Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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)$.

References [Enhancements On Off] (What's this?)

  • [1] R. Askey, Orthogonal expansions with positive coefficients, Proc. Amer. Math. Soc. 16 (1965), 1191-1194. MR 0185331 (32:2799)
  • [2] C. A. Micchelli and R. A. Willoughby, On functions which preserve the class of Stieltjes matrices, Linear Algebra and Appl. (to appear). MR 520618 (80b:15022)
  • [3] William F. Trench, Proof of a conjecture of Askey on orthogonal expansions with positive coefficients, Bull. Amer. Math. Soc. 81 (1975), 954-956. MR 0374798 (51:10994)
  • [4] D. V. Widder, The Laplace Transform, Princeton Univ. Press, Princeton, N.J., 1946. MR 0005923 (3:232d)
  • [5] M. W. Wilson, Nonnegative expansions of polynomials, Proc. Amer. Math. Soc. 24 (1970), 100-102. MR 0287244 (44:4451)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42A52

Retrieve articles in all journals with MSC: 42A52


Additional Information

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

American Mathematical Society