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WAVELETS
and OTHER
ORTHOGONAL
SYSTEMS
SECOND EDITION

Studies in Advanced Mathematics
Series Editor
STEVEN G. KRANTZ
Washington University in St. Louis
Editorial Board
R. Michael Beats
Rutgers University
Gerald B. Folland
University o f Washington
Dennis de Turck
University o f Pennsylvania
William Helton
University of California at San Diego
Ronald DeVore
University o f South Carolina
Norberto Salinas
University o f Kansas
Lawrence C. Evans
University o f California at Berkeley
Michael E. Taylor
University o f North Carolin
Titles Included in the Series
Steven R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping
John J . Benedetto, Harmonic Analysis and Applications
John J . Benedetto and Michael W. Frazier, Wavelets: Mathematics and Applications
Albert Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex
Goong Chen and Jianxin Zhou, Vibration and Damping in Distributed Systems,
Vol. 1: Analysis, Estimation, Attenuation, and Design.
Vol. 2: W K B and Wave Methods, Visualization, and Experimentation
Carl C. Cowen and Barbara D. MacCluer, Composition Operators on Spaces o f Analytic Functions
John P. D'Angelo, Several Complex Variables and the Geometry o f Real Hypersurfaces
Lawrence C. Evans and Ronald F. Gariepy, Measure Theory and Fine Properties o f Functions
Gerald B. Folland, A Course in Abstract Harmonic Analysis
Jose Garcia-Cuerva, Eugenio Hernandez, Fernando Soria, and Jose-Luis Torrea,
Fourier Analysis and Partial Differential Equations
Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem,
2nd Edition
Alfred Gray, Modern Differential Geometry o f Curves and Surfaces with Mathematica, 2nd Edition
Eugenio Hernandez and Guido Weiss, A First Course on Wavelets
Steven G. Krantz, Partial Differential Equations and Complex Analysis
Steven G. Krantz, Real Analysis and Foundations
Kenneth L . Kuttler, Modern Analysis
Michael Pedersen, Functional Analysis in Applied Mathematics and Engineering
Clark Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition
John Ryan, Clifford Algebras in Analysis and Related Topics
Xavier Saint Raymond, Elementary Introduction to the Theory o f Pseudodifferential Operators
John Scherk, Algebra: A Computational Introduction
Robert Strichartz, A Guide to Distribution Theory and Fourier Transforms
Andre Unterberger and Harald Upmeier, Pseudodifferential Analysis on Symmetric Cones
James S. Walker, Fast Fourier Transforms, 2nd Edition
James S. Walker, Primer on Wavelets and their Scientific Applications
Gilbert G. Walter and Xiaoping Shen, Wavelets and Other Orthogonal Systems, Second Edition
Kehe Zhu, A n Introduction to Operator Algebras

GILBERT G. WALTER
Department of Mathematical Sciences
The University ofWisconsin-Milwaukee
Milwaukee, Wisconsin
XIAOPING SHEN
Department of Mathematics and Computer Sciences
Eastern Connecticut State University
Willimantic, Connecticut
WAVELETS
and OTHER
ORTHOGONAL
SYSTEMS
SECOND EDITION
CHAPMAN & HALL/CRC
Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data
Walter, Gilbert G.
Wavelets and other orthogonal systems / Gilbert G.
Walter, Xiaoping Shen.— 2nd ed.
p. c m . — (Studies in advanced mathematics)
Includes bibliographical references and index.
I S B N 1-58488-227-1 (alk. paper)
1. Wavelets (Mathematics) I . Shen, Xiaoping. I I . Title. I I I . Series.
QA403 .3 .W34 2000
515,
.2433—dc21 00-050874
This book contains information obtained from authentic and highly regarded sources. Reprinted material
is quoted with permission, and sources are indicated. A wide variety o f references are listed. Reasonable
efforts have been made to publish reliable data and information, but the author and the publisher cannot
assume responsibility for the validity o f all materials or for the consequences o f their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic
or mechanical, including photocopying, microfilming, and recording, or by any information storage or
retrieval system, without prior permission in writing from the publisher.
The consent o f CRC Press L L C does not extend to copying for general distribution, for promotion, for
creating new works, or for resale. Specific permission must be obtained in writing from C R C Press L L C
for such copying.
Direct all inquiries to C R C Press L L C , 2000 N . W . Corporate Blvd., Boca Raton, Florida 33431.
T r a d e m a r k Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation, without intent to infringe.
© 2001 by Chapman & Hall/CRC
N o claim to original U.S. Government works
International Standard Book Number 1-58488-227-1
Library of Congress Card Number 00-050874
Printed in the United States o f America 1 2 3 4 5 6 7 8 9 0
Printed on acid-free paper

V
Preface to first edition
The subject of wavelets has evolved very rapidly in the last five or
six years—so rapidly that many articles and books are already obso-
lete. However, there is one portion of wavelet theory that has reached
a plateau, that is, the subject of orthogonal wavelets. The major con-
cepts have become standard, and further development will probably be
at the margins. In one sense they are no different than other orthogonal
systems. They enable one to represent a function by a series of orthogo-
nal functions. But there are notable differences: wavelet series converge
pointwise when others don't, wavelet series are more localized and pick
up edge effects better, wavelets use fewer coefficients to represent certain
signals and images.
Unfortunately, not all is rosy. Wavelet expansions change excessively
under arbitrary translations—much worse than Fourier series. The same
is true for other operators such as convolution and differentiation.
In this book wavelets are presented in the same setting as other orthog-
onal systems, in particular Fourier series and orthogonal polynomials.
Thus their advantages and disadvantages can be seen more directly.
The level of the book is such that it should be accessible to engineering
and mathematics graduate students. It will for the most part assume a
knowledge of analysis at the level of beginning graduate real and com-
plex analysis courses. However, some of the later chapters are more
technical and will require a stronger background. The Lebesgue integral
will be used throughout. This has no practical effect on the calculation
of integrals but does have a number of theoretical advantages.
Wavelets constitute the latest addition to the subject of orthogonal
series, which are motivated by their usefulness in applications. In fact,
orthogonal series have been associated with applications from their in-
ception. Fourier invented trigonometric Fourier series in order to solve
the partial differential equation associated with heat conduction and
wave propagation. Other orthogonal series involving polynomials ap-
peared in the 19th century. These too were closely related to problems
in partial differential equations. The Legendre polynomials are used to
find solutions to Laplace's equation in the sphere, the Hermite poly-
nomials and the Laguerre polynomial for special cases of Schrodinger
wave equations. These, together with Bessel functions, are special cases

vi
of Sturm-Liouville problems, which lead to orthogonal series, which are
used to solve various partial differential equations.
The arrival of the Lebesgue integral in the early 20th century allowed
the development of a general theory of orthogonal systems. While not
oriented to applications, it allowed the introduction of new systems such
as the Haar and Walsh systems, which have proven useful in signal
processing. Also useful in this subject are the sine functions and their
translates, which form an orthogonal basis of a Paley-Wiener space.
These are related to the prolate spheroidal functions, which are solutions
both to an integral equation and a Sturm-Liouville problem.
The orthogonal sequences of wavelets, which are generalizations of
the Haar system and the sine system, have a number of unique prop-
erties. These make them useful in data compression, in image analysis,
in signal processing, in numerical analysis, and in acoustics. They are
particularly useful in digitizing data because of their decomposition and
reconstruction algorithms. They also have better convergence properties
than the classical orthogonal systems.
While the Lebesgue integral made a general theory of orthogonal sys-
tems possible, it is insufficiently general to handle many of the appli-
cations. In particular, the delta "function" or impulse function plays
a central role in signal processing but is not a square integrable func-
tion. Fortunately a theory that incorporates such things appeared in
the middle of the 20th century. This is the theory of "distributions,"
due mainly to L. Schwartz. It also is related to orthogonal systems in
that it allows representation of distributions by orthogonal systems and
also allows representations of functions by orthogonal distributions.
The body of the book is divided into 13 chapters of which the first 7
are expository and general while the remaining are more specialized and
deal with applications to other areas. Each will be concerned with the
use of or properties of orthogonal series.
In Chapter One we present two orthogonal systems that are prototypes
for wavelets. These are the Haar system and the Shannon system, which
have many, but not all, of the properties of orthogonal wavelets. They
will be preceded by a section on general orthogonal systems. This is a
standard theory that contains some results that will be useful in all of
the particular examples.
Chapter Two will give a short introduction to tempered distributions.
This is a relatively simple theory and is the only type of generalized
function needed for much of orthogonal series. Many engineers still seem
to apologize for their use of a*"delta function". There is no need to do

vii
so since these are well defined proper mathematical entities. Included
here also is the associated theory of Fourier transforms that enables one
to take Fourier transforms of things like polynomial and trigonometric
functions.
Chapter Three contains an introduction to the general theory of or-
thogonal wavelets. Their construction by a number of different schemes
is given as are a number of their properties. These include their multires-
olution property in which the terms of the series are naturally grouped
at each resolution. The decomposition and reconstruction algorithms of
Mallat, which give the coefficients at one resolution in terms of others,
are presented here. Some of these properties are extended to tempered
distributions in Chapter Five.
In Chapter Four we return to trigonometric Fourier series and discuss
more detailed properties such as pointwise convergence and summability.
These are fairly well known and many more details may be found in
Zygmund's book. A short presentation on expansion of distributions in
Fourier series is also presented.
In Chapter Five we also consider orthogonal systems in Sobolev spaces.
These can be composed of delta functions as well as ordinary functions.
In the former case we obtain an orthonormal series of delta function
wavelets.
Chapter Six is devoted to another large class of examples, the ortho-
gonal polynomials. The classical examples are defined and certain of
their properties discussed. The Hermite polynomials are naturally as-
sociated with tempered distribution; properties of this connection are
covered. Other orthogonal series are discussed in Chapter Seven.
Various kinds of convergence of orthogonal series are discussed in
Chapter Eight. In particular, pointwise convergence of wavelet series is
compared to that of other orthogonal systems. Also, the rate of conver-
gence in Sobolev spaces is determined. Gibbs' phenomenon for wavelet
series is compared to that for other series.
Chapter Nine deals with sampling theorems. These arise from many
orthogonal systems including the trigonometric and polynomial systems.
But the classical Shannon sampling theorem deals with wavelet sub-
spaces for the Shannon wavelet. This can be extended to other wavelet
subspaces as well. Both regular and irregular sampling points are con-
sidered.
In Chapter Ten we cover the relation between the translation operator
and orthogonal systems. Wavelet expansions are not very well behaved
with respect to this operator except for certain examples.

