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Walsh Equiconvergence of Complex Interpolating Polynomials (Springer Monographs in Mathematics)

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Synopses & Reviews

Publisher Comments:

This monograph is centered around the following simple and beautiful observation of J.L. Walsh. If a function is analytic in a disc, then the difference between the Lagrange interpolant of the function at the roots of unity and the partial sums of the Taylor series about the origin, tends to zero in a larger disc than the disk of convergence of the Taylor series; while both operators converge to the function only in the original disc.This result was stated by Walsh in 1932 in a short paper and proved later. A precise formulation of this interesting result appears in 1935 in the first edition of his book Interpolation and Approximation by Rational Functions in the Complex Domain.In this monograph various results stemming from this theorem of Walsh are collected which appeared in the literature, and some new results as well.

Synopsis:

This book is a collection of the various old and new results, centered around the following simple and beautiful observation of J.L. Walsh - If a function is analytic in a finite disc, and not in a larger disc, then the difference between the Lagrange interpolant of the function, at the roots of unity, and the partial sums of the Taylor series, about the origin, tends to zero in a larger disc than the radius of convergence of the Taylor series, while each of these operators converges only in the original disc. This book will be particularly useful for researchers in approximation and interpolation theory.

Table of Contents

Dedication. Preface. Lagrange Interpolation and Walsh Equiconvergence.- Hermite and Hermite-Birkhoff Interpolation and Walsh Equiconvergence.- A generalization of the Taylor Series to Rational Functions and Walsh Equiconvergence.- Sharpness Results.- Converse Results.- Padé Approximation and Walsh Equiconvergence for Meromorphic Functions with v-Poles.- Quantitative Results in the Equiconvergence of Approximation of Meromorphic Functions.- Equiconvergence for Functions Analytic in an Ellipse.- Walsh Equiconvergence Theorems for the Faber Series.- Equiconvergence on Lemniscates.- Walsh Equiconvergence and Summability.- References.

Product Details

ISBN:
9781402041747
Author:
Jakimovski, Amnon
Publisher:
Springer
Author:
Sharma, Ambikeshwar
Author:
Szabados, Jzsef
Author:
Szabados, Jozsef
Author:
Szabados, Jszsef
Subject:
Mathematical Analysis
Subject:
Algebra - Elementary
Subject:
Functional Analysis
Subject:
Polynomials
Subject:
Approximations and Expansions
Subject:
Analysis
Subject:
Functions of a complex variable.
Subject:
Sequences, Series, Summability
Subject:
Several Complex Variables and Analytic Spaces
Subject:
Several Complex Variables and Analytic Spaces Only work on this subject
Subject:
Mathematics-Analysis General
Copyright:
Edition Number:
1
Edition Description:
Book
Series:
Springer Monographs in Mathematics
Publication Date:
April 2006
Binding:
HARDCOVER
Language:
English
Pages:
309
Dimensions:
235 x 155 mm 1360 gr

Related Subjects

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Science and Mathematics » Mathematics » Numeric Analysis

Walsh Equiconvergence of Complex Interpolating Polynomials (Springer Monographs in Mathematics) New Hardcover
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Product details 309 pages Springer - English 9781402041747 Reviews:
"Synopsis" by , This book is a collection of the various old and new results, centered around the following simple and beautiful observation of J.L. Walsh - If a function is analytic in a finite disc, and not in a larger disc, then the difference between the Lagrange interpolant of the function, at the roots of unity, and the partial sums of the Taylor series, about the origin, tends to zero in a larger disc than the radius of convergence of the Taylor series, while each of these operators converges only in the original disc. This book will be particularly useful for researchers in approximation and interpolation theory.
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