Table Of ContentReal Analysis with an Introduction
to Wavelets
Don Hong
Jianzhong Wang
Robert Gardner
Author address:
(D. Hong, R. Gardner) Department of Mathematical and
Statistical Sciences, East TennesseeState University, John-
son City, Tennessee
(J.Wang)Department of Mathematics and Statistics, Sam
Houston State University, Huntsville, Texas
E-mail address, D. Hong: [email protected]
E-mail address, J. Wang: mth [email protected]
E-mail address, R. Gardner: [email protected]
.
To our wives: Ying, Hui-Lan, and Kathryn.
Contents
1. Preface v
Chapter 1. Fundamentals 1
1. Elementary Set Theory 1
2. Relations and Orderings 5
3. Cardinality and Countability 12
4. The Topology of Rn 18
Chapter 2. Measure Theory 31
1. Classes of Sets 32
2. Measures on a Ring 36
3. Outer Measures and Lebesgue Measure 42
4. Measurable Functions 50
5. Convergence of Measurable Functions 56
Chapter 3. The Lebesgue Integral 59
1. The Riemann Integral and the Lebesgue Integral 59
2. The General Lebesgue Integral 65
3. Convergence and Approximation of Lebesgue Integrals 70
4. Lebesgue Integrals in the Plane 76
Chapter 4. Special Topics of Lebesgue Integral and Applications 81
1. Differentiation and Integration 81
2. Mathematical Models for Probability 93
3. Convergence and Limit Theorems 99
Chapter 5. Vector Spaces, Hilbert Spaces, and the L2 Space 105
1. Groups, Fields, and Vector Spaces 105
2. Inner Product Spaces 114
3. The Space L2 120
4. Projections and Hilbert Space Isomorphisms 124
5. Banach Spaces 132
Chapter 6. Fourier Analysis 141
1. Fourier Series 142
iii
iv CONTENTS
2. Parseval’s Formula 152
3. The Fourier Transform of Integrable Functions 160
4. Fourier Transforms of Square Integrable Functions 172
5. The Poisson Summation Formula 179
Chapter 7. Orthonormal Wavelet Bases 189
1. Haar Wavelet Basis 191
2. Multiresolution Analysis 198
3. Orthonormal Wavelets From MRA 205
4. Orthonormal Spline Wavelets 215
5. Fast Wavelet Transforms 225
6. Biorthogonal Wavelet Bases 236
Chapter 8. Compactly Supported Wavelets 245
1. Symbols of Orthonormal Scaling Functions 245
2. The Daubechies Scaling Functions 258
3. Computation of Daubechies Scaling Functions 266
4. Wavelet Packets 275
5. Compactly Supported Biorthogonal Wavelet Bases 281
Chapter 9. Wavelets in Signal Processing 289
1. Signals 289
2. Filters 297
3. Coding Signals by Wavelet Transform 308
4. Filter Banks 316
Appendix A. List of Symbols 323
Bibliography 325
Index 328
1. PREFACE v
1. Preface
Real Analysis is based on the real numbers, and it is naturally
involved in practical mathematics. On the other hand, it has taken on
subject matter from set theory, harmonic analysis, integration theory,
probability theory, theory of partial differential equations, etc., and
hasprovided these areas withimportant ideas andbasic concepts. This
relationshipcontinues eventoday. Thisbook,containsthebasicmatter
of “real analysis” and an introduction to “wavelet analysis,” a popular
topic in “applied real analysis.”
Nowadays, studentsstudyingrealanalysiscomefromdifferentback-
groundsandhave diverse objectives. Acourse inthissubject mayserve
both undergraduates and graduates, students not only in mathematics
but also in statistics, engineering, etc., and students who are seek-
ing master’s degrees and those who plan to pursue doctoral degrees in
the sciences. Written with the student in mind, this book is suitable
for students with a minimal background. In particular, this book is
written as a textbook for the course usually called Real Analysis as
presently offered to first or second year graduate students in American
universities with a master’s degree program in mathematics.