viii
Chapter Eleven deals with analytic representation based on both Fou-
rier series and wavelet. These are used to solve boundary value problems
for harmonic functions in a half-plane with specified values on the real
line.
Chapter Twelve covers probability density estimation with various
orthogonal systems. Both Fourier series and Hermite series have been
used, but wavelets come out the best.
Finally in the last chapter we cover the Karhunen-Loeve theory for
representing stochastic processes in terms of orthogonal series. An al-
ternate formulation based on wavelets is developed.
Some of this text material was presented to a graduate course of mixed
mathematics and engineering students. While not directly written as a
text, it can serve as the basis for a modern course in Special Functions
or in mathematics of signal processing. Problems are included at the
end of each chapter. For the most part these are designed to aid in the
understanding of the text material.
Acknowledgment s
Many persons helped in the preparation of the manuscript for this
book, but two deserve special mention: Joyce Miezin for her efficient
typing and ability to convert my handwriting into the correct symbols,
and Bruce O'Neill for catching many of my mathematical misprints.
G. G. Walter

ix
Preface to second edition
In the years since the first edition of this book appeared, the subject
of wavelets has continued its phenomenal growth. Much of this growth
has been associated with new applications arising out of the multiscale
properties of wavelets. Another source has been the widespread use of
threshold methods to reduce the data requirements as well as the noise in
certain signals. But in the area of wavelets as orthogonal systems, which
is the main theme of this book, the growth has not been as marked. The
principal new material has been in the area of multiwavelets, which,
however, have not found their way into as many applications as the
original theory. In addition, there seems to a resurgence of interest in
nontensor product higher dimensional wavelets, but this area still needs
some time to sort itself out.
In this new edition we have tried to correct many of the misprints and
errors in the first edition (and in the process, have probably introduced
others). We have reviewed the problems and introduced others in an
effort to make their solution possible for average graduate students. We
have also introduced a number of illustrations in an attempt to further
clarify some of the concepts and examples. The first and fourth chap-
ters remain approximately the same in this edition. The second chapter
on distribution theory has been rewritten in order to make it somewhat
more readable and self contained. Chapter three on orthogonal wavelet
theory has been expanded with some additional examples: the raised
cosine wavelets in closed form, and other Daubechies wavelets and their
derivation. In Chapter five on wavelets and distributions, a section on
impulse trains has been added. Chapter six on orthogonal polynomials
remains essentially the same, while in Chapter seven a new section on
an alternate approach to periodic wavelets has been added. In Chap-
ter eight on pointwise convergence, an additional section on positive
wavelets and their use in avoiding Gibbs' phenomenon is new. Chapter
nine has been extensively revised and, in fact, has been split into two
chapters, one devoted primarily to the Shannon sampling theorem and
its properties and the new Chapter ten which concentrates more on sam-
pling in other wavelet subspaces. New topics include irregular sampling
in wavelet subspaces, hybrid wavelet sampling, Gibbs' phenomenon for
sampling series in wavelet subspaces, and interpolating multiwavelets.

X
Chapter eleven on translation and dilation has only minor changes as
does most of Chapter twelve except for a few pages on wavelets of en-
tire analytic functions. In Chapter thirteen on statistics a number of
new topics have been added. These include positive wavelet density es-
timators, density estimators with noisy data, and threshold methods.
Some additional calculations involving some of these estimators are also
included. Chapter fourteen, which deals with stochastic processes, has
some new material on cyclostationary processes.
Acknowledgements.
The contributions of many individuals appear in this new edition. In
particular the authors wish to acknowledge the work of Youming Liu,
Hong-tae Shim, and Luchuan Cai which is covered in more detail here.
Gilbert G. Walter and Xiaoping Shen

Contents
Preface to first edition v
Preface to second edition ix
List of Figures xvii
1 Orthogonal Series 1
1.1 General theory 1
1.2 Examples 5
1.2.1 Trigonometric system 6
1.2.2 Haar system . 10
1.2.3 The Shannon system 12
1.3 Problems 15
2 A Primer on Tempered Distributions 19
2.1 Intuitive introduction 20
2.2 Test functions 22
2.3 Tempered distributions 25
2.3.1 Simple properties based on duality 27
2.3.2 Further properties 29
2.4 Fourier transforms 30
2.5 Periodic distributions 32
2.6 Analytic representations 33
2.7 Sobolev spaces 35
2.8 Problems 35
3 A n Introduction to Orthogonal Wavelet Theory 37
3.1 Multiresolution analysis 38
xi

xii
3.2 Mother wavelet 44
3.3 Reproducing kernels and a moment condition 53
3.4 Regularity of wavelets as a moment condition 55
3.4.1 More on example 3 59
3.5 Mallat's decomposition and reconstruction algorithm . . . 64
3.6 Filters 65
3.7 Problems 70
4 Convergence and Summability of Fourier Series 73
4.1 Pointwise convergence 73
4.2 Summability 79
4.3 Gibbs phenomenon 81
4.4 Periodic distributions 84
4.5 Problems 87
5 Wavelets and Tempered Distributions 91
5.1 Multiresolution analysis of tempered distributions . . . . 92
5.2 Wavelets based on distributions 95
5.2.1 Distribution solutions of dilation equations . . . . 95
5.2.2 A partial distributional multiresolution analysis . . 99
5.3 Distributions with point support 100
5.4 Approximation with impulse trains 104
5.5 Problems 107
6 Orthogonal Polynomials 109
6.1 General theory 109
6.2 Classical orthogonal polynomials 114
6.2.1 Legendre polynomials 115
6.2.2 Jacobi polynomials 119
6.2.3 Laguerre polynomials 120
6.2.4 Hermite polynomials 121
6.3 Problems 126
7 Other Orthogonal Systems 129
7.1 Self adjoint eigenvalue problems on finite intervals . . . . 130
7.2 Hilbert-Schmidt integral operators 132
7.3 An anomaly: the prolate spheroidal functions . . .134
7.4 A lucky accident? 135
7.5 Rademacher functions 140
7.6 Walsh function 142
7.7 Periodic wavelets 143

7.7.1 Periodizing wavelets 144
7.7.2 Periodic wavelets from scratch 146
7.8 Local sine or cosine basis 150
7.9 Biorthogonal wavelets 154
7.10 Problems 159
8 Pointwise Convergence of Wavelet Expansions 161
8.1 Reproducing kernel delta sequences 162
8.2 Positive and quasi-positive delta sequences 163
8.3 Local convergence of distribution expansions 169
8.4 Convergence almost everywhere 172
8.5 Rate of convergence of the delta sequence 173
8.6 Other partial sums of the wavelet expansion 177
8.7 Gibbs phenomenon 178
8.8 Positive scaling functions 181
8.8.1 A general construction 181
8.8.2 Back to wavelets 182
8.9 Problems 186
9 A Shannon Sampling Theorem in Wavelet Subspaces 187
9.1 A Riesz basis of V m 188
9.2 The sampling sequence in V m 189
9.3 Examples of sampling theorems 191
9.4 The sampling sequence in T m 195
9.5 Shifted sampling 197
9.6 Gibbs phenomenon for sampling series 199
9.6.1 The Shannon case revisited 201
9.6.2 Back to wavelets 201
9.7 Irregular sampling in wavelet subspaces 212
9.8 Problems 214
10 Extensions of Wavelet Sampling Theorems 217
10.1 Oversampling with scaling functions 218
10.2 Hybrid sampling series 223
10.3 Positive hybrid sampling 225
10.4 The convergence of the positive hybrid series 228
10.5 Cardinal scaling functions 232
10.6 Interpolating multiwavelets 240
10.7 Orthogonal finite element multiwavelets 242
10.7.1 Sobolev type norm 244
10.7.2 The mother multiwavelets 245

xiv
10.8 Problems 252
11 Translation and Dilation Invariance in Orthogonal
Systems 255
11.1 Trigonometric system 255
11.2 Orthogonal polynomials 256
11.3 An example where everything works 257
11.4 An example where nothing works 258
11.5 Weak translation invariance 259
11.6 Dilations and other operations 265
11.7 Problems 267
12 Analytic Representations V i a Orthogonal Series 269
12.1 Trigonometric series 270
12.2 Hermite series 274
12.3 Legendre polynomial series 280
12.4 Analytic and harmonic wavelets 282
12.5 Analytic solutions to dilation equations 286
12.6 Analytic representation of distributions by wavelets . . . . 287
12.7 Wavelets analytic in the entire complex plane 291
12.8 Problems 293
13 Orthogonal Series in Statistics 295
13.1 Fourier series density estimators 296
13.2 Hermite series density estimators 299
13.3 The histogram as a wavelet estimator 301
13.4 Smooth wavelet estimators of density 305
13.5 Local convergence 309
13.6 Positive density estimators based on
characteristic functions 310
13.7 Positive estimators based on positive wavelets 312
13.7.1 Numerical experiment 316
13.8 Density estimation with noisy data 318
13.9 Other estimation with wavelets 322
13.9.1 Spectral density estimation 322
13.9.2 Regression estimators 324
13.10 Threshold Methods 324
13.11 Problems 326

X V
14 Orthogonal Systems and Stochastic Processes 329
14.1 K-L expansions 329
14.2 Stationary processes and wavelets 332
14.3 A series with uncorrected coefficients 335
14.4 Wavelets based on band limited processes 341
14.5 Nonstationary processes 345
14.6 Problems 349
Bibliography 351
Index 363

List of Figures
1.1 The scaling function and mother wavelet for the Haar
system 11
1.2 The scaling function for the Shannon system 14
1.3 The mother wavelet for the Shannon system 15
2.1 A test function in the space S (the Hermite function h^(x)). 23
2.2 Some approximations to the delta function in S' 28
3.1 Typical functions in the subspaces Vb of the multiresolu-
tion analysis for the Haar scaling function 39
3.2 Typical functions in the subspaces V of the multiresolu-
3.3 Typical functions in the subspaces V2 of the multiresolu-
3.4 The scaling function of the Franklin wavelet arising from
the hat function 47
3.5 The Daubechies scaling function of example 3 47
3.6 The mother wavelet of example 3 48
3.7 The mother wavelet of Example 5 in the time domain. . . 49
3.8 The scaling function and absolute value of the mother
wavelet of Example 7 in frequency domain 51
3.9 The scaling function of Figure 3.7 in the time domain. . . 52
3.10 The reproducing kernel q(x,t) for Vo in the case of Haar
wavelets 55
3.11 Daubechies scaling function and mother wavelet (N = 4). 63
3.12 The system functions of some continuous filters: low-pass,
high-pass and band-pass 66
3.13 The system function of discrete lowpass (halfband) filter. 68
xvii

xviii
3.14 The decomposition algorithm 69
3.15 The reconstruction algorithm. . . 69
4.1 The Dirichlet kernel of Fourier series (n= 6) 77
4.2 The Fejer kernel of Fourier series (n= 6) 81
4.3 The saw tooth function 83
4.4 Gibbs phenomenon for Fourier series; approximation to
the saw tooth function using Dirichlet kernel 83
4.5 The approximation to the saw tooth function using Fejer
kernel 84
5.1 A mother wavelet with point support. The vertical bars
represent delta functions 104
5.2 A continuous function and its impulse train 105
6.1 Some Legendre polynomials (n= 2, 3, and 6) 115
6.2 Some Laguerre polynomials (alpha= l/2, n = 5, 7, and 10). 120
6.3 Some Hermite polynomials (modified by constant multi-
ples, n = 4, 5, and 7) 122
7.1 The Haar mother wavelet 141
7.2 One of the Rademacher functions 143
7.3 Two orthogonal Walsh functions 144
7.4 A bell used for a local cosine basis 151
7.5 Three elements in the local cosine basis with bell of Figure
7.4 153
7.6 Two additional elements of the local cosine basis showing
the bell 154
7.7 Two biorthogonal pairs of scaling functions with the same
MRA 156
7.8 A biorthogonal pair of scaling functions and wavelets with
compact support 159
8.1 The delta sequences from Fourier series - the Dirichelet
kernel 164
8.2 The delta sequences from Fourier series - the Fejer kernel. 165
8.3 The quasi-positive delta sequence for the Daubechies
wavelet 20 , m = 0 168
8.4 The summability function pr
(x) for Daubechies wavelet
2 0 ( x ) , m = 0 183

xix
8.5 The positive delta sequence fcrjm(x, y) for Daubechies wavelet
2(/>(x),m = 0 184
9.1 The sampling function for the Daubechies wavelet 20(0
with 7 = -± 194
9.2 The function h of Proposition 9.3 202
9.3 The partial sum of the Shannon series expansion 202
10.1 An example of the scaling function of a Meyer wavelet
at scale m= 0 and the sum of 5 terms of its sampling
expansion in the next scale m =l 221
10.2 The positive summability function for the Coiflet of de-
gree 2 with r = 0.22 227
10.3 The dual of the positive summability function in Figure
10.2 228
2
10.4 A non-negative function f(x) = e~^~[-1/2,1/2] 229
10.5 The positive hybrid series (m= 4) using Coiflet of degree
2 for the function in Figure 10.4 229
10.6 The hybrid sampling series for the function in Figure 10.4
(m = 4) using Coiflet of degree 2 230
10.7 An example of cardinal scaling function of Theorem 10.5,
type two raised cosine wavelet 239
10.8 The mother wavelet for the scaling function in
Figure 10.7 239
10.9 The scaling function and wavelet of interpolating multi-
wavelets for n = l 250
10.10 The two scaling functions of interpolating multiwavelets
for n = 2 250
10.11 The first pair of wavelets of interpolating multiwavelets
for n= 2 251
11.1 The approximation of the shifted scaling function by Haar
series at scale m =l 259
12.1 The kernel Kr(t) given in Lemma 12.1 with m — 2. . . . 271
12.2 The Hermite function h^ix) and the real part of the Her-
mite function of the second kind fi2(x + i • 0) 278
12.3 A Legendre polynomial (n = 3) 281
12.4 The real and imaginary parts of the analytic representa-
tions at y= 2 for the Legendre polynomial in Figure 12.3. 281

