Mean convergence of Lagrange interpolation for Freud's weights with application to product integration rules (Q1096816)

The authors investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials associated with Freud weights on \({\mathbb{R}}\). The results apply to the weights exp \((-x^ m/2)\), \(m=2,4,6,..\). and for the Hermite weight \((m=2)\) extend results of \textit{P. Nevai} [J. Approximation Theory 30, 263-276 (1980; Zbl 0469.41004)] and \textit{S. S. Bonan} [Ph. D. Dissertation, The Ohio State University, Columbus, OH (1982)]. The results are sharp in \(L_ p\), \(1<p\leq 2\). As a consequence, the authors improve results of \textit{W. E. Smith, I. H. Sloan} and \textit{A. H. Opie} [Math. Comp. 40, 519-536 (1983; Zbl 0542.65013)] on convergence of product integration rules based on the zeros of the orthogonal polynomials associated with the Hermite weight. Also they prove a new Markov-Stieltjes inequality for Gauss quadrature sums involving even weights and integrants and solve a problem posed by P. Nevai [Approximation Theory II, Proc. Int. Symp., Austin 1976, 163-201 (1976; Zbl 0343.41002)] on how to estimate certain quadrature sums.