Table Of ContentPreface
Real Analysis is based on the real numbers, and 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, the theory of partial
differential equations, etc., and has provided these areas with important ideas
and basic concepts. This relationship continues even today. This book contains
the basic matter of "real analysis" and an introduction to "wavelet analysis," a
popular topic in "applied real analysis."
Today, students studying real analysis come from different backgrounds and
have diverse objectives. A course in this subject may serve both undergrad
uates and graduates, students not only in mathematics but also in statistics,
engineering, etc., and students who are seeking master's degrees and those
who plan to pursue doctoral degrees in the sciences. Written with the stu
dent 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, U spaces, Fourier analysis, and wavelet theory,
as well as on applications. It contains many features 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 be used for a one-semester course only on Real
Analysis (I) by covering Chapters 1-3 and selected material in Chapters 4 and
Xfff
xiv Preface
5 or a one-semester course on Applied Analysis or Introduction to Wavelets using
material in Chapters 5-9. Here are a few other 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.
• In contrast to the classical real analysis books, this text covers a number of
applied topics related to measure theory and pure analysis. Some topics in basic
probability theory, the Fourier transform, and wavelets provide interesting
subjects in applied analysis. Many projects can be developed that could lead
to quality undergraduate/graduate research theses.
• 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 class notes and, a primary version of this text has
been tested in the classroom of our universities on several occasions in the
courses of Real Analysis, Topics in Applied Mathematics, and Special Topics on
Wavelets.
Though the text is basically self-contained, it will be very helpful for the
reader to have some knowledge of elementary analysis — for example, the
material in Walter Rudin's Principles of Mathematical Analysis [25] or Kirkwood's
An Introduction to Analysis [16].
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 discussing mea
sure on the rings of sets and then the Lebesgue theory on the line.
Chapter 3 discusses Lebesgue integration and its fimdamental properties.
This material is prerequisite to subsequent chapters.
Chapter 4 explores the relationship of differentiation and integration 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.
Preface xv
Chapters 7 and 8 include basic wavelet theory by starting with the Haar basis
and multiresolution analysis and then discussing orthogonal wavelets and the
construction of compactly supported wavelets. Smoothness, convergence, and
approximation properties of wavelets are also discussed.
Chapter 9 provides applications of wavelets. We examine digital signals and
filters, multichannel coding using wavelets, and filter banks.
A Web site is available which contains a list of known errors and updates for
this text:
http://www.etsu.edu/math/real-analysis/updates.htm.
Please e-mail the authors if you have any input.
We are deeply grateful to Professor Charles K. Chui for his constant encour
agement, kind support, and patience. Our thanks are due to, in particular,
Jiansheng Cao, Bradley Dyer, Renee Ferguson, Wenyin Huang, Joby Kauff-
man, Jun Li, Anna Mu, Chris Wallace, David Atkins, Panrong Xiao, and Qingbo
Xue for giving us useful suggestions from students' point of view. We thank
our colleagues who have reviewed the manuscript: Doug Hardin (Vanderbilt
University), Pete Johnson (Auburn University), Ram Mohapatra (University of
Central Florida), Dave Ruch (Metropolitan State College of Danver), and Wasin
So (San Jose State University). It is a pleasure to acknowledge the great support
given to us by Robert Ross, Barbara Holland, Mary Spencer, Kyle Sarofeen, and
Tom Singer at Elsevier/Academic Press.
Don Hong and Robert Gardner
Johnson City, TN
Jianzhong Wang
Huntsville, TX
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].
I Elementary Set Theory
We start with the assumption that the reader has an intuitive feel for the con
cept 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), in the late 1800s wrote: "A set is a collection
into a whole 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 collection
will be called elements of the set. For example, A = {x, y, z} is a set with three
elements x, y, and z. We use the notation x G 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
Chapter I Fundamentals
c- ft^?''\--iy^&0.^ ^^w^^'f'fe? 'm!^m^'$^^^sm^m^^^mm>m
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
lowercase letters: a, b, c, li a does not belong to A, we write a ^ A. We
sometimes describe sets by displaying the elements in braces. For example, A =
{2,4,8,16}. If A is the collection of all x that satisfy a property P, we indicate this
briefly by writing A = {x | x satisfies P}. A set containing no elements, denoted
by 0, is called an empty set. For example, {x | x is real and x^ + 1 = 0} = 0. We
say that two sets are equal if they have the same elements.
