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Automorphic Orthogonal and Extremal Polynomials

Published online by Cambridge University Press:  20 November 2018

A. L. Lukashov  and
F. Peherstorfer
A. L. Lukashov
Affiliation:
Institut für Mathematik, Johannes Kepler Universität Linz, A-4040 Linz, Austria e-mail: alexei.loukachov@jk.uni-linz.ac.at
F. Peherstorfer
Affiliation:
Institut für Mathematik, Johannes Kepler Universität Linz, A-4040 Linz, Austria e-mail: franz.peherstorfer@jk.uni-linz.ac.at
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Abstract

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It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szegö polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szegö polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients etc. are derived.

Keywords

42C0530F3531A1541A2141A50
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 3 , 01 June 2003 , pp. 576 - 608
Copyright
Copyright © Canadian Mathematical Society 2003

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