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On Orthogonal Polynomials With Respect to a Positive Definite Matrix of Measures

Published online by Cambridge University Press:  20 November 2018

Antonio J. Duran
Antonio J. Duran*
Affiliation:
Departamento de Análisis Matemático Universidad de Sevilla Apdo.1160 41080-Sevilla, Spain
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Abstract

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In this paper, we prove that any sequence of polynomials (pn)n for which dgr(pn) = n which satisfies a (2N + l)-term recurrence relation is orthogonal with respect to a positive definite N × N matrix of measures. We use that result to prove asymptotic properties of the kernel polynomials associated to a positive measure or a positive definite matrix of measures. Finally, some examples are given.

Keywords

42C0515A57orthogonal polynomialsrecurrence relationpositive definite matrices
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 1 , 01 February 1995 , pp. 88 - 112
Copyright
Copyright © Canadian Mathematical Society 1995

References

[A] Akhiezer, N.I., The classical moment problem, Oliver and Boyd, Edinburgh, 1965.Google Scholar
[Ch] Chihara, T.S., An introduction to orthogonal polynomials, Gordon and Breach, New York, 1978.Google Scholar
[DS] Dunford, N. and Schwartz, J.T., Linear operators II, Interscience publishers, New York, 1963.Google Scholar
[D] Duran, A.J., A generalization ofFavard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74(1993), 83–109.Google Scholar
[F] Favard, J.A., Sur les polynômes de Tchebicheff, C.R. Acad. Sci. Paris 200(1935), 2052–2053.Google Scholar
[HJ] Horn, R.A. and Johnson, C.A., Matrix analysis, Cambridge University Press, Cambridge, 1985.Google Scholar
[R] Rosenberg, M., The square-integrability of matrix-valued functions with respect to a non-negative hermitian measure, Duke Math. J. 31(1964), 291–298.Google Scholar