A characterization of M. W. Wilson’s criterion for nonnegative expansions of orthogonal polynomials

Abstract

Given a nonnegative function f(x), M. W. Wilson observed that if Formula Presented then the polynomials Pn(x), Pn(0) = 1, orthogonal relative to f(x)dμ(x), have an expansion Formula Presented with nonnegative coefficients akn ≥ 0 where Qn(x), Qn(0) =1, are orthogonal relative to dμ(x). Recently it was shown that (1) holds for f(x) = xc, 0 < c > 1. In this paper we characterize those functions f(x) for which (1) is valid for all positive measures dμ(x). © 1978 American Mathematical Society.