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Proceedings of the American Mathematical Society

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

 

 

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


Author: Charles A. Micchelli
Journal: Proc. Amer. Math. Soc. 71 (1978), 69-72
MSC: Primary 42A52
DOI: https://doi.org/10.1090/S0002-9939-1978-0481893-7
MathSciNet review: 0481893
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Abstract: Given a nonnegative function $ f(x)$, M. W. Wilson observed that if

$\displaystyle \int_0^\infty f(x) Q_i(x) Q_j(x)d\mu(x) \leqslant 0,\quad i \ne j, \quad$ ($ 1$)

then the polynomials $ {P_n}(x),{P_n}(0) = 1$, orthogonal relative to $ f(x)d\mu (x)$, have an expansion

$\displaystyle {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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DOI: https://doi.org/10.1090/S0002-9939-1978-0481893-7
Article copyright: © Copyright 1978 American Mathematical Society