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This book covers the basic elements of difference equations and the
tools of difference and sum calculus necessary for studying and
solv ing, primarily, ordinary linear difference equations. Examples
from various fields are presented clearly in the first chapter,
then discussed along with their detailed solutions in Chapters 2-7.
The book is in tended mainly as a text for the beginning
undergraduate course in difference equations, where the
"operational sum calculus" of the di rect use of the discrete
Fourier transforms for solving boundary value problems associated
with difference equations represents an added new feature compared
to other existing books on the subject at this introductory level.
This means that in addition to the familiar meth ods of solving
difference equations that are covered in Chapter 3, this book
emphasizes the use of discrete transforms. It is an attempt to
introduce the methods and mechanics of discrete transforms for solv
ing ordinary difference equations. The treatment closely parallels
what many students have already learned about using the opera
tional (integral) calculus of Laplace and Fourier transforms to
solve differential equations. As in the continuous case, discrete
operational methods may not solve problems that are intractable by
other meth ods, but they can facilitate the solution of a large
class of discrete initial and boundary value problems. Such
operational methods, or what we shall term "operational sum
calculus," may be extended eas ily to solve partial difference
equations associated with initial and/or boundary value problems."
This book represents the first attempt at a unified picture for the
pres ence of the Gibbs (or Gibbs-Wilbraham) phenomenon in
applications, its analysis and the different methods of filtering
it out. The analysis and filtering cover the familiar Gibbs
phenomenon in Fourier series and integral representations of
functions with jump discontinuities. In ad dition it will include
other representations, such as general orthogonal series
expansions, general integral transforms, splines approximation, and
continuous as well as discrete wavelet approximations. The mate
rial in this book is presented in a manner accessible to
upperclassmen and graduate students in science and engineering, as
well as researchers who may face the Gibbs phenomenon in the varied
applications that in volve the Fourier and the other approximations
of functions with jump discontinuities. Those with more advanced
backgrounds in analysis will find basic material, results, and
motivations from which they can begin to develop deeper and more
general results. We must emphasize that the aim of this book (the
first on the sUbject): to satisfy such a diverse audience, is quite
difficult. In particular, our detailed derivations and their
illustrations for an introductory book may very well sound repeti
tive to the experts in the field who are expecting a research
monograph. To answer the concern of the researchers, we can only
hope that this book will prove helpful as a basic reference for
their research papers."
This book represents the first attempt at a unified picture for the
pres ence of the Gibbs (or Gibbs-Wilbraham) phenomenon in
applications, its analysis and the different methods of filtering
it out. The analysis and filtering cover the familiar Gibbs
phenomenon in Fourier series and integral representations of
functions with jump discontinuities. In ad dition it will include
other representations, such as general orthogonal series
expansions, general integral transforms, splines approximation, and
continuous as well as discrete wavelet approximations. The mate
rial in this book is presented in a manner accessible to
upperclassmen and graduate students in science and engineering, as
well as researchers who may face the Gibbs phenomenon in the varied
applications that in volve the Fourier and the other approximations
of functions with jump discontinuities. Those with more advanced
backgrounds in analysis will find basic material, results, and
motivations from which they can begin to develop deeper and more
general results. We must emphasize that the aim of this book (the
first on the sUbject): to satisfy such a diverse audience, is quite
difficult. In particular, our detailed derivations and their
illustrations for an introductory book may very well sound repeti
tive to the experts in the field who are expecting a research
monograph. To answer the concern of the researchers, we can only
hope that this book will prove helpful as a basic reference for
their research papers."
This book covers the basic elements of difference equations and the
tools of difference and sum calculus necessary for studying and
solv ing, primarily, ordinary linear difference equations. Examples
from various fields are presented clearly in the first chapter,
then discussed along with their detailed solutions in Chapters 2-7.
The book is in tended mainly as a text for the beginning
undergraduate course in difference equations, where the
"operational sum calculus" of the di rect use of the discrete
Fourier transforms for solving boundary value problems associated
with difference equations represents an added new feature compared
to other existing books on the subject at this introductory level.
This means that in addition to the familiar meth ods of solving
difference equations that are covered in Chapter 3, this book
emphasizes the use of discrete transforms. It is an attempt to
introduce the methods and mechanics of discrete transforms for solv
ing ordinary difference equations. The treatment closely parallels
what many students have already learned about using the opera
tional (integral) calculus of Laplace and Fourier transforms to
solve differential equations. As in the continuous case, discrete
operational methods may not solve problems that are intractable by
other meth ods, but they can facilitate the solution of a large
class of discrete initial and boundary value problems. Such
operational methods, or what we shall term "operational sum
calculus," may be extended eas ily to solve partial difference
equations associated with initial and/or boundary value problems."
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