The subject matter of the book focuses on measure theory, the
Lebesgue integral, topics in probability, Lp spaces, Fourier analysis,
and wavelet theory as well as on applications. It contains many fea-
tures that are unique for a real analysis text. This one-year course
provides students the material in Fourier analysis, wavelet theory and
applications and makes a very natural connection between classical
pure analysis and the applied topic — wavelet theory and applications.
The text can also beused fora one-semester course only onReal Analy-
sis (I) by covering Chapters 1-3 and selected material in Chapters 4
and 5 or a one-semester course on Applied Analysis or Introduction to
Waveletsusing material inChapters 5–9. Here area fewother features:
• The text is relatively elementary at the start, but the level of
difficulty increases steadily.
• The book includes many examples and exercises. Some exercises
form a supplementary part of the text.
• Different from the classical real analysis books, this text covers
a number of applied topics related to measure theory and pure analy-
sis. Some topics in basic probability theory, the Fourier transform,
and wavelets provide interesting subjects in applied analysis. Many
projects can be developed which could lead to qualitative undergradu-
ate/graduate research theses.
vi CONTENTS
• The text is intended mainly for graduates pursuing a master’s
degree, so only a basic background in linear algebra and analysis is
assumed. Wavelet theory is also introduced at an elementary level,
but it leads to some updated research topics. We provide relatively
complete references on relevant topics in the bibliography at the end
of the text.
This text is based on our classnotes and a primary version of this
text has been tested in the classrooms of our universities on several oc-
casions in the courses of Real Analysis, Topics in Applied Mathematics,
and Special Topics on Wavelets.
Though the text is basically selfcontained, it will be very helpful
for the reader to have some knowledge of elementary analysis — for
example, the material in Walter Rudin’s Principal of Mathematical
Analysis [25].
The Outline of the book is as follows.
Chapter 1 is intended as reference material. Many readers and
instructors will be able to quickly review much of the material. This
chapter is intended to make the text as self-contained as possible and
also to provide a logically ordered context for the subject matter and
to motivate later development.
Chapter 2 presents the elements of measure theory by first dis-
cussing measure on rings of sets and then the Lebesgue theory on the
line.
Chapter 3 discusses Lebesgue integrationanditsfundamental prop-
erties. This material is prerequisite to subsequent chapters.
Chapter 4 explores the relationship of differentiation and integra-
tion and presents some of the main theorems in probability which are
closely related to measure theory and Lebesgue integration.
Chapters 5 and 6 provide the fundamentals of Hilbert spaces and
Fourier analysis. These two chapters become a natural extention of the
Lebesgue theory and also a preparation for the later wavelet analysis.
Chapters 7 and 8 include basic wavelet theory by starting with the
Haar basis and multiresolution analysis and then discussing orthog-
onal wavelets and the construction of compactly supported wavelets.
Smoothness, convergence, andapproximationpropertiesofwaveletsare
also discussed.
Chapter9providesapplicationsofwavelets. Weexaminedigitalsig-
nals and filters, multi-channel coding using wavelets, and filter banks.
A website is available which contains a list of known errors and
updates for this text:
http://www.etsu.edu/math/real-analysis/updates.htm.
1. PREFACE vii
Please e-mail the authors if you have any input for this webpage.
Wearedeeply gratefultoProfessor Charles K.Chui forhisconstant
encouragement, kind support, and his patience. Our thanks are due to,
in particular, Jiansheng Cao, Bradley Dyer, Rene´e Ferguson, Wenyin
Huang, Joby Kauffman, Jun Li, Anna Mu, Chris Wallace, Panrong
Xiao, and Qingbo Xue for giving us useful suggestions from the stu-
dents’ point of view. It is a pleasure to acknowledge the great support
given to us by Robert Ross, Barbara Holland, and Mary Spencer in
Academic Press.
Don Hong & Robert Gardner
Johnson City, TN
Jianzhong Wang
Huntsville, TX
viii CONTENTS
CHAPTER 1
Fundamentals
The concepts of real analysis rest on the system of real numbers and
on such notions as sets, relations, and functions. In this chapter we
introduce some elementary facts from set theory, topology of the real
numbers, and the theory of real functions that are used frequently
in the main theme of this text. The style here is deliberately terse,
since this chapter is intended as a reference rather than a systematic
exposition. For a more detailed study of these topics, see such texts as
[16] and [25].
