Mean convergence of Lagrange interpolation for Freud's weights with application to product integration rules

Abstract

The connection between convergence of product integration rules and mean convergence of Lagrange interpolation in L p (1 < p < ∞) has been thoroughly analysed by Sloan and Smith [37]. Motivated by this connection, we investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials associated with Freud weights on R . Our results apply to the weights exp(− x m /2), m = 2, 4, 6 …, and for the Hermite weight ( m = 2) extend results of Nevai [28] and Bonan [2] in at least one direction. The results are sharp in L p , 1 < p ⩽ 2. As a consequence, we can improve results of Smith, Sloan and Opie [38] on convergence of product integration rules based on the zeros of the orthogonal polynomials associated with the Hermite weight. In the process, we prove a new Markov-Stieltjes inequality for Gauss quadrature sums, and solve a problem of Nevai on how to estimate certain quadrature sums.