# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## A characterization of M. W. Wilson’s criterion for nonnegative expansions of orthogonal polynomialsHTML articles powered by AMS MathViewer

by Charles A. Micchelli
Proc. Amer. Math. Soc. 71 (1978), 69-72 Request permission

## Abstract:

Given a nonnegative function $f(x)$, M. W. Wilson observed that if $$\int _0^\infty f(x) Q_i(x) Q_j(x)d\mu (x) \leqslant 0,\quad i \ne j, \quad \tag {1}$$ then the polynomials ${P_n}(x),{P_n}(0) = 1$, orthogonal relative to $f(x)d\mu (x)$, have an expansion ${P_n}(x) = \sum \limits _{k = 0}^n {{a_{kn}}{Q_k}(x)}$ with nonnegative coefficients ${a_{kn}} \geqslant 0$ where ${Q_n}(x),{Q_n}(0) = 1$, are orthogonal relative to $d\mu (x)$. Recently it was shown that (1) holds for $f(x) = {x^c},0 < c < 1$. In this paper we characterize those functions $f(x)$ for which (1) is valid for all positive measures $d\mu (x)$.
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