1. Elementary Set Theory
We start with the assumption that the reader has an intuitive feel
for the concept of a “set.” In fact, we have no other choice! Since
we can only define objects or concepts in terms of other objects and
concepts, we must draw the line somewhere and take some ideas as
fundamental, intuitive starting blocks (almost atomic in the classical
Greek sense). In fact, one of the founders of set theory, Georg Cantor
(1845–1918),inthelate1800’swrote: “Aset isacollectioninto awhole
of definite, distinct objects of our intuition or our thought” [13]. With
this said, we begin ...
A set is a well-defined collection of objects. The objects in the
collectionwillbecalledelementsoftheset. Forexample, A = {x,y,z}
is a set with three elements x,y, and z. We use the notation x ∈ A
to denote that x belongs to A or equivalently, x is in A. The set A,
in turn, will be said to contain the element x. By convention, a set
contains each of its elements exactly once (as opposed to a multiset
which can contain multiple copies of an element).
Sets will usually be denoted by capital letters: A,B,C,... and
elements by lower-case letters: a,b,c,.... If a does not belong to A, we
write a ∈/ A. We sometimes designate sets by displaying the elements
in braces. For example, A = {2,4,8,16}. If A is the collection of
all x which satisfy a property P, we indicate this briefly by writing
A = {x | x satisfies P}. A set containing no elements, denoted by ∅, is
1
2 1. FUNDAMENTALS
called an empty set. For example, {x | x is real and x2+1 = 0} = ∅.
We say that two sets are equal if they have the same elements.
Usually, we use R to denote the set of real numbers, i.e. R = {x |
−∞ < x < ∞}. Using the interval notation, we have the real line
R = (−∞,∞), the closed interval [a,b] = {x | a ≤ x ≤ b}, open
interval (a,b) = {x | a < x < b), and half-open and half-closed
intervals [a,b) = {x | a ≤ x < b} and (a,b] = {x | a < x ≤ b}.
As usual, the set of natural numbers is denoted by N, the set of
integers by Z, the set of rational numbers by Q, and the set of complex
numbers by C:
N = {1,2,3,...},
Z = {... ,−2,−1,0,1,2,...},
(cid:1) (cid:2) (cid:3)
m (cid:2)
Q = (cid:2) m,n ∈ Z,m (cid:5)= 0 ,
n
and
C = {z | z = x+yi for x,y ∈ R and i2 = −1}.
Given two sets A and B, we say A is a subset of B, denoted by
A ⊂ B, if every element of A is also an element of B. That is, A ⊂ B
means that if a ∈ A then a ∈ B. We say A is a proper subset of B if
A ⊂ B andA (cid:5)= B. NoticethatA = B ifandonlyifA ⊂ B andB ⊂ A.
Theemptyset∅isasubset ofanyset. TheunionoftwosetsAandB is
defined asthe set of all elements that arein Aor inB, and it is denoted
by A∪B. Thus, A∪B = {x | x ∈ A or x ∈ B}. The intersection
of two sets A and B is the set of all elements that are in both A and
B, denoted by A ∩ B. That is, A ∩ B = {x | x ∈ A and x ∈ B}.
If A∩ B = ∅, then A and B are called disjoint. We can generalize
the ideas of unions and intersections of sets from a pair of sets to an
arbitrary collection of sets. For a family of infinitely many sets A ,
λ
λ ∈ Λ (Λ is called the indexing set), union and intersection are
defined as:
(cid:4)
A = {x | x ∈ A for some λ ∈ Λ} and
λ λ
λ∈Λ
(cid:5)
A = {x | x ∈ A for all λ ∈ Λ}.
λ λ
λ∈Λ
We define the relative complement of B in A, also called the
difference set and denoted A\B, to be A\B = {x | x ∈ A and x ∈/
B}. The symmetric difference of A and B, denoted by A∆B, is
defined as A∆B = (A\B)∪(B\A). The following properties are easy
to prove.
Theorem 1.1.1. Let A, B, and C be subsets of X. Then
