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.
From the reviews: "This book, written by three leading experts in interpolation and approximation theory, is a collection of new and old results stemming from the ... theorem of Walsh (1932) ... . The book is quite accessible ... . There are 11 chapters, each ending with some historical remarks, very appropriate for a monograph of this character. ... this book is a valuable source of information about this nice topic of complex approximation theory, containing many results that so far were available only in research papers." (Andrei Martínez Finkelshtein, Zentralblatt MATH, Vol. 1093 (19), 2006) "This elegant and interesting book treats a topic that grew out of a clever observation of Joseph Walsh, for many years one of America's pre-eminent approximators. ... The authors of the book are prominent contributors to this topic, and so especially suited to chronicle its development. ... All in all, this is a nice book, and readers can dip into it to learn about many different topics, through the lens of equiconvergence." (D. S. Lubinsky, Mathematical Reviews, Issue 2007 b)
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.