Usually, we use M to denote ttie set of real numbers, i.e., E = {x| -oo<x<
oo}. Using the interval notation, we have the real line E = (—oo, oo), 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,-l,0,l,2,...}
Q= I- I m,neZ,nio\,
V n \ J
and
C = {z I z = X + y/ for x,i/ G M and f- = -1}.
Given two sets A and B, we say A is a subset of B, denoted by A C B, if every
element of A is also an element of B. That is, A C B means that it a e A then
a eB.VSfe say A is a proper subset of B if A C B and A 7^ B. Notice that A = B if
and only if A C B and B c A. The empty set 0 is a subset of any set. The union
of two sets A and B is defined as the set of all elements that are in A or in B, and
it is denoted by A U B. Thus, A U B = {x | x G A or x G B}. The intersection of
two sets A and B is the set of all elements that are in both A and B, denoted by
AflB. Thatis,AnB = {x | x G Aandx G B}. If AflB = 0, 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 Ax, A G A (A is called the indexing set), union and intersection are
defined as
(^ AA = {x I X G AA for some A G A} and
A€A
PI AA = {x I X G AA for all A G A}.
AeA
I Elementary Set Theory 3
We define the relative complement of B in A, also called the difference set and
denoted A\B,iohe A\B = {x \ x e A and x ^ B}. The symmetric difference of
A and B, denoted by AAB, is defined as AAB = (A \ B) U (B \ A). The following
properties are easy to prove.
Theorem 1.1.1 Let A, B, and C be subsets of X. Then
(i) AUB = BUA,AnB = BnA;
(ii) (A U B) U C = A U (B U C), (A n B) n C = A n (B n C);
(iii) (A U B) n C = (A n C) U (B n C), (A n B) U C = (A U C) n (B U C).
When it is clearly understood that all sets in question are subsets of a fixed
set (called a universal set) X, we define the complement A^ of a set A (in X) as
A^ = X \ A. In this situation we have de Morgan's Laws:
(A U Bf = A^ n B^ and (A n Bf = A'U B'.
In general, we have the following.
Theorem 1.1.2 For any collection of sets Ax, A G A, we have:
[[JA\ =n^Aand (1.1)
\AGA / AGA
{[^A^ =U^A- (1-2)
VAGA / AGA
Proof: We only show (1.1). The proof of (1.2) is similar. Let P = (UAGAAA)""
and Q = HAGAA'A-
For X G P, we have that x t UAGA^A- Thus, for every A G A, x ^ AA and so
X G A\. Therefore, xeQ. This implies that P C Q.
On the other hand, for x G Q, we have that for any A G A, x ^ AA. Thus,
X t UAGA^A. Therefore, x G (UAGA^A)^ = P. This means that Q C P. Hence,
P = Q> •
Let {An) = {An}neN be a sequence of subsets of X. (A„) is said to be in
creasing if
Ai C A2 C A3 C • • • .
(An) is called decreasing if
Ai D A2 D A3 D • • • .
