Bounds for orthogonal polynomials for exponential weights

Abstract

Orthogonal polynomials pn(W2,cursive chi) for exponential weights W2 = e -2Q on a finite or infinite interval I, have been intensively studied in recent years. We discuss efforts of the authors to extend and unify some of the theory; our deepest result is the bound |pn(W2,cursive chi)\W(cursive chi)\(cursive chi - a-n)(cursive chi - an)l1/4 ≤ C, cursive chi ∈ I with C independent of n and x. Here a±n are the Mhaskar-Rahmanov-Saff numbers for Q and Q must satisfy some smoothness conditions on I. © 1998 Elsevier Science B.V. All rights reserved.