Strong Asymptotics for Extremal Polynomials Associated with Weights on R

by Doron S. Lubinsky, Edward B. Saff

Overview

0. The results are consequences of a strengthened form of the following assertion: Given 0 <p- f Lp ( ) and a certain sequence of positive numbers associated with Q(x), there exist polynomials Pn of degree at most n, n = 1,2,3..., such that if and only if f(x) = 0 for a.e. 1. Auxiliary results include inequalities for weighted polynomials, and zeros of extremal polynomials. The monograph is fairly self-contained, with proofs involving elementary complex analysis, and the theory of orthogonal and extremal polynomials. It should be of interest to research workers in approximation theory and orthogonal polynomials.

Product Details

ISBN-13:	9783540189589
Publisher:	Springer Berlin Heidelberg
Publication date:	04/15/1988
Series:	Lecture Notes in Mathematics , #1305
Edition description:	1988
Pages:	156
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

Table of Contents

Notation and index of notation.- Statement of main results.- Weighted polynomials and zeros of extremal polynomials.- Integral equations.- Polynomial approximation of potentials.- Infinite-finite range inequalities and their sharpness.- The largest zeros of extremal polynomials.- Further properties of Un, R(x).- Nth root asymptotics for extremal polynomials.- Approximation by certain weighted polynomials, I.- Approximation by certain weighted polynomials, II.- Bernstein's formula and bernstein extremal polynomials.- Proof of the asymptotics for Enp(W).- Proof of the asymptotics for the Lp extremal polynomials.- The case p=2 : Orthonormal polynomials.