Convergence properties of discrete analogs of orthogonal polynomials

Abstract

If (,) is an inner product on [a, b], and if [,] N is a discrete inner product analogous to (,), and such that [1, 1] N =(1, 1), then, a sufficient condition that the discrete orthogonal polynomials converge to the corresponding continuous orthogonal polynomials likeN −p , is that [1,t k ] N =(1,t k )+O(N −p ),k=1, 2, ... A similar result holds for correspondingFourier segments.