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Automorphic Orthogonal and Extremal Polynomials |
Canad. J. Math. 55(2003), 576-608
Published:2003-06-01
Printed: Jun 2003
Printed: Jun 2003
- A. L. Lukashov
- F. Peherstorfer
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 |