Introduction to the theory of weighted polynomial approximation

393 Pages·1996·20.55 MB·English

by  Mhaskar

, INTRODUCTION TO
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APPROXIMATIONS AND DECOMPOSITIONS
Editor-in-Chief: CHARLES K. CHUI
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pproUNI Wavelets and Multi-level Approximation - Vol. 2
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ynomiWEAL Vol. 7: HIn.t rNod. uMcthioans ktoa rth e Theory of Weighted Polynomial Approximation
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eory of WeightRGINIA COM F. Fontanella, K. Jetter and P. J. Laurent, eds.
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Series in Approximations and Decompositions — Vol. 7
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Singoarplodre i London • Hong Kong
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on 6/1 volumes, text books, edited review volumes, and conference proceedings.
mation 0 This seventh volume is a monograph on the approximation of functions on
xiV the whole real line. Written by an expert in approximation theory, this book
pproUNI differs from the existing texts and survey articles on this and related topics in
AH that the subject is presented from the point of view of approximation theory,
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World Scientific Series in
APPROXIMATIONS AND DECOMPOSITIONS
Editor-in-Chief: CHARLES K. CHUI
Texas A&M University, College Station, Texas
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m
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mation 0 spaces.
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miAL worked on this problem for at least 40 years. In 1970's G. Freud started a
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ThVI
e y In this book, we have attempted to explain a variety of different techniques
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on t refinements during the last 25 years. There are other books and surveys re
ucti viewing the ideas from the perspective of either potential theory or orthogonal
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ntro polynomials. The main thrust of this book is to introduce the subject from
I an approximation theory point of view. Thus, the main motivation is to study
analogues of results from classical trigonometric approximation theory, intro
ducing other ideas as needed. It is not our objective to survey the most recent
results, but merely to introduce to the readers the thought processes and ideas
ix
x Preface
as they are developed. Some results in the book are new, and we have often
pursued some of the old results from a new perspective, seeking to clarify the
plethora of conditions under which the results appear in journal articles.
In Chapter 1, we review some elementary facts from the theory of orthog
m onal polynomials, which are used throughout the book. Next, we start with
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ntifi between the accuracy of approximation and the constructive properties of the
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we explored in Chapters 4 and 5. Chapter 3 develops many technical estimates
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xiV relevant facts are summarized in these two chapters as needed. Chapter 6 is
pproUNI one of the central chapters of the book, where many important technical re
AH sults are proved. Although the results were first developed using the language
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ThVI results to the study of orthogonal polynomial expansions, polynomials of La
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o thb grange interpolation, and quadrature processes. Chapters 10 and 11 are of a
n t more advanced nature. In Chapter 10, we utilize most of the results in Chap
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cti ters 3, 6, and 8 to study the closure of certain "weighted polynomials", and
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od the asymptotic behavior of the leading coefficients of the Freud polynomials.
Intr In Chapter 11, we study miscellaneous subjects, which could not be integrated
into the rest of the book. Thus, we briefly present the potential theory ideas,
and their application to incomplete polynomials and related topics. We also
apply the theory to the complexity problem in the theory of neural networks,
or more precisely, Gaussian networks. We make some remarks about the con-