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# Automorphic Orthogonal and Extremal Polynomials

Published:2003-06-01
Printed: Jun 2003
• A. L. Lukashov
• F. Peherstorfer
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## Abstract

It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szeg\"o 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-Szeg\"o 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 {\it etc.} are derived.
 MSC Classifications: 42C05 - Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45] 30F35 - Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx] 31A15 - Potentials and capacity, harmonic measure, extremal length [See also 30C85] 41A21 - Pade approximation 41A50 - Best approximation, Chebyshev systems