Table Of ContentFundamentals of
Approxi1nation Theory
Fundamentals of
Approxiination Theory
Hrushikesh N. Mhaskar
Devidas V. Pai
CRC Press
Boca Raton London New York Washington, D.C.
'
N arosa Publishing House
New Delhi Chennai Mumbai Calcutta
Hrushikesh N. Mhaskar
Professor of Mathematics, California State University
Los Angeles. USA
Devidas V. Pai
Professor of Mathematics
Indian Institute of Technology, Bombay
Mumbai-400 067. India
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Preface
The subject of approximation theory has attracted the attention of several
mathematicians during the last 130 years or so. With the advent of computers
the research in this area has become even more vigorous. By now, the field has
become so vast that it has significant intersections with every other branch of
analysis. Moreover, it plays an increasingly important role in applications to
many branches of applied sciences and engineering. The present book aims at
treating certain basic topics in approximation theory which we find particularly
interesting in view of their connections with other branches of analysis as well
as their role in applications. It is primarily intended as a text for a variety of
courses in approximation theory both introductory as well as advanced. Typi
cally, it is felt that these courses would fit well in the curriculum of the second
year of Master's level in mathematics. In addition, a judicious selection of ad
vanced level material from the book would render it suitable for a course at
M.Phil./Ph.D. level. The coverage of topics in this book is also likely to be of
interest to many scientists and engineers.
Plan and special features. We shall concentrate on four major problems in
approximation theory. This can be best illustrated with the aid of the theory of
uniform approximation of continuous functions by trigonometric polynomials.
The first question is whether it is possible to find a trigonometric polynomial,
of however high order, arbitrarily close to a given continuous function in the
uniform sense. Theorems relating to such problems are classified as density
theorems. Chapter I illustrates different ways of answering such questions not
only for trigonomettic polynomials, but also for more general systems of ap
proximants.
Having settled such questions, we begin to probe further. Thus, the next
question which arises is to examine how well one can approximate the given func
tion from the class of trigonometric polynomials of a fixed order. Obviously, one
turns
cannot do arbitrarily well in general, but it out that for every continuous
271'-periodic function, there is a unique trigonometric polynomial which does the
best job among its colleagues of the same order. The problem is to obtain crite
ria to recognize what is the best and then to develop algorithms to compute the
same. Such problems concerning existence, uniqueness, characterization, etc.
of best members from general systems as well as their computability are dealt
with in Chapter II.
The next, or perhaps a concurrent question is to say something quantitative
about how well one can approximate a continuous function by trigonometric
polynomials of a specified order. It turns out that this degree of approximation
!iii Preface
has such a close connection with the smoothness of the function that one may
even take the sequence of degrees of best approximation of a function as a
measure of its smoothne.ss. These ideas are illustrated in Chapter Ill with the
aid of approximation by trigonometric and algebraic polynomials.
· In some applications, the best approximation is not the most desirable one; it
may be computationally expensive, or might not share certain essential features
(e.g., monotonicity) with the function. So, the next problem is how well one
can do with concrete processes for constructing a good, but not necessarily the
best approximant. Two such processes are commonly used. One is to find an
interpolatory approximant, the other is to take partial sums of a fixed expansion
of the function, such as, the Fourier series for continuous 211"-periodic functions
or Taylor series for analytic functions, etc. Study of such processes constitute
Chapters IV & V. Chapter IV deals with interpolation and Chapter Vis devoted
to Fourier series.
In our opinion, this gives the core of information with which a student can
begin his research in this domain. The next three chapters are devoted to special
topics, viz., spline functions, orthogonal polynomials and best approximation in
normed linear spaces. The choice of these topics was made mostly due to our
own research interests in these areas. Of course, each of these topics deserves
separate treatises in its own right and there do exist many such. The treatment
in the book is meant only to give a flavor of these topics.
While the material presented here is not generally our original research, there
are quite a few new features of the present book as an intended text.
First, we make a special effort to emphasize the connections between different
branches of analysis with approximation theory: between the 'classical' and the
'abstract'. The student will find the techniques used here over and over again
in most of the modern literature on approximation theory.
As regards the choice of the topics, the first four chapters are standard in
almost any text book on the subject. Some of the noteworthy features of the
present book are: the inclusion of the notion of K-functionals and their use in
obtaining Brudnyi's theorem, the inclusion of the topic Hermite-Birkhoff inter
polation and some of the results on Lagrange-Hermite-Fejer-type interpolation,
as well as inclusion of the complex methods in the treatment of Fourier series.
Chapter VI which deals with spline functtons in an extensive manner pro
vides a 'case study' to illustrate how the ideas in the first four chapters apply
in other contexts. Indeed, there are numerous treatises on spline functions, but
very few books on approximation theory include a discussion of spline functions.
The use of spline functions has become widespread even in the study of classical
problems in approximation theory. Neverthless, it is hoped that the treatment
in this chapter which is aimed to be self-contained, will introduce the topic in
a fairly rigorous manner.
Chapter VII which deals with orthogonal polynomials serves two purposes:
one is, of course, to introduce the student to the rudiments of this fascinating
and rapidly growing field. The other is to expose him to the use of complex
analytic methods in approximation theory. While the importance of orthogonal
polynomials in approximation theory is long well known, it is surprising that no
IX
?reface
introductory text deals with the subject in any detail.
Chapter VIII which deals with best approximation in normed linear spaces
is meant to illustrate techniques of functional analysis in treating problems
more general than the ones dealt with in Chapter II. Although this topic is
covered in monographs/lecture notes and some books on functional analysis, at
present no text book on approximation theory deals with it except in a very
rudimentary fashion. The notable novel features of this chapter are inclusion
of the topics : connections between geometry of Banach space concepts and
approximative properties of sets, rudiments of multifunctions and continuity of
metric projection, relative Chebyshev centers of sets and optimal recovery of
functions.
It is impossible to really separate the different sections in a· mutually exclu
sive fashion. For example, Chebyshev polynomials appear over and over again
in almost all contexts. The writing of this book has spanned almost a decade.
The odd and even numbered chapters have been written for the most part in
dependently by the two authors at different periods of time. We have tried to
write the sections in such a way that the teacher can safely select those which
interest him most and omit some others without seriously hampering the study.
Courses and their prerequisites. The reader of this book is expected to
know set theoretic concepts, elements of linear algebra, metric spaces and ad
vanced calculus. In addition, for some of the advanced courses suggested below
an exposure to elements of general topology, measure theory, functional analy
sis and complex analysis would be helpful. Depending on the number of hours
available for instruction and the level of students, the following courses are
suggested.
(i) An introductory course on approximation theory along traditional lines
can be planned as a one semester course meeting four hours per week based on
Chapters I to VI. Only the first two sections from Chapter II need be covered
and the last sections from Chapters III and IV may be dropped for such a course.
(ii) A course on the theory of best approximations can be planned as a
one-semester course meeting three hours per week (as a sequel to a traditional
introductory course on functional analysis for instance) consisting of Chapter II
( except the first two sections) and the entire Chapter VIII.
(iii) A course on interpolation and spline functions can be planned as a one
semester course meeting three hours per week covering Chapters IV and VI
entirely.
(iv) A course on approximation processes can be planned as a one-semester
course meeting three hours per week based on Chapters IV, V and VII.
A judicious choice from the wealth of material included in each of the chap
ters is very crucial for successful organization of the courses suggested above.
Bibliography. Several standard books on approximation theory, monographs
on the special topics and research articles on topics of interest have been con
sulted while preparing this book. An exhaustive bibliography listing 302 items
has been compiled. A particular reference is mentioned in two ways. For exam-