X X
13.1 Histogram for data in Example 13.1 305
13.2 Smooth wavelet estimator for the same data as in Figure
13.1 307
13.3 Daubechies scaling function (N — 4) 314
13.4 The associated summability function for the scaling func-
tion in Figure 13.3 315
13.5 The positive kernel associate with Daubechies scaling func-
tion (N — 4) 315
13.6 Density estimate for the Old Faithful geyser data using
the reproducing kernel associated with Daubechies scaling
function (N = 4) 317
13.7 Density estimate for the Old Faithful geyser data using
the positive kernel of Figure 13.4 318

Chapter 1
Orthogonal Series
Orthogonal series play an important part in many areas of mathematics
as well as in applications. They constitute an easy way of representing a
function in terms of a series and may replace complicated operators on
the function by simpler ones on the coefficients of the series. The most
familiar orthogonal systems are the trigonometric and the various or-
thogonal polynomials. Not so familiar but becoming increasingly widely
used are the Haar, the Shannon, and wavelet systems.
The basic theory of orthogonal series is deceptively simple, but its
detailed study contains many surprisingly difficult questions. In this
chapter we skip the latter and present only a few elements of the theory.
We first present a little of the general theory and then discuss a few
of the principal examples. One, the trigonometric system, will be an
important tool in subsequent chapters. The other two, the Haar and
the Shannon systems, will serve as prototypes for the construction of
wavelets.
1.1 General theory
While there exist many different orthogonal systems, they all have
a number of properties in common, which we present here. We shall
restrict ourselves to L2
(a,6): the set of square (Lebesgue) integrable
functions on (a, 6), a real interval. The theory is the same if we introduce
a weight function or even if we consider general separable Hilbert spaces.
See[R-N].
A nontrivial sequence {fn}^Lo °f r e a
l ( o r
complex) functions in L2
(a, b)
1

2 1. Orthogonal Series
is said to be orthogonal if
(fn,fm)= / fn(x)fm(x)dx = 0, n / m , 71,771 = 0,1,2,...
Ja
and orthonormal if in addition ( / n , / n ) = l , n = 0,l,2,---. For example,
/n(^) — sin(77 + l ) x is orthogonal on (0, n). Another example is
f ( — , x _ J l , 7 1 < X < 7 1 + 1
J n W - X[n,n+1) W ~ j q ? 0 < X < 7 7 , 77 + l < X
which in fact is orthonormal on [0, oo).
The idea is to expand a given function f{x) <
E L2
(a,6) in an ortho-
normal series
oo
}(•••) - £ V „ / „ { . , - ) . (1.1)
This is not always possible (e.g., take f{x) — X[o.5,i){x
) m
^ n e
second
example), but if it is, then the cn 's must have a special form. We shall
use the usual notation for the L 2
norm, ||/|| = (/,
Proposition 1.1
Let {cn} be a sequence such that the series in (1.1) converges in the
sense of L2
(a,b) to f(x); then cn = (/, f n ) .
The proof is immediate. We multiply both sides of (1.1) by fm(x) and
then integrate. Because of the orthogonality all the terms in the series
drop out except cm. There is no problem with interchanging the integral
and the summation because of the continuity of the inner product with
respect to the norm.
Convergence in the sense of L2
(a,6) is also known as mean square
convergence, and the error
eN
N
f ^ ^ Cnfn
is called the mean square error. The coefficients appearing in Proposi-
tion 1.1 are called the Fourier coefficients of / with respect to { f n } and
have another property that makes them useful.

1.1. General theory 3
Proposition 1.2
Let {cn} be the Fourier coefficient of f G L2
(a,6) and {an} any other
sequence; then we have
N 2
N
f ~~ ^ ^ c
t i fn <
7 1 = 0 7 1 = 0
i.e., the mean square error is minimized for the series with Fourier co-
efficients.
The proof is obtained by adding and subtracting the series with the
Cm S.
N
7 1 = 0
N N
N
N
7 1 = 0
7 1 = 0 7 1
= 0
N N
(/> /) - 52 a n
^ n
' ^ ~ 5 2 ^ ^ +
XI ia n
i
7 1 = 0 7 1
= 0
AT
^ 7 1 ^ 7 1 H
~ ^ T l ^ T l )
7 1 = 0
A T iV 7
+ 52 la
nP + X] l C n
! 2
" 52 l C n
! 2
7 1 = 0 7 1
= 0 7 1
= 0
TV A T
(/,/) + £ | a „ - C „ | 2
- y j | c „ | 2
7 1 = 0 7 1 = 0
iV
/ 5 ^ £71/71
7 1 = 0
AT
+ ^ |an - cn |2
. (1.2)
7 1 = 0
Since J2n=o l a
n — c n | 2
> 0, the conclusion follows.
Another way of thinking of this is that the Fourier coefficients give the
orthogonal projection of / onto the subspace VN spanned by (/o, / 1 , / 2 ,
. . / A T ) . Indeed by another simple calculation we see that
•
N N
( 5 2 f - 52 =
°-
7 1 = 0 7 1 = 0
Thus, not only is the best approximation to / in Vjv given by this sum,
but the error is orthogonal to VN-

Similar calculations lead to Bessel's inequality
CO
£ M 2
< I I / I I 2
(1.3)
since
and therefore
0 <
f N
Ln=0
n=0
N
f ~ ^ ^ c
nfn
n=0
N
- E i
n=0
is a monotone sequence bounded by
Thus the series of (1.3) converges and has the same bound. A simple
consequence of Bessel's inequality is that { c n } G £2
and cn —» 0 as
n —
> oo.
To round out our theory, we should like to have the series with Fourier
coefficients ] P c n / n converge to / . By Bessel's inequality the partial
sums are a Cauchy sequence in L2
(M), which because of the completeness
of this space must converge in the L 2
sense but not to / necessarily.
To ensure this we need to add another condition, the completeness of
the orthogonal system (not to be confused with the completeness of the
space). The orthonormal system {fn} is said to be complete in L2
(a, b) if
no nontrivial / E L2
(a, b) is orthogonal to all the /n 's, i.e., if (/, f n ) = 0 ,
n = 0,1, 2,..., for / 6 L2
(a, b) then / = 0, a.e.
THEOREM 1.1
Let {fn} be an orthonormal system in L2
(a,6); let f G L2
(a,6) with
Fourier coefficients {cn}; then
N
f ^ ^ c
nfn
n=0
0 as N —• oo
if and only if {fn} is complete.
P R O O F By Bessel's inequality we know that the series (1.1) converges
to some g G L2
(a, 6), and hence
N
f ~Y1 Cn
^n
f 9-
71=0

1.2. Examples 5
Now the Fourier coefficients of g are given by
N
(gjm) = Jim (S^CnfnJm) = Cm
n=0
and hence are the same as those of / . Thus f — g has all zero coefficients,
and, if the system is complete, / — g — 0 a.e. Since the series converges
to g it must also converge to / .
On the other hand if the series converges to / , and all the coefficients
are zero, then f — 0 a.e. as well. •
The conclusion of the theorems can be restated as Parseval's equality
oo
ll/ll2
= £ W 2
(1-4)
n=0
since
N
f ~ 52 C n
f n
An alternate form is given by
oo
52cfcdfc =
(f>9)
where dk = (g, fk)- This is obtained by applying (1.4) to f + g and f — g
and then subtracting.
Either of these results can be taken as test for completeness. However,
it is sufficient to check (1.4) for a set of functions {h^} whose closed linear
K
span (i.e., the closure in the sense of L2
(a, b) of ^ a>kh>k) is L2
(a, b). In
particular we may take to be the set of characteristic functions of
subintervals of (a, b).
N
l / l l 2
- £ ! c
<
n=0
1.2 Examples
There are many examples of orthonormal systems in the mathematical
literature (see [Al], [0], [Sa]). The earliest and most widely studied
is the trigonometric system which we consider in more detail below.

We also consider two more recent systems, the Haar and the Shannon,
which form prototypes of the newest system, the orthogonal wavelets.
In a later chapter we study some aspects of orthogonal polynomials.
But there are many others, e.g., Sturm-Liouville systems, which are
used to solve partial differential equations; the Walsh functions, which
are piecewise constant; and the eigenfunctions of a compact symmetric
integral operator, which will not be covered in detail.
1.2.1 Trigonometric system
The trigonometric system is a complete orthogonal system in L2
(—7r, TT)
given by
fo(x) = 1/2, fi(x) = sinx, f2(x) = cosx,
• • -f2n-i(x) = sinnx, f2n(x) = cosnrr,
It is usually not normalized, since | | / n | | 2
= ^ ^ 7^ 0- The series is
usually written in the form
0 0
n=l
If (1.5) is the Fourier series of a function / 6 L2
(—7r,7r), the coefficients
are given by
1 r
an = — / f (x) cos nx dx, n = 0,1, • • • (1-6)
^ J-IX
1 r
K 7 - 7 T
f(x) sinnx dx, n = l , 2 ,
The orthogonality of this system is easy to prove by using a few
trigonometric identities. However, the completeness, though well known,
is not so obvious. In the interest of completeness we present a proof.
It involves first showing that the Fourier series converges uniformly for
certain functions. It should be remarked that this is not true for all
continuous functions; there are examples where the Fourier series fails
to converge on a dense set of points [Z, p. 298].
If the series (1.5) is to converge uniformly, then the limit function
must be continuous and periodic of period 27r, which we assume / to be.
We shall need an expression for the partial sums of the series, obtained

1.2. Examples 7
by substituting (1.6) into the partial sums of (1.5).
Sn(x) a0
+ 52a
k c o s
kx + bk sin kx
k=l
If71
{ 1 n
— / f(t) < - - f cos H cos kx + sin H sin
I2
£i
1 r ajn(n + l ) ( x - t ) d t
2 s i n ( x - t ) / 2
. sin (n + | )
/
Q U I 1 I V | r) . —
-7T 27T sinw/2
The expression
Dn(u)
+52c
°s
^
/ c = i
sin (n + I )
27r sinix/2
(1.7)
(1.8)
is called the Dirichlet kernel, and it plays a central role in the study
of pointwise convergence of Fourier series. It may be shown true by
multiplying both sides of
1 n
7rDn(u) = - + cos ku
k=l
by sin | and then forming a telescoping sum.
Proposition 1.3
Let f be a 2TT periodic function in C2
(M); then
W S U
P * G M Sn(x) ~ f(x) 0
(n) Sn-f^0
as n —
> oo.
P R O O F Since f*^ Dn(u)du = 1, the difference between Sn and /
may be expressed as
Sn(x)-f(x) = f {f(x-u)- f(x)}Dn(u)du
J —IT

