Pointwise convergence of Lagrange interpolation based at the zeros of orthonormal polynomials with respect to weights on the whole real line

Abstract

Under various assumptions on a weight W 2, with support R , we obtain rates for the pointwise convergence of Lagrange interpolation based at the zeros of the orthonormal polynomials with respect to W 2, in the case of a uniformly continuous function ƒ(x). The weights considered include W m(x) = exp(− 1 2 ¦x¦ m), m an even positive integer. The technique used generalizes that of Freud, who considered pointwise convergence of Lagrange interpolation in the case of the Hermite weight. However, even for the Hermite weight, our results refine and extend the upper and lower bounds of Freud. We establish as well, as preliminary results, upper and lower bounds for generalized Lebesgue functions and for absolute values of the orthogonal polynomials associated with W m 2( x).