Convergence properties of discrete analogs of orthogonal polynomials

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 like N-p, is that [1, tk]N=(1, tk)+O(N-p), k=1, 2, ... A similar result holds for corresponding Fourier segments. © 1970 Springer-Verlag.