— — I < — — > sin [ n + - u du
7rJ_n 2sinu/2 j 2J
I F ( 1
1 cos (n + i ) w . x ,
= - - i 9{x,u)%
where g(x,u) = ^ ^ l ' / ^ •
Since both g(x,u) and its derivative are uniformly bounded, the last
expression gives us
C
for some constant C. Hence (i) must hold and by squaring and integra-
tion of (i), so must (ii). •
Since twice differentiate functions are dense in L2
(—7r,7r), S O are
trigonometric polynomials by this proposition and therefore it follows
that the trigonometric system is complete by Theorem 1.1.
An alternative form for the trigonometric series (1.5) is the exponen-
tial form
oo
s{x)= Y, C
^INX
72= — OO
where convergence is with respect to the symmetric partial sums. If
(1.10) is a Fourier series the coefficients are
1 F
f(x)e-inx
dx. (1.11)
Of course expression (1.10) is reducible to (1.5) by using
e±mx __ c o g n x ± i g m n x
Another way of looking at these expressions is as transforms.
The finite Fourier transform of the periodic function f(x) is given by
(1.11) with the inverse transform given by (1.10). It converts a differen-
tial operator into a multiplication operator,
(Df(x))e-*nx
dx = incn,

1.2. Examples 9
which makes it a useful tool in differential equations. Its absolute value
is also shift invariant,
^ f f(x - a)e-inx
dx
The infinite Fourier transform of the function / e Lx
(—oo, oo) is the
expression
/
oo
f{t)e-iwt
dt, weR. (1.12)
-OO
The image of the transform in this case is a continuous function on
which, if it is also in leads to the inverse [B-C, p. 19]
i r°° ~
/(') = 2^ J _ fWetwt
dw, teR. (1.13)
Versions of Parseval's equality (1.4) also exist for Fourier transforms.
They are, for f,ge L2
(R), [B-C, p. 105]
II/I|2
= ^ I I / | | 2
(f,9) = ^(f,9)-
This requires a more general definition of Fourier transform, which to-
gether with other properties are found in the next chapter.
One can also go the other way and approximate (1.11) by a discrete
sum. This gives us the discrete Fourier transform,
y> = jfEf (jf) e
~vki
"/N
- k = o,...,N-i, (i.i4)
with the inverse given by
N-l
k=0
This is the form that leads to the fast Fourier transform which, by group-
ing terms in (1.14), reduces the computation time considerably. This has
made transform methods much more useful in partial differential equa-
tions, image processing, time series, and other applied problems [Stl, p.
448].

1.2.2 H a a r s y s t e m
The Haar orthogonal system begins with </>(£), the characteristic function
of the unit interval
• 4>(t) =X[o,i)(*)-
It is clear that c/)(t) and cj)(t — n), n ^ 0, n G Z are orthogonal since
their product is zero. It is also clear that {(p(t — ri)} is not a complete
orthogonal system in L2
(R) since its closed linear span Vb consists of
piecewise constant functions with possible jumps only at the integers.
The characteristic function of [0,1/2), for example, with a jump at 1/2,
cannot have a convergent expansion.
In order to include more functions we consider the dilated version
of (j)(t) as well, (j)(27n
t) where m G Z. Then by a change of variable
we see that { 2 m
/ 2
0 ( 2 m
t - n)} is an orthonormal system. Its closed
linear span will be denoted by Vm- Since any function in L 2
( R ) may be
approximated by a piecewise constant function fm with jumps at binary
rationals, it follows that (J Vm is dense in L2
(R). Thus the system { 0 m n }
m
where
<pmn{t) = 2m
'2
^{2m
t - n)
is complete in L2
(R), but, since cj){t) and 4>(2t) are not orthogonal, it
is not an orthogonal system. We must modify it somehow to convert it
into an orthogonal system.
Fortunately the cure is simple; we let ijj(t) = 0(2£) — 4>(2t — 1). Then
everything works; {i/j(t — n)} is an orthonormal system, and ip(2t — k)
and ip(t — n) are orthogonal for all k and n. This enables us to deduce
that {ipmn}m,nez where
i;mn(t) = 2 ^ ( 2 m
t - n )
is a complete orthonormal system in L2
(R). This is the Haar system;
the expansion of / G L 2
(R) is
oo oo
• W ) = E E </'^n)^mn(<), (1-16)
m=—oo n = —oo
with convergence in the sense of I? (R). The standard approximation is
the series given by
m— 1 oo
/«»(*)= E E (MknWknit) . (1.17)
fc——oo n=—oo

1.2. Examples 11
-1 J
F I G U R E 1.1
The scaling function and mother wavelet for the Haar system.
It converges to a piecewise constant function with jumps at 2 _ m
n , n G Z,
at most.
Hence, fm G Vm and since, by Parseval's equality,
772—1 OO
(/m,0mn) = ^ ^ if ^kj) {^kj Aran) = {f Aran) ,
k=—oo j=—oo
it is the projection of / on T/m , i.e., / m = </w)</w-
n
This enables us to get a pointwise convergence theorem.
Proposition 1.4
Let f be continuous on R and have compact support; then f7n —• /
uniformly.
P R O O F Since / is uniformly continuous, it follows that for each e > 0
there exists an m such that
f(x)-f(y) < e when x - y < 2~m
.
For x e [n2~m
, (n + l)2~m
) we have
/ . 2 - " * ( n + l )
fm(x) = 2m
'2
/ f(t)dt 2m
/2
4>(2m
x - n)

since all the other terms in the series are zero in this interval. Therefore
by the mean value theorem
fm(x) = f((m)2-m
2m
4>(2m
x -n)
= f(Cm),
for some £ m in this interval, and since x — £ m | < 2~m
, fm(x) — f(x) <
e. •
This Haar system is our first prototype of a wavelet system, and we
shall return to it several times later. At this point it should be observed
that the uniform convergence of fm to / is a property not shared by the
trigonometric system. The uniform convergence of (1.16) also follows
since the inner series has only a finite number of terms, and the partial
sums of the outer series converge uniformly since they are of the form
fm(%) ~ f-p(x
) a n
d f - p converges to zero uniformly as p —> oo.
The <fi(t) is usually called the scaling function in wavelet terminology
while ip(t) is the mother wavelet.
1.2.3 T h e S h a n n o n s y s t e m
A second prototype also begins with the characteristic function of an
interval. Now, however, it is the Fourier transform of the scaling func-
tion, taken to be
< W < 7T
i . W .
Its inverse Fourier transform is
10 o.w.
The orthogonality of (p(t) and cf)(t — n) is based on properties of the
Fourier transform, of Parseval's equality, and the fact that {<p{t — OL)){W)
f
J — c
1 f°° - •
4>(t)c/>(t - n)dt = — / (j)(w)(t)(w)elwn
dw
2TT J _ 0 0
1
r iwni sin7rn
= — / elwn
dw = = 0 , n / 0.
27T 1^ rcn
Let f(t) be a function that is square integrable and whose Fourier
transform f(w) vanishes for w > ir. It has a Fourier series given by
/ H = yjC n e -«, H < T T (i.i8)

1.2. Examples 13
where cn — f(w)e lwn
dw. By the Fourier integral theorem (1.13)
this is just /(—n). This theorem applied to both sides of (1.18) yields
fit) = ^ f f(w)e^dw = E / ( ~ n
) ^ f elWn
etWt
dw
/ H ^ . (1.19)
n[t - f n)
We denote by Vb the set of all such functions. This is a linear space
and is closed as well since limits (in the square integrable sense) of the
sequences of functions in Vb are also in Vb-
The formula (1.19) is referred to as the Shannon sampling theorem
[Sh]. It enables one to recover a band-limited function in Vb from its
values on the integers. This is used by engineers to convert a digital
to an analog signal (as in compact discs). See [B] and [Za] for more
properties.
By changing the scale in (1.19) by a factor of 2, we can obtain a
sampling theorem on the half integers. (Let 2x — t and let g(x) =
f(2x)). The space with the new scale is V and it will consist of functions
whose Fourier transforms vanish outside of [—27r,27r]. We may repeat
this as often as we want and get thereby an increasing sequence of spaces.
We can also stretch the scale instead of shrinking it to obtain a sequence
{ K n C = _ o o satisfying
• • • c y _ m c . . . c y _ 1 c % c 7 i c . . . c y m c . . . ,
As we go to the left in this sequence the support of the Fourier transform
(the set outside of which it vanishes) shrinks to 0. As we go to the right,
it expands to all of R. Thus we have
(i) H K n = {0}and
m
(ii) each / £ L2
(M) can be approximated by a function in Vm for m
sufficiently large.
The sequence {Vm} is called a "multiresolution analysis" associated
with (f)(t).
Just as with the other prototype, we can introduce a function ip(t) in
V which is orthogonal to (j)(t — n), ip(t) — 2(j)(2t) — 4>{t). Its Fourier
transform is given by
and has support on [—2TT, —7r] U [TT, 2TT}. Since the supports of <j) and -0
are disjoint, the needed orthogonality follows. Thus, the inverse Fourier

F I G U R E 1.2
The scaling function for the Shannon system.
transform ip(t) is another example of a "mother wavelet", and leads to
another orthonormal sequence {ipm,n(t)}- As in the case of the Haar sys-
tem, it is also complete and the expansion of certain continuous functions
converges uniformly. Rather than require this function to have compact
support, we assume only that it is continuous and that its Fourier trans-
form / e LX
(R).
Proposition 1.5
Let f G L2
(R) be continuous and have a Fourier transform f £ L1
(M);
then fm—>f uniformly on R.
P R O O F If fm = <f>mn)</>mn, then
n
£ » H = E f " /(C)^(2-r o
C)ei C n 2
"m
dC 4(2-m
w)e-™n2
-m
.
r, J— 2 m
7 T
The right hand side is the Fourier series of f(w) on [—2m
7r,2m
7r] and
hence is equal to the restriction of f(w) to this interval,
/ m H = / M X 2 - 7 r H (1-20)
where X2m
?r(w) is the characteristic function of [—2m
7r, 2 m
7 r ] . The error
then may be given by
/(*) - fm(t) = ^ (fH - / m H ) elwt
dw

1.3. Problems 15
.1 "
A
/ , /
^ • v — r ~
4
-4
F I G U R E 1.3
The mother wavelet for the Shannon system.
Hence we obtain
f(t)-fm(t)<-
COO n — 2'U
7T |
+ / fWdw
/2m
7T OO
which, since / G L 1
^ ) , must converge to 0 as m oo. •
The astute reader will recognize VQ as the Palev-Wiener space of TT
band limited functions [P-W]. It is a reproducing kernel Hilbert space
with reproducing kernel q(x,t) = S 1
^ , ^ ~ ^ . The expansion of / G Vo
with respect to <j)(x — n) given by (1.19) not only converges in L2
(K) but
converges uniformly on all of R. The proof is implicit in (1.19).
1.3 Problems
1. The following are orthogonal systems in L2
(0,7r):
(a) { c o s n x } ^= 0 ,
(b) {sinnx}£°= 1 ,
(c) {sin(n + £ M ~ 0 .