Chapter I Fundamentals
For a given sequence (A„) of subsets of X, we construct two new sequences (B„)
and (C„) of sets as follows:
oo oo
B„ = [JAkand C„^f]Ak.
k=n k=n
Clearly, (B„) is decreasing and (C„) is increasing. The set of intersection of
B«, n G N is called the limit superior of (A„) and is denoted by lim„_oo^n or
lim sup^^^A„. The set of union of C„, n G N, is called the limit inferior of (An)
and is denoted by lim„^^A„ or lim inf„_oo^«- Therefore,
o o / o o\ oo / oo \
hmn-.ooAn = PI I [JAk j and lim„_^^A„ = [j f f^Aj, J .
n=l \k=n J n=l \k=n J
It can be seen that the limit superior is the set of those members that belong
to An for infinitely many values of n, while the limit inferior is the set of those
members that belong to A„ for all but a finite number of subscripts. For
this description it can be seen that lim„^^A„ C lim„^oo^n- H lim„_^A^ =
lim„^ooAn/ then we say the limit of (A„) exists and is denoted by lim A„.
n-^oo
EXAMPLE 1.1.1: Let E and F be any two sets. Define a sequence
(Ak) of sets by
^ r EE,, iiff /: is odd,
" I F, ^ if"i^ Zeis even.
Then, B„ = U^„ A^ = E U F, so limk^^Ak = p| B„ = E U F From the
n=l
fact that Cn = CCn ^it = E H F, n = 1,2,3,... , we obtain lunjt^oo^fc =
oo / cx) \
\jif]AA=EnF.
n=l \k=n )
For a given set A, the power set of A, denoted by P{A) or 2^, is the collection
of all subsets of A. For example, if A = {1,2,3}, then V{A) = {0, {1}, {2}, {3},
{1,2}, {1,3}, {2,3}, {1,2,3}}. There are 8 distinct elements in the power set of
A = {1,2,3}. In general, for any finite set A containing n distinct elements, we
find that A has 2" subsets and therefore, V{A) has 2" elements in it.
I Elementary Set Theory
Exercises
1. For sets A, B, and C, the operations Pi and U satisfy the properties of
commutation, association, and distribution:
(a)AUB = BUA,AnB = BnA;
(b) (A U B) U C = A U (B U C), (A n B) n C = A n (B n C);
(c) (A u B) n c = (A n C) u (B n C), (A n B) u c = (A u C) n (B u C).
2. Show that S and T are disjoint if and only if S U T = SAT.
3. Prove that AA{?>AC) = {AAB)AC and A n {BAQ = (A n B)Z\(A n C).
4. For any sets A, B, and C, prove that
(a) AAB C (AziC)U(BZiC), and show by an example that the inclusion
may be proper.
(b) (A \ B) U B = (A U B) \ B if and only if B = 0.
5. For a given integer n > 1, let A„ = <^ — m G Z >. Show limn_oo^n =
Q, the set of rational numbers and lim^ ^^An = Z, the set of integers.
6. Let (An) be a sequence of sets. Prove the following properties.
(a) X e lim„_oo^n if and only if for any N G N, there exists n > N
such that X G A„.
(b) X G lim„^^A„ if and only if there exists Nx G N, such that for all
^ > Nj, we have x G A„.
(c) If (A„) is increasing, then lim A„ exists and
lim A„ = l JA„.
n=l
(d) If (A„) is decreasing, then lim A„ exists and
CX)
lim A„ = nA„.
«=1
7. If A = lim„ ^^A„ and B = lim„^oo^n- Show that B"^ = ]im„_^^A^„ and
A' = lim„^oo^^
*oo-'^n-
8. Let {A„) be a sequence of sets defined as follows:
1
0,2- ,fc = 0,l,2,---
^2Jt+l =
2A: + 1
1
A2k^ ,k^l,lr
'''^2k
Show that lim„_^A„ = [0,1] and lim„_,oo^n = [0,2).
Chapter I Fundamentals
9. Let (An) be a sequence of sets which are mutually disjoint, i.e., Ak fl A^ =
0 for A: 7/ L Then Jim_ A„ = IhnAn = 0.
«—s-00 n—>-oo
10. Let A be a nonempty finite set with n elements. Show that there are
2" elements in its power set V{A).
2 Relations and Orderings
Let X and Y be two sets. The Cartesian product, or cross product, of X and Y,
denoted by X x Y, is the set given by
Xx Y = {(x,y) \xeX,yeY}.
