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

摘要

The connection between convergence of product integration rules and mean convergence of Lagrange interpolation in Lp (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(−xm/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 Lp, 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.