Show they are also complete in L2
(0,7r) by using the symmetry
of these functions and the completeness of the full trigonometric
system in L2
(—7r,7r).
2. Show that any orthogonal system { / n } is also linearly independent.
3. Let {fn}^Lo be a linearly independent sequence in L2
(a, b); define
0o - /o, 0 1 - / i - ||^0 ||2 , 02 - / 2 - R ^ l F -
' ' ' '•
Show that { 0 n } is an orthogonal system.
4. Let { f n } be linearly independent and complete in L2
(a, b) (i.e., the
closed linear span of { / n } is L2
(a,6)). Then { f n } is said to be a
Riesz basis if there are positive constants A and B such that
2 = 1 Z =l 2 = 1
for each sequence { Q } of complex numbers. Take { g n } to be a
biorthogonal sequence to { / n } ((gn, fk) — fink) hi L2
(a,6). Show
that
(a) {<7n} is the unique biorthogonal sequence to { / n } ,
(b) If {cn} G £2
then Y2c
nfn converges in L2
(a,6),
(c) For each / G L 2
M ) , { ( / , g n ) } G ^ 2
,
(of) For each / G L2
(a,6), / = Yln
= o(f'9n)fn.
5. Let fi(x) = x and / 2 ( # ) = TT2
- 3 x 2
, —TT < x < TT. Find the Fourier
series of f i and f% o n
( —
^ 7 7
) a n
d u s e
them to sum the series
n=l
oo
(*)
n=l
n=l
oo
id)
n=l
6. Show that the Haar system
{0(t), 2 m
/ V ( 2 m
t - n ) } , m = 0 , l , 2 r - - , n = 0,1, • • •, 2 m
- 1,
is an orthonormal basis of L2
(0,1).

1.3. Problems 17
7. Find the expansion with respect to {(p(t — n)}, the Shannon system
in Vb, of the functions given by
(a) fi(x) = ± f we™w
dw
(b) f2(x) = ^ J (TT2
- 3w2
)eixw
dw.

Chapter 2
A Primer on Tempered
Distributions
For many applications it is necessary to work with a larger class of
functions than L1
(M). It is also desirable to have a more general theory
of the Fourier transform, which would enable us to calculate Fourier
transforms of periodic functions or polynomials. This is impossible with
the L l
(or the usual L 2
) theory, which does not even work for constant
functions.
The simplest theory that extends the Fourier transform to all functions
of polynomial growth is the theory of "tempered distributions". These
include, in addition to functions, certain objects such as the "delta func-
tion," which is not strictly a function. Nonetheless, it is widely used and
is central to some applications such as the study of transfer functions of
linear filters in engineering.
These tempered distributions may be considered limits in some sense
of sequences of ordinary L1
(R) functions. In fact, one approach is to
define them as limits of Cauchy sequences of such functions. In this case
the familiar process of extending the rational numbers to obtain the real
numbers is imitated (see [K]). However, there are other approaches that
can be used for the formal definitions, but they will still be limits of
sequences of functions and are best thought of as such.
In our approach, which is due to L. Schwartz [S], these ideal elements
are defined as continuous linear functionals on a space of "test func-
tions". The resulting objects will include all locally integrable func-
tions of polynomial growth and all tempered measures as well as their
derivatives of every order. The set of such objects is closed under the
Fourier transform and provides an appropriate setting for much of signal
processing.
We shall omit many details. The reader completely unfamiliar with
19

20 2. A Primer on Tempered Distributions
the subject should consult one of the many excellent texts devoted to it,
2.1 Intuitive introduction
One of the difficulties with the L l
theory of functions is that we cannot
always perform operations that we would like. We cannot, in general,
differentiate such a function, nor can we even multiply it by a polyno-
mial and get another function in this class. We should like to extend
our class of functions to a larger class of objects (no longer necessarily
functions) for which these operations are always possible. These will be
our "tempered distributions".
We begin with the "tempered functions". These are functions which
are locally in L 1
and are of at most polynomial growth. This may be
expressed by requiring that
for any a and some positive constants C and p. The class of such tem-
pered functions includes
• all polynomials
• all piecewise continuous functions of polynomial growth
• all classical orthogonal functions of Chapter 1
• all piecewise continuous periodic functions
• all scaling functions and wavelets.
However, functions of exponential growth are excluded, and hence so
are solutions of certain differential equations (e.g., y' = y) on the real
line.
Our goal is to extend the class of these tempered functions in such a
way that the following operations hold for / in the extended class:
• differentiation
• dilation f(ax),a £ M +
,
e.g., [Kl], [S], [Br], [Ze], [G-S].

2.1. Intuitive introduction 21
• translation f(x — b),b £ R,
• multiplication by Q(x), a C°° function of polynomial growth, f(x)9(x)
• convolution with a function g of compact support ( / * g)(x).
One example that will recur is the Heaviside function given by
H(x) = J x
x > 0
Its derivative does not exist in the usual sense at x — 0, but is equal to
0 for all x / 0. We denote by 8 the formal derivative of the Heaviside
function. If it is to behave as a usual derivative, then since 8 — 0 a.e.,
the integral
6(t)dt = 0,
J — e
but on the other hand
6(t)dt = H{e) - H (-e) = 1 - 0 = 1,
for any positive e. Therefore 6 cannot be an ordinary function, but is
some different sort of object.
Even though it is not a function, we can define all the operations listed
above for 6. The intuitive way to think of S is as approximately a spike,
a function <5A given by
f 0, | x | > l / A
X [ X )
A ( 1 - A | x | ) , x < 1/A'
where A is some large positive number. Then some of the properties
of 6 can be found by performing operations on <5A to determine the
expected behavior of the same operations on 6. For example, 8(ax) is
given approximately by
c ( °' a x
> * M 1
r /
6 x { a x ) =
X{l-Xax), H < l / A = a 6 M
'
Thus we might expect 6 to satisfy 8(ax) = ~b(x
) a
n d we would be right.
Similarly the integral of 8

J 6x(t)dt = 1 for 1/A < e,
as expected. Other properties follow similarly, in particular, the fact
that
9(x)6(x) = 6{0)6(x)
for any C°° function of polynomial growth.
Other tempered distributions are obtained by differentiating different
tempered functions. One such example is the function f(x) — | x | a
where
a > — 1. Its derivative is axa
~1
sgn(x), which is an ordinary function
for x| ^ 0, but is not a tempered function when a < 0. Similarly the
function g(x) = xa
sgn(x) has derivativeo|x|a - 1
. These derivatives and
others lead to various principal value distributions.
If we treat these tempered distributions as ordinary functions as far
as the operations are concerned, we will not usually be far wrong. The
exception is multiplication. The product of two tempered functions is
not a tempered function necessarily, nor is the product of two tempered
distributions a tempered distribution. Even the product of 6 with itself
does not exist.
The next sections will present a more rigorous approach in which the
tempered distributions are defined as continuous linear functionals on a
space of test functions.
2.2 Test functions
The device for defining our objects (the tempered distributions) is
in terms of a certain space of test functions. These functions test our
objects by averaging them or "smearing" them. Since our objects are
not functions for which we know the exact values at points, the next best
thing is to know the average or smeared values which we do know. These
smeared values correspond to integrals or more generally to continuous
linear functionals, i.e., linear functions from this space of test functions
to the complex numbers which are continuous. The properties of the
tempered distributions are based on the properties of these test functions
which we must first define and study.

2.2. Test functions 23
A 2
'
/
-2 V
F I G U R E 2.1
A test function in the space S (the Hermite function h^{x)).
Our test functions will belong to the space S of rapidly decreasing
C°° functions on R, i.e., functions that satisfy
8{k)
(t) <C'pk(l + t)-p
, p,k = 0,1,2,...,teR. (2.1)
S is clearly a linear space, that is, linear combinations of elements of
S also belong to S. But in order to define continuity of the linear
functionals, we need a notion of convergence in S. This is given by the
semi-norms
l p k = s u P ( i + tr ov°t) p,k = 0,l,2,. (2.2)
This is used to define that 6U —> 0 in S whenever
(l + t)PDk
(9u(t)-9(t))^0
uniformly in t for each p and k as v —>• oo. (D is the derivative operator.)
The space S is dense in L 2
(R) in the sense that each / GL2
(R) may
be approximated by some 0 G S in the norm of L 2
(R). This may be
shown by observing that S contains the Hermite functions {hn} given
by
e-x2
/2
ho(x) = 777-, X G
r V 4
and recursively
(x - D)hn(x) = /2n + 2hn+i(x), n = 0 , 1 , . . . ,x G

These constitute an orthonormal basis of L2
(M) (see Chapter 6). Hence
linear combinations of these {hn} also belong to S and may be used to
approximate a given / G L2
(R) in the I? norm.
The C°° functions of compact support are also contained in S since
they trivially satisfy the decay condition (2.1). But polynomials and
trigonometric functions do not belong to S since they do not converge
to 0 as t —• oo.
Some of the properties of S are as follows:
1. S is complete with respect to the convergence consistent with (2.2),
i.e., every Cauchy sequence in this sense converges to an element
of S.
2. Differentiation is a continuous operation in 5, i.e., if 6V 9 in the
sense of (2.2) then D6V —> DO in the same sense.
3. Multipliction by a polynomial is a continuous operation in S.
4. S is closed under dilations and translations.
5. The Fourier transform given by (1.12) is a 1-1 mapping of S onto
itself.
The two operators of property 4 are important for wavelets. These
are the dilation operator Da, a > 0 (Daf(t) = f(at)) and translation
operator Tp (Tpf(t) = f(t — (3)). Our space S is not only closed under
these operators but both are continuous with respect to (2.2).
The Fourier transform T is given by
/
oo
e-iwt
d(t)dt. (2.3)
-oo
That it maps S into S follows from the fact that T converts differenti-
ation into multiplication by a multiple of w and vice versa. In fact by
the Fourier integral theorem [B-C, p. 10],
0(t) = — / elwt
0{w)dw (2.4)
2 7 r
J - o o
and hence T can be seen to be one to one and onto as well.
For some of our examples we will need a related smaller space than the
space of all tempered distributions. This makes it necessary to define
a corresponding test function space which will be larger. We denote by

2.3. Tempered distributions 25
Sr the space containing S consisting of all Cr
functions such that (2.1)
is satisfied for k < r but for all p. The convergence is the same but with
this restriction on k. Its Fourier transform will be denoted Sr and will
consist of all C°° functions such that (2.1) is satisfied for all p < r and
all k.
2.3 Tempered distributions
A tempered distribution is an element of the dual space Sf
of S. This
space is composed of all continuous linear functionals on S; functions
from S to C (denoted by (T, 0), the value of T on which are linear
(T, M i + a292) = a2{T, 9X) + a2 (T, 92)
and continuous
9n -> 0 in S (T, 0n) -> 0 in C.
These are the ideal elements mentioned at the start of the chapter; they
inherit many of the properties of S.
The limit of any bounded sequence of L 1
functions convergent in S'
(i.e., convergent for each 9 <
G S) defines such an element. Indeed, let
{fn} be a sequence in L1
(M) such that
/
oo
l/n| < C
-OO
and { ( / n , 9)} {= { J fn9}) is a Cauchy sequence of complex numbers for
each 9 G S. Then we may define T as
(T,6) - lim (fn,9).
n—>oo
This will be a functional since the complex numbers are complete; it will
be linear since each term in the sequence is linear; it will be continuous
as well since if 9m —> 9 in S then 9m —> 9 uniformly on E and
9(x) < C sup em(x) ~9(x)
x
(fn,0m-6) = I J fn(6m-9)
< / fn SUp 9m(x)

which converges to 0. Hence we can interchange the limits in
lim (T,9m) = lim lim ( / n , 0 m ) = lim (fn,0) = (T,9),
m—>-oo m—>oo n—>oo n—>oo
thereby proving continuity and membership in Sf
. This last equality
corresponds to convergence in Sf
. We say that fn —» / in the sense of
S' if it converges in the sense that
lim (fn,0) = (f,e)
n—>oo
for each 0 € S.
We could go further than this to prove that each Cauchy sequence
converges in S' so that this space is complete. We shall not do so (see,
however [S, II, p.94]).
Each locally integrable function / of polynomial growth belongs to S'.
The functional corresponding to it is
/
oo
f(t)9(t)dt
-oo
where we have used the same symbol / for the functional in Sf
and the
function f(t). This functional clearly exists and is linear. Furthemore,
since
f(t)/(t2
+ iy eLR)
for some integer r, we see that
= | ~ j ^ y ( t 2
+ i)r
en(t)dt.
This is continuous since if 6n —• 0 in the sense of 5, then (t2
+ l)r
0n{t) —»
0 uniformly on R and hence (/, 0n) —
^ 0.
Each T G S' has a derivative defined by
(DT,0) = -(T,ef
), OeS.
Thus even if T corresponds to a nowhere differentiable function, its
derivative exists as a tempered distribution. By combining this with the
previous example we see that each derivative of any order of a locally
integrable function of polynomial growth belongs to S''. In fact this
characterizes Sf
.

