Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T13:09:37.499Z Has data issue: false hasContentIssue false
  • English
  • Français

Zeros of expansions in orthogonal polynomials

Published online by Cambridge University Press:  04 October 2011

A. Iserles  and
E. B. Saff
A. Iserles
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW
E. B. Saff
Affiliation:
Institute for Constructive Mathematics, University of South Florida, Tampa, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

The theory of bi-orthogonal polynomials is exploited to investigate the location of zeros of truncated expansions in orthogonal polynomials. It turns out that, subject to additional conditions, these zeros can be confined to certain real intervals. Two general techniques are being used: the first depends on a theorem that links strict sign consistency of a generating function to loci of zeros and the second consists of re-expression of transformations from [3] in an orthogonal basis.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 559 - 573
Copyright
Copyright © Cambridge Philosophical Society 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Chihara, T. S.. An Introduction to Orthogonal Polynomials (Gordon and Breach, 1978).Google Scholar
[2] Hille, E.. Analytic Function Theory (Blaisdell, 1962).Google Scholar
[3] Iserles, A. and Nørsett, S. P.. Zeros of transformed polynomials. University of Cambridge Technical Report NA12/1987.CrossRefGoogle Scholar
[4] Iserles, A. and Nørsett, S. P.. On the theory of bi-orthogonal polynomials. Trans. Amer. Math. Soc. 306 (1988), 455–474.CrossRefGoogle Scholar
[5] Karlin, S. and Studden, W. J.. Tchebycheff Systems: with Applications in Analysis and Statistics (Wiley, 1966).Google Scholar
[6] Marden, M.. Geometry of Polynomials (American Mathematical Society, 1966).Google Scholar
[7] Pólya, G. and Szegö, G.. Aufgaben und Lehrsätze aus der Analysis, vol. 2 (Berlin, 1925).Google Scholar
[8] Rainville, E. D.. Special Functions (Macmillan, 1967).Google Scholar
[9] Slater, L. J.. Generalized Hypergeometric Functions (Cambridge University Press, 1966).Google Scholar
[10] Szegö, G.. Orthogonal Polynomials, 4th edition (American Mathematical Society, 1975).Google Scholar
[11] Turán, P.. Hermite-expansion and strips for zeros of polynomials. Arch. Math. (Basel) 5 (1954), 148–152.CrossRefGoogle Scholar