Automorphic Orthogonal and Extremal Polynomials

Lukashov, A. L. ; Peherstorfer, F.

Canadian journal of mathematics, Tome 55 (2003) no. 3, pp. 576-608

Résumé

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.

Détail

DOI : 10.4153/CJM-2003-024-6

Mots-clés : 42C05, 30F35, 31A15, 41A21, 41A50

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