Proposition 2.1
Let T G Sf
; then there exists a locally integrable function of polynomial
growth F(x) and an integer p such that
T = DP
F
(see [Ze, p.111]).
Thus, we may interpret elements of Sf
as generalizations of functions
(in fact, they constitute examples of "generalized functions"), and shall
adopt the same notation as for functions. The symbol (/, 8) will stand for
either the value of the functional or, in the case of tempered distributions
given by functions, the integral.
We already have the notation (/, 8) for the latter and hence (/, 9) =
(/.*>.
2.3.1 Simple properties based on duality
In addition to differentiation, we have a number of other operations
defined on S'. Each is defined by the same device. Any operation in S
is translated to a corresponding operation in S' by first observing what
happens to (/, 8) when / is an ordinary function and (/, 8) is an integral.
Then this behavior is extended to all of S'. For example, translation Ta
is defined on S' as
(Taf,9) :
= (/,T_a 0)
since for ordinary functions
(Taf, 0) = J f(t - a)6{t)dt = J f(t)9(t + a)dt = (/, T_a9).
Similarly for dilation by a positive quantity a, we have
(Daf,9) :
= -(f,D±0)
a a
and for multiplication by a C°° function of polynomial growth F(t),
(Ff,6):=(f,F0).
Of course we need to check for each definition that the result is in fact
in S'. But that again is clear in each of the cases.
The "delta function", 6ai which we discussed in the first section, is
properly defined simply as
(6a,6) :
= 9(a),

4
F I G U R E 2.2
Some approximations to the delta function in 5'.
i.e., the unit point mass at a. It can also be given as the derivative of
the Heaviside function
6a = DHa
where
H a
® = {o, t < a '
Indeed, if 6 G 5,
— (DHa, 9) = (Ha,9') = / = -0(a) = -(6a,6).
J a
Still a third way of defining SA is as the limit of a sequence of functions
S = 6Q = lim nxro,n-M
where X[v,n-l
){t) l s
^ n e
characteristic function of [ 0 , n _ 1
) . This limit
must be taken in the sense of S' and not pointwise. Then 6a —Ta6. It
is easy to check that this is consistent with the other definitions.
Other elements of 5' include the pseudofunctions such aspv(|) defined
by
9{t)
pv
-f9
)=cpv
jZ t
•dt
where cpv stands for Cauchy principal value.

2.3.2 Further properties
In addition to having a derivative, each f & S' has an antiderivative
6 S'. Let 9 e S and define 9x(t) = 9(t) - (jZo°) h
o(t) w h e r e
ho is the Hermite function of order 0. Then 6 — 0 and hence
/*«, ° € S. We now define by
(f^Ke) : = -(/, /* +
J — oo
where C is arbitrary. Then
(£>/(-1)
,e) = - ( / ( - 1
) ) 0 /
)
= (/,/ 9x(d)) + c(i,e')
J —oo
= (/,/" 0') = (/,0)
J —oo
since (1,0') =0 and the 9 corresponding to 9' equals 9'.
The support of a function 9 G S is the smallest closed set outside of
which 9 is identically zero. Two distributions / and g G Sf
are said to be
equal in an open set f i if their difference is zero on each 9 with support
in J], i.e., if
(f,0) = (g,9), s u p p l e a
In particular / could be zero on some such Q. The support of / then
is the smallest closed set K such that / = 0 on the complement of K.
Clearly 6 has support on the single point {0}, as do all of its derivatives.
In fact, the only f € S' with point support are linear combinations of 6
and its derivatives ([Ze, p.81]).
The delta function also satisfies the useful sifting property. For F(t)
a continuous function at t = a, we have
F(t)Sa(t) = F(a)6a(t).
This is clear for F G C°° and of polynomial growth, but may be shown
for other functions by using the sequence approach to 8.
There is also an intimate connection between 6 and convolution. For
a tempered distribution g with compact support, the convolution with
any / G Sf
can be defined. (The convolution of two I/(IR) functions is
defined to be / * g(t) := f(t - s)g(s)ds).

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Mr. Coleman. Yes.
Miss Waterman. Outside the Passport Office?
Mr. Coleman. Yes; I am just asking you for the record.
Miss Waterman. I know. But you mentioned—such as who?
Mr. Coleman. Did anyone call you up and say, "Miss Waterman,
this is the way you have to resolve this case"?
Miss Waterman. Oh, no. Oh, no.
Mr. Coleman. And you made the decisions you made based upon
the record and your judgment as to what you thought the law was
and what the facts were?
Miss Waterman. Certainly.
Mr. Dulles. Did you consult anyone in connection with reaching
that decision in the Oswald case?
Miss Waterman. Well, Mr. Dulles, in preparing this
correspondence, as I have told you, the correspondence was
prepared for the signature of my superiors, and if they didn't agree
with what I wrote, that was all right with me. But that was my
impression, and I believed there had been discussion among persons
in our immediate office. And while——
Mr. Dulles. Your decision, then, is not final. It is subject to review
by your superiors in matters of this kind?
Miss Waterman. That is right.
But in no event—I don't know of any—as I say, my connection
with the case closed, and I never heard in the press or any other
place that indicated that Oswald expatriated himself and that he
wasn't entitled to a passport.
Mr. Coleman. Your decision wasn't in any way influenced by the
fact that Miss James told you that this was a decision that would
have to be made or anything like that?

Miss Waterman. Certainly not. They have absolutely nothing to do
with citizenship—nothing.
Mr. Coleman. I have no further questions, Mr. Chairman.
Mr. Dulles. Off the record.
(Discussion off the record.)
Mr. Coleman. Mr. Chairman, before we close the testimony of Miss
Waterman, I would like to move for the admission of Commission
Exhibits No. 957 through Commission Exhibit No. 983, which were
the documents that we marked.
Mr. Dulles. They shall be admitted.
(The documents heretofore marked for identification as
Commission Exhibits Nos. 957–983, were received in evidence.)
Mr. Coleman. I would like to thank Miss Waterman for coming in.
Mr. Dulles. We thank you very much, Miss Waterman.
(Whereupon, at 12:50 p.m., the President's Commission
recessed.)

Afternoon Session
TESTIMONY OF THE HON. DEAN
RUSK, SECRETARY OF STATE
The President's Commission reconvened at 3:30 p.m.
The Chairman. Mr. Secretary Dean Rusk, we wanted to ask you a
few questions about this matter in any particular detail you wanted
to answer. Mr. Rankin would you inform the Secretary the areas we
intend to cover before we ask the questions.
Mr. Rankin. Mr. Chief Justice, I think the particular area that we
would be interested in with the Secretary is just as to whether, or his
knowledge of whether there was any foreign political interest in the
assassination of President Kennedy?
We have been getting the information in regard to other matters
concerning the State Department from other of his associates and
colleagues and employees of the Department, and we are going to
complete that and it has been helpful to us and I think we can rather
limit the inquiry to that area.
The Chairman. Yes; very well.
Mr. Secretary, would you rise and be sworn, please. Do you
solemnly swear the testimony you are about to give before this
Commission shall be the truth, the whole truth, and nothing but the
truth, so help you God?
Secretary Rusk. I do.

The Chairman. Will you be seated, please, and Mr. Rankin will ask
you the questions, Mr. Secretary.
Secretary Rusk. Mr. Chief Justice, may I ask one question?
The Chairman. Yes, indeed.
Secretary Rusk. I would like to be just as helpful as possible to
the Commission. I am not quite clear of testimony in terms of future
publication. There may be certain points that arise where it might be
helpful to the Commission for me to comment on certain points but
there—it would be a very grave difficulty about publication, so I
wonder what the Commission's view on that is.
The Chairman. Well, Mr. Secretary, our purpose is to have
available for the public all of the evidence that is given here. If there
is any phase of it that you think might jeopardize the security of the
Nation, have no hesitation in asking us to go off the record for a
moment, and you can tell us what you wish.
Secretary Rusk. Thank you, sir, I am at your disposal.
Mr. Dulles. Mr. Chief Justice, could I make a suggestion in that
connection?
The Chairman. Yes.
Mr. Dulles. Would it be feasible to have a discussion here of the
points that are vital from the point of view of our record, and so
forth, and maybe a little informal conversation afterward to cover
the other points.
The Chairman. We will have a recess for a few moments then.
Mr. Dulles. I thought between the two wouldn't that be easier
than put the two together.
The Chairman. Back on the record.
Mr. Rankin. Mr. Secretary, will you give us your name and
address, please?

Secretary Rusk. Dean Rusk, 4980 Quebec Street, Washington,
D.C.
Mr. Rankin. And you are the Secretary of State for the United
States?
Secretary Rusk. That is correct.
Mr. Rankin. You have occupied that position for some time?
Secretary Rusk. Since January 22, 1961.
Mr. Rankin. In that position you have become familiar with our
foreign relations and the attitude and interest in some degree of
other countries that we deal with?
Secretary Rusk. Yes; within the limitations of the possibilities, it is
at least my task to be as familiar as possible with those things.
Mr. Rankin. In your opinion, was there any substantial interest or
interests of the Soviet Union which would have been advanced by
the assassination of President Kennedy?
Secretary Rusk. I would first have to say on a question of that
sort that it is important to follow the evidence. It is very difficult to
look into the minds of someone else, and know what is in someone
else's mind.
I have seen no evidence that would indicate to me that the
Soviet Union considered that it had an interest in the removal of
President Kennedy or that it was in any way involved in the removal
of President Kennedy. If I may elaborate just a moment.
Mr. Rankin. If you will, please.
Secretary Rusk. As the Commission may remember, I was with
several colleagues in a plane on the way to Japan at the time the
assassination occurred. When we got the news we immediately
turned back. After my mind was able to grasp the fact that this
event had in fact occurred, which was the first necessity, and not an
easy one, I then, on the plane, began to go over the dozens and

dozens of implications and ramifications of this event as it affects our
foreign relations all over the world.
I landed briefly in Hawaii on the way back to Washington, and
gave some instructions to the Department about a number of these
matters, and learned what the Department was already doing. But
one of the great questions in my mind at that time was just that
question, could some foreign government somehow be involved in
such an episode.
I realized that were this so this would raise the gravest issues of
war and peace, but that nevertheless it was important to try to get
at the truth—to the answer to that question—wherever that truth
might lead; and so when I got back to Washington I put myself
immediately in touch with the processes of inquiry on that point, and
as Secretary of State had the deepest possible interest in what the
truthful answer to those questions would be, because it would be
hard to think of anything more pregnant for our foreign relations
than the correct answer to that question.
I have not seen or heard of any scrap of evidence indicating that
the Soviet Union had any desire to eliminate President Kennedy nor
in any way participated in any such event.
Now, standing back and trying to look at that question
objectively despite the ideological differences between our two great
systems, I can't see how it could be to the interest of the Soviet
Union to make any such effort.
Since I have become Secretary of State I have seen no evidence
of any policy of assassination of leaders of the free world on the part
of the Soviets, and our intelligence community has not been able to
furnish any evidence pointing in that direction.
I am sure that I would have known about such bits of evidence
had they existed but I also made inquiry myself to see whether there
was such evidence, and received a negative reply.
I do think that the Soviet Union, again objectively considered,
has an interest in the correctness of state relations. This would be

