The weighted Lp-norms of orthonormal polynomials for freud weights

Abstract

Let W(colon equals) e-Q, where Q: R → R is even, continuous in R, Q" is continuous in (0, ∞), and Q′ > 0 in (0, ∞), while for some A, B > 1, [formula]. For example, W(x) = exp(-|x|α), α > 1, satisfies these hypotheses. Let pn(W2, x) denote the nth orthonormal polynomial for the weight W2, n ≥ 1, and let an = an(Q) denote the nth Mhaskar-Rahmanov-Saff number for Q. We show that for 0 < p < ∞, and n ≥ 2, [formula] The results are based on bounds for pn(W2,.) established recently by A. L. Levin and one of the authors. © 1994 Academic Press, Inc.