43117-7 (cloth)
I.
Wavelets (Mathematics) 2.
Fourier analysis.
I. Narcowich, Francis J. II. Title.
QA403.3.B64 2009
515'.2433-dc22
2009013334
Printed in United States of America.
IO
9 8 7 6 5 4 3 2 I
CONTENTS
Preface and Overview
0
Inner Product Spaces
ix
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Motivation, 1
Definition of Inner Product, 2
The
.l
Spaces
L2
ti
and
4
4
0.3 Defini ons,
2
0.3.2 Convergence in
L
Versus Uniform Convergence,
8
Schwarz and Triangle Inequalities, 11
Orthogonality,
tio
13
and Examples, 13
0.55..12 Orthogonal Projections, 15
Defini ns
0.
5.3 Gram-Schmidt Orthogonalization, 20
0.
Lin
6
ear Operators and Their
21
Adjoints, 21
0.
6.
.
2
1
Adjoints, 23
Linear Operators,
0.
Least1 Squares and Linfor Data, c25ve Coding, 25
ti
0.
7
7.
.2
Best-Fit
Least
ear Predi
Algorithm, 29
Line
Squares
0.7.3 GeneralPredictive Coding, 31
0. Linear
Exercises, 34
12,
vi
CONTENTS
1
Fourier Series
38
1 . 1 Introduction, 38
I
. 1 .1 Historical Perspective, 38
1.1. 2 Signal Analysis, 39
Partial Differential Equations, 40
I
.1.3
1 .2 Computation of Fourier Series, 42
x :S
42
1.2. l On the Interval
1.2.2 Other Intervals, 44
1.2.3 Cosine and Sine Expansions, 47
1 .2.4 Examples, 50
1.2.5 The Complex Form of Fourier Series, 58
-rr :S
rr,
1 .3 Convergence Theorems for Fourier Series, 62
1 .3.1 The Riemann-Lebesgue Lemma, 62
1.3.2 Convergence at a Point of Continuity, 64
1.3.3 Convergence at a Point of Discontinuity, 69
1.3.4 Uniform Convergence, 72
1 .3.5 Convergence in the Mean, 76
Exercises, 83
2
The Fourier Transform
92
2.1 Informal Development of the Fourier Transform, 92
2.1 . 1 The Fourier Inversion Theorem, 92
2.1.2 Examples, 95
2.2 Properties of the Fourier Transform, 101
2.2. 1 Basic Properties, 101
2.2.2 Fourier Transform of a Convolution, 107
2.2.3 Adjoint of the Fourier Transform, 109
2.2.4 Plancherel Theorem, 109
2.3 Linear Filters, 1 10
2.3.1 Time-Invariant Filters, 1 10
2.3.2 Causality and the Design of Filters, 115
2.4 The Sampling Theorem, 120
2.5 The Uncertainty Principle, 123
Exercises, 127
3
Discrete Fourier Analysis
132
3.1 The Discrete Fourier Transform, 132
3. 1 .
l
Definition of Discrete Fourier Transform, 134
3.1. 2 Properties of the Discrete Fourier Transform, 135
3. 1 .3 The Fast Fourier Transform, 138
京公网安备 11010802033920号
Copyright © 2005-2024 EEWORLD.com.cn, Inc. All rights reserved
评论