The elements of X x Y are called ordered pairs. For (xi, yi), fe, yi) EXX Y, we
have (xi, yi) = (x2, yz) if and only if x\ = X2 and yi = y2.
Any subset i? of X x Y is called a relation from X to Y. We also write (x, y) G i?
as xRy. For nonempty sets X and Y, a function, or mapping/ from X to Y, denoted
by/ : X i-> Y, is a relation from X to Y in which every element of X appears
exactly once as the first component of an ordered pair in the relation. We often
write y = f{x) when (x, y) is an ordered pair in the function. In this case, y is
called the image (or the value) of x under/ and x the preimage of y. We call X
the domain of/ and Y the codomain of/. The subset of Y consisting of those
elements that appear as second components in the ordered pairs of/ is called
the range of/ and is denoted by/(X) since it is the set of images of elements of
X under/. If / : X 1-^ Y and A c X, the image of A under/ is the set
f{A) = {y G Y I y =f{x) for some x G A}.
If B C Y, the inverse image of B is the set
/-\B) = {X G X I y =/(x) for some y G B}.
A real-valued function on A is a function / : A 1-^ R, and a complex-
valued function on A is a function/ : A H-» C.
The proofs of the following properties are left to the reader as exercises.
Theorem 1.2.1 Let/ : X i-> Y and A a collection of subsets of X, iB a collection
of subsets of Y. Then
(i) /(U^e^A) = U^e^f (A);
(ii) /(riAe^A) C n^e^f (A);
2 Relations and Orderings
(iii) rH^BeBB) = UBesf-HB);
(iv) r'inBesB) = HBeBf-'iB).
A function / : X i-^ Y is called one-to-one, or injective, if f{xi) =
f{x2) implies that Xi = X2. Therefore, by the contrapositive of the definition,
/ : X H^ y is one-to-one if and only if xi 7^ X2 implies that/(xi) T^/fe). A
function/ : X 1-^ Y is called onto, or surjective, if/(X) = Y. Thus,/ : X ^
Y is onto if and only if for any y eY, there is x G X such that/(x) = y,f:X\-^Y
is said to be bijective or a one-to-one correspondence if/ is both one-to-one and
onto.
Let/ be a relation from a set X to a set Y and g a relation from Y to a set Z.
The composition of/ and g is the relation consisting of ordered pairs {x, z), where
X e X,z e Z, and for which there exists an element 1/ G Y such that {x, y) ^f
and {y,z) G g. We denote the composition of/ and ghygof. If/ : X 1-^ Y
and g :Y \-^ Z are functions, then g of : X ^-^ Zis called the composite function
or composition of/ and g. Iif:A\-^Bis bijective, there is a unique inverse
function/"^ : B h^ A such that/~^ o/W = ^, for all x e A, and/ o/~^(i/) = 1/
for all y G B.
Definition 1.2.1 For E C E, a function/ : E H-^ R is said to be continuous at a
point XQ G E, if for every e > 0, there exists a 5 > 0 such that
\f{x)-f{x,)\<e
for all X G E with |x — Xo| <5. The function/ is said to be continuous on E if and
only if/ is continuous at every point in E. The set of continuous functions on
E is denoted by C{E), When E = [a, b] is an interval, we simply denote C{E) by
C[a,b],
EXAMPLE 1.2.1: Consider the functions/ : Z ^ N,g : Z ^ Z,h :
Z 1-^ Z, defined by/(n) = n^ + hg{n) = 3n,h{n) = 1 - n. Then
/ is neither one-to-one nor onto, g is one-to-one but not onto, h is
bijective, h~^{n) = 1 - n, and/ o h{n) = n^ - 2n + 2. All of these
functions are continuous on their domains.
EXAMPLE 1.2.2: A real-valued polynomial function of degree n is a
function p{x) of the form
p{x) = ^0 + aix + • • • + anX^
defined on M, where n is a fixed nonnegative integer and the coeffi
cients ak eR,k = 0,1,' ",n with an 7^ 0. We usually denote by Vn
