Synopses & Reviews
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.
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.