particularly true among the great powers, with which the major
interests of the Soviet Union are directly engaged.
Mr. Rankin. Could you expand on that a little bit so that others
than those who deal in that area might understand fully what you
mean?
Secretary Rusk. Yes; I think that although there are grave
differences between the Communist world and the free world,
between the Soviet Union and other major powers, that even from
their point of view there needs to be some shape and form to
international relations, that it is not in their interest to have this
world structure dissolve into complete anarchy, that great states and
particularly nuclear powers have to be in a position to deal with each
other, to transact business with each other, to try to meet problems
with each other, and that requires the maintenance of correct
relations and access to the leadership on all sides.
I think also that although there had been grave differences
between Chairman Khrushchev and President Kennedy, I think there
were evidences of a certain mutual respect that had developed over
some of the experiences, both good and bad, through which these
two men had lived.
I think both of them were aware of the fact that any Chairman
of the Soviet Union and any President of the United States
necessarily bear somewhat special responsibility for the general
peace of the world.
Indeed without exaggeration, one could almost say the existence
of the Northern Hemisphere in this nuclear age.
So that it would be an act of rashness and madness for Soviet
leaders to undertake such an action as an active policy. Because
everything would have been put in jeopardy or at stake in
connection with such an act.
It has not been our impression that madness has characterized
the actions of the Soviet leadership in recent years.

I think also that it is relevant that people behind the Iron
Curtain, including people in the Soviet Union and including officials in
the Soviet Union, seemed to be deeply affected by the death of
President Kennedy.
Their reactions were prompt, and I think genuine, of regret and
sorrow. Mr. Khrushchev was the first to come to the Embassy to sign
the book of condolences. There were tears in the streets of Moscow.
Moscow Radio spent a great deal of attention to these matters.
Now they did come to premature conclusions, in my judgment,
about what this event was and what it meant in terms of who might
have been responsible for it—and ideological effect has crept into
that.
But I had the impression that the regret was genuine and that
the ordinary Soviet citizen joined with ordinary people in other parts
of the world in feeling the loss of the President in a very genuine
sense.
Mr. Rankin. There has been some suggestion that possibly the
leadership of the Soviet Union would not have been politically
interested in the death of the President but possibly a distant wing
of the Party might have been so involved.
Can you give us any light on that, Mr. Secretary.
The Chairman. By suggestion you mean rumor?
Mr. Rankin. In the newspapers, and things of that kind, rumor.
Secretary Rusk. I haven't been able to put a rational structure
behind that possibility. If there are dissident elements their primary
problem is within the Soviet Union.
If these dissident elements were aiming to change the present
Government of the Soviet Union or its leadership or to return to an
early range of policy by the elimination of present leadership or
seizure of control, I don't quite see how the elimination of the
President of the United States could contribute to that purpose.

I would also suppose that in their kind of system such elements
would be under pretty close supervision and surveillance and they
would have limited opportunities for the kind of action that would be
organized in a way in this direction, although that is a matter of
some speculation.
But, I would doubt very much that such dissident elements
would have a motive or very much of an opportunity. Again, I have
seen no evidence pointing in that direction.
Mr. Rankin. How could you tell us in regard to Cuba in the same
general way, your opinion and knowledge of any information or
credible evidence?
Secretary Rusk. Well, I would again repeat that the overriding
consideration is to make every possible effort to find evidence and
follow the evidence to wherever it leads.
I think it is, at least for me, more difficult to try to enter into the
minds of the present leadership in Cuba than, perhaps, even of the
present leadership of the Soviet Union. We have had very few
contacts, as the Commission knows, with the present Government of
Cuba.
But again, I have seen no evidence that seems to point in that
direction.
There were some exchanges, with which the Commission is
familiar, that seemed to be—seemed to come to another conclusion.
But I would think that objective considerations would mean that it
would be even greater madness for Castro or his government to be
involved in any such enterprise than almost for anyone else, because
literally the issue of war and peace would mean the issue of the
existence of his regime and perhaps of his country might have been
involved in that question.
We were under the impression that there was very considerable
concern in Cuba as to whether they would be held responsible and
what the effect of that might be on their own position and their own
safety.

But I have seen no evidence that points to involvement by them,
and I don't see objective facts which would seem to make it in their
interests to remove Mr. Kennedy.
You see, this embarks upon, in any event it would embark upon,
an unpredictable trail for them to go down this path, but I would
think again the Commission would wish to examine the evidence as
it has been doing with meticulous care and follow the evidence in
these matters.
Mr. Rankin. After the assassination, did you have direct
communications with Ambassador Thomas Mann while he was still
Ambassador at Mexico?
Secretary Rusk. Yes; we had a number of exchanges with
Ambassador Mann connected with the presence in Mexico of Mr.
Oswald.
I say those messages, and over a period of some days had daily
consultations about them with our Deputy Under Secretary for
Political Affairs, Mr. U. Alexis Johnson. Mr. Johnson is my principal
representative in our dealings with the various intelligence and
security agencies of the government and with the Pentagon, and he
has an office very near mine on the seventh floor of the Department
of State.
These exchanges raised questions of the most far-reaching
character involving the possibility of the implications of another
government, and so I had a very deep personal interest in that at
the time.
Our principal concern was to be sure that the FBI and the CIA
who were the principal agencies investigating this matter would have
every possible facility at their disposal, and would—and that our
Ambassador would be given the fullest support from us in facilitating
the investigation at the Mexican end.
So I was for a period, until this particular trail ran its course,
very much involved in those exchanges.

Mr. Rankin. Do you have any commentary that you want to make
about those exchanges other than what you have given us?
Secretary Rusk. I think not, sir. I think that the materials, the
information developed in those exchanges are before the
Commission, and I believe the Commission has had a chance to
inquire into them both as I understand both here and in Mexico with
the appropriate agencies and I would think that the Commission's
conclusions on that would be more valuable than mine because I
have not put together all the pieces to draw finished conclusions
from them.
Mr. Rankin. One of the Commissioners saw a newspaper story
shortly after the assassination saying "The Voice of America beaming
its message into Russia immediately blamed the reactionary
rightwing movements after Kennedy's death."
Do you know anything about that matter or what the source of it
might have been?
Secretary Rusk. No; I have not anticipated that question so that I
could have a chance to investigate it, but I will, if I may, Mr. Chief
Justice, file a report with the Commission on that point.
I can say now that there was never any policy guidance from the
Department of State or from the leadership of the Voice of America
suggesting that any broadcasters take that line.
It is possible, and this is purely speculative at the moment, that
the Voice of America in repeating a great many news accounts, as it
frequently does in its overseas broadcasts, may have repeated some
news accounts from this country, among which might have been a
story to that effect from one source or another, but I would like if I
may, sir, an opportunity to investigate that point and make a report
to the Commission.
The Chairman. You may do that, Mr. Secretary.
Representative Ford. May I ask a question? Have we received in
the Commission all of the Voice of America broadcasts that were

made over a period of 2 to 7 days involved in this incident?
Mr. Rankin. I don't know of any.
Representative Ford. I think the Commission ought to have them
for our own analysis as well as the analysis of the Secretary of State.
Mr. Rankin. Is that under your jurisdiction?
Secretary Rusk. Yes; indeed I could provide that.
Mr. Rankin. If you will, please.
Secretary Rusk. The Commission might also be interested in
either digests or the fuller materials on world reactions to the
President's assassination.
I have here, for example, a daily summary of the 26th of
November 1963, on foreign radio and press reaction which gives
some interesting treatment about this behind the Iron Curtain.
I would be happy to furnish the Commission with any material of
that sort which you might wish.
Mr. Rankin. We would appreciate having that.
The Chairman. Very well, thank you, Mr. Secretary.
Representative Ford. Would that include the Voice of Moscow or
whatever they call it over there?
Secretary Rusk. Yes, sir.
Representative Ford. From the outset of the events that took
place?
Secretary Rusk. Yes, sir; you might just wish to look at the first
two or three paragraphs here to get a sample of the kind of
summary that that involves.
Mr. Dulles. Was that prepared in the Department or by the
Foreign Broadcast Information Service?
Secretary Rusk. This particular one is from the Foreign Broadcast
Information Service. We also have another one. We also have

another one from within the Department which is also available in
terms.
Representative Ford. I think it would be useful to have both for a
period of about a week or so. I realize this is a summary covering
several days. I think I saw that at the time.
Mr. Rankin. There was another statement in the paper apparently
purporting to be official that one of the Commissioners asked me to
ask about and that was the Washington Post, Sunday, November 24,
1963, which was quoted by the Commissioner as, "Today in
Washington State Department officials said they have no evidence
indicating involvement of any foreign power in the assassination."
Do you know anything about that or can you give us any
information?
Secretary Rusk. That was the view which we took at the time in
consultation with the investigative agencies. We did not then have
evidence of that sort nor do we now, and the implications of
suggesting evidence in the absence of evidence would have been
enormous.
Representative Ford. I don't understand that.
Secretary Rusk. Well, for us to leave the impression that we had
evidence that we could not describe or discuss, when in fact we
didn't have the evidence on a matter of such overriding importance
could have created a very dangerous situation in terms of——
Representative Ford. Wouldn't it have been just as effective to
say no comment?
Secretary Rusk. Well, unfortunately, under the practices of the
press, no comment would have been taken to confirm that there was
evidence. I mean, that would have been the interpretation that
many would have put upon no comment.
But, Mr. Ford, I think the key thing is that at the time that
statement was made we did not have such evidence. I mean, this
was a factual statement at that time.

Representative Ford. But, at that time, this was 2 days after the
assassination, you really didn't have much time to evaluate all of the
evidence.
Secretary Rusk. Well, that is correct. But if the evidence or the
known facts had changed certainly that type of statement would
have changed.
In other words, such statements are based upon the situation as
known at the time the statements are made.
Representative Ford. This statement then appeared in the
Sunday morning, November 24 issue or edition of the Washington
Post. That was a statement issued certainly on the 23d of November
because it had to be in order to get in the Sunday edition of the
Post. So, that is 24 hours after the assassination.
Secretary Rusk. That is correct, sir, and this statement was made
on the basis of such information as was available to us in the first 24
hours.
Mr. Rankin. I was also asked to inquire whether that was an
official statement if under your responsibility or if you could tell me
who would be responsible for it?
Secretary Rusk. Well, I would have to check the actual source of
the statement. But I would have no present doubt that it was an
officer of the Department who was authorized to make that and for
which I would be fully responsible.
Mr. Rankin. That is all I have.
Mr. Dulles. Could I ask a question in connection with that?
The Chairman. Mr. Dulles.
Mr. Dulles. There was some evidence presented here quite
recently when the district attorney of Dallas was here with regard to
a message from Washington, from the White House to the attorney
general of Texas, who was also here the other day before the
Commission, on this point: A rumor had reached Washington that in

preparing the indictment there, they were going to put in some
reference to an international conspiracy. As a matter of fact, when
that was run down it was not a correct rumor. But when that
reached Washington, the reaction was rather strong and I think
entirely understandable, and word went back to Dallas from high
quarters that that should not, hoped that that would not be included
in the legal proceedings and papers that were filed in connection
with the assassination of the President and charging——
Mr. Rankin. Unless there was evidence to support it.
Mr. Dulles. Unless there was evidence to support it. And the
district attorney, who was here, testified that he had never
considered adding that into it, putting that in the proceedings
because if you put it in you had to prove it, and it is not necessary at
all. All you need to do is allege a murder with intent, and so forth,
and so on. So that that was all pretty well cleared up.
Mr. Dulles. Did that ever reach your attention, did you know
anything about that?
Secretary Rusk. I don't personally recall that particular message.
I do recall——
Mr. Dulles. That took place, I think before you got back, because
that took place on the evening of the 22d.
Secretary Rusk. I didn't arrive until——
Mr. Dulles. You didn't get back until the 23d?
Secretary Rusk. Until the early morning of the 23d.
Mr. Dulles. Yes.
Secretary Rusk. I do recall being concerned if several different
authorities and agencies undertook investigations that would cut
across each other's bow or make it difficult to elicit the cooperation
of people outside the United States whose cooperation we might
need in matters of that sort, I felt myself at that time there ought to
be a complete and absolutely thorough investigation by the most

responsible authorities and I was glad to see that brought into some
order at the time but I don't remember the particular message you
are talking about.
Representative Ford. Could you check to see if somebody in the
Department of State made such a call or made such a contact?
Secretary Rusk. Yes; I will be be glad to.
Representative Ford. And if so so report it for the proceedings?
Secretary Rusk. Yes, indeed; I will be glad to.
Mr. Chayes. I may be able to supply some information to the
Commission on this point because during the night of the 22d when
we were examining the data in my office, the files, I did receive a
call from Mr. Katzenbach who said that they had heard at the Justice
Department, that there was a possibility that this kind of an element
would get into the indictment, and said that—I can't remember the
exact words that he used—but he conveyed to me that he regarded
this as not very good, in the absence of evidence to support it, and
said that he was seeking to have Mr. Saunders, who is the U.S.
attorney in Dallas, admitted to the councils of the State officials
there so that they could discuss these matters as time went on. And
that he would try to, I don't know exactly again what he said, but
that he would try to see that in the absence of evidence no such
allegation was made in the indictment.
I didn't in any sense authorize, and I certainly couldn't direct him
to do anything of this kind but my recollection of my reaction is that
I acquiesced fully in what he was proposing to do, and raised no
objection to it.
I think at sometime during that evening I reported this
conversation to Mr. Ball. I am less clear about this part of the
recollection, but I think I did report the conversation to Mr. Ball,
much in the same way as I am reporting it to you, and he saw no
objection either.

I think that is the entire State Department side of that particular
transaction.
Representative Ford. Would you check, however, Mr. Secretary,
to see if there is anything further in this regard?
Secretary Rusk. Yes; I will.
Representative Ford. Do I understand that you or somebody for
you is to summarize the USIA Voice of America broadcast that went
out for the first 3 or 4 days subsequent to the assassination and that
would be submitted for the record?
Secretary Rusk. Yes, indeed. And we can, of course, have
available to the Commission such tapes or transcripts as we have of
all those broadcasts in full, but I think we can start with the
summary and then you can have the other materials if you wish to
follow up particular points.
Representative Ford. Would they be voluminous, the originals?
Secretary Rusk. I would think they would be fairly voluminous,
but not unmanageably so.
Representative Ford. I would say for at least the first 24 hours it
might be well to have the full text of the USIA Voice of America
material that was sent out.
Secretary Rusk. Right.
Representative Ford. Do I also understand for the record that we
are to have this or others like it showing what the press reaction was
throughout the world?
Secretary Rusk. Yes, sir.
Now, the Foreign Broadcast Information Service material would
be much more voluminous because there we are receiving
broadcasts in the clear from most broadcasting countries. But we will
be in touch with your staff to show them everything that we have,
and they can have any part of it they wish or we will be glad to give
any help in terms of digesting or summarizing.

Mr. Rankin. We have been furnished some information,
considerable information, about the attitude of the foreign press as it
was recited and has come to the attention of the people from time
to time, but I don't believe we have right close, the Voice of America
we don't have right close to the date of the assassination.
The Chairman. I read a sizable file on that that came from the
State Department and very early in the life of the Commission that
seemed to encompass all of the statements that were made around
the world at that time.
Secretary Rusk. Yes.
Representative Ford. This document which you handed me, Mr.
Secretary, is for Tuesday, 26 November 1963. Are these done on a
daily basis?
Secretary Rusk. I think that one was a summary of the first 2 or
3 days, but I would——
Mr. Dulles. Summaries are done from time to time and there are
daily reports from Foreign Broadcasting Information Service covering
the Soviet Union and the satellites and another volume covering
China and southeast Asia, and so forth and so on.
Mr. Rankin. Mr. Secretary, could you give us a brief description of
that, we have been calling it this and these.
Secretary Rusk. Yes; this is a daily report or rather a supplement
to the daily report put out by the Foreign Broadcast Information
Service in what is called its world reaction series.
This apparently is a supplement to the foreign radio and press
reaction to the death of President Kennedy, and the accession of
President Johnson, prepared on 26 November 1963.
This is a daily report, the subject matter of which varies from
day to day, but I will be glad to draw together not only such digests
as we have, but also to see what we have retained in terms of the
actual broadcasts from other countries so that although it may be

voluminous it might have some material of interest to the
Commission or its staff.
Representative Ford. I think it would be particularly pertinent as
far as the Soviet Union or any of the bloc countries or Cuba,
anything in this area that could be pulled together and included in
the record, which I think would be very helpful.
Secretary Rusk. All right, sir.
Representative Ford. I have the recollection that some people
have alleged that Castro either prior to or subsequent to the
assassination, made some very inflamatory speech involving
President Kennedy.
Do you have any recollection of that?
Secretary Rusk. I don't have a recollection of a speech specially
related to time. He has made more than his share of inflamatory
speeches about this country and its leaders. But I will be glad to
furnish the Commission a schedule of his speeches, and the
character of these speeches and the texts if we have them during
this period.
Representative Ford. There was one that I vaguely recall, either
prior to or subsequent to the assassination that some people
construed to be directed specifically at President Kennedy, and I
think if there was such a speech that the Commission ought to have
it and it ought to be analyzed by the staff and by the Commission.
Secretary Rusk. We will be very glad to look into that and furnish
you with speeches made during this period or during a substantial
part of the period on both sides of the November 22 date.
I gather the Commission has Mr. Danielle's interview with Mr.
Castro on the subject. You have the published report of that.
Mr. Dulles. Was that the long interview with Castro?
Secretary Rusk. Yes; that was as close to any reflection of a thing
that he might have said personally about this that went beyond the

kind of broadcast speeches you referred to that I have seen, but——
Mr. Dulles. Do you have that available?
Secretary Rusk. We certainly can get it.
Mr. Dulles. It was in the press I guess at the time. Maybe you
have a fuller copy than we have.
Secretary Rusk. Yes; it was a rather extensive interview.
Mr. Chayes. I think the staff has it already.
Secretary Rusk. I see.
Mr. Rankin. I think Commissioner Ford is referring to that speech
of Mr. Castro which is sometimes called the slip-of-the-tongue
speech that referred in a way that may have some implications in it.
I think that might help you to identify it, Mr. Secretary.
Secretary Rusk. It might be well for me, just to complete the
sense of the atmosphere, to accompany that with the timing and the
nature of statements and speeches that were being made on our
side as a part of this continuing rather acrimonious discourse with
Cuban leadership. But I will provide full information on this.
Mr. Rankin. We would appreciate it so it would give a complete
picture.
Secretary Rusk. Yes.
Representative Ford. Do I understand now, Mr. Rankin, that what
the Secretary provides will be put in the record as exhibits?
Mr. Rankin. Mr. Chairman, I would like to offer to do that if that is
satisfactory, as a part of this record.
The Chairman. Yes, sir; it might be admitted.
Representative Ford. There is one question that I think ought to
be cleared up, you mentioned Mr. Mann who was our Ambassador at
Mexico at that time. The way the record stands now it could be
construed by somebody who wanted to so construe it that the

country in which he served us was involved in what he was
reporting. I think it ought to be made clear that is not the case.
Secretary Rusk. That is absolutely correct, sir. We never had the
slightest view that Mexico was involved in this. The problem, the
question arose because Mr. Oswald had been in Mexico, and was
known to have been in touch with some Cubans at the Cuban
Embassy in Mexico. But the Mexican authorities gave us complete
and the most helpful cooperation in full investigation of this matter.
The Chairman. Are there any further questions? Mr. Dulles.
Mr. Dulles. Had you finished?
Mr. Rankin. Yes; I have.
The Chairman. Are we ready to go back on the record?
All right, the Commission will be in order.
Mr. Rankin. Mr. Chief Justice, I should like to offer in evidence at
this point Commission Exhibit No. 984 being the communication
from yourself as Chairman of the Commission to the Secretary of
State, dated March 11, 1964, and the Note Verbale in regard to the
inquiries of the Soviet Union.
And Commission Exhibit No. 985 being the responses of the
Soviet Union, including all of the medical as well as all other
responses together with the transmittal letters from the Soviet Union
and from the State Department.
The Chairman. They may be admitted under those numbers.
(Commission Exhibits Nos. 984 and 985 were marked for
identification and received in evidence.)
Mr. Rankin. I would like to assign, Mr. Chief Justice, Commission
Exhibit No. 986, if I may, to those prior communications from the
files of the Soviet Embassy in Washington that were furnished to us
by the State Department.

The Chairman. They may be admitted under that number.
(Commission Exhibit No. 986 was marked for identification and
received in evidence.)
Mr. Rankin. Commission Exhibit No. 986 will be the copies of the
records from the Soviet Embassy in Washington that were supplied
to the Commission earlier by the State Department as a part of the
records that were furnished to us by the State Department.
The Chairman. Those were the ones that were voluntarily offered
by the Russians before any request was made of them?
Mr. Rankin. Yes, Mr. Chairman.
The Chairman. They may be admitted under that number.
Mr. Rankin. Mr. Secretary, will you tell us whether you know of
any credible evidence to show or establish or tending to show any
conspiracy either domestic or foreign involved in the assassination of
President Kennedy?
Secretary Rusk. No; I have no evidence that would point in that
direction or to lead me to a conclusion that such a conspiracy
existed.
Mr. Rankin. That is all I have.
The Chairman. Are there any further questions, gentlemen?
If not, thank you very much, Mr. Secretary.
Secretary Rusk. Thank you very much, Mr. Chief Justice and
gentlemen.

TESTIMONY OF FRANCES G.
KNIGHT
The Chairman. The Commission will be in order.
Mr. Coleman, will you state to Miss Knight, please, the reason we
asked her to come here today?
Mr. Coleman. Miss Frances G. Knight is the head of the Passport
Office of the State Department.
Miss Knight. Yes, sir.
Mr. Coleman. We want to ask her concerning the standard
operating notice with respect to the lookout card system which was
in effect as of November—as of February 28, 1962, and we also
wanted to ask her concerning the decision of the Passport Office
that Mr. Oswald had not expatriated himself and, therefore, he
should be reissued his passport.
Miss Knight. Yes, sir.
The Chairman. Would you raise your right hand and be sworn,
Miss Knight?
Do you solemnly swear the testimony you are about to give
before the Commission shall be the truth, the whole truth, and
nothing but the truth, so help you God?
Miss Knight. I do.
The Chairman. Be seated. Mr. Coleman will ask you the questions.
Mr. Coleman. Miss Knight, will you state your name for the
record?

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