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Asymptotic representation of weighted L∞- and L1-minimal polynomials

Published online by Cambridge University Press:  01 January 2008

ANDRÁS KROÓ  and
FRANZ PEHERSTORFER
ANDRÁS KROÓ
Affiliation:
Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Realtanoda 13-15, H-1053, Hungary e-mail: kroo@renyi.hu
FRANZ PEHERSTORFER
Affiliation:
Abteilung fur Dynamische Systeme, Und Approximationstheorie, Institute fur Analysis, J.K. Universitat Linz, Altenberger Str. 69, A-4040 Linz, Austria e-mail: Franz.peherstorfer@iku.at
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Abstract

In 1858 Chebyshev, and some years later his students Korkin and Zolotarev, determined the polynomial which deviates least from zero among all polynomials of degree n with leading coefficient one with respect to the maximum- and the L1-norm, respectively; these are now called the Chebyshev polynomial of first and second kind.

The next natural step which is to find, at least asymptotically, the minimal polynomial with respect to a given weight function has not been settled until today. Indeed, Bernstein gave asymptotics for the minimum deviation of weighted minimal polynomials, Fekete and Walsh found nth root asymptotics and, recently, Lubinsky and Saff provided asymptotics outside [−1, 1]. But the main point of interest: the asymptotic representation of the weighted minimal polynomials on the interval of approximation [−1, 1] remained open. Here we settle this problem with respect to the maximum norm for weight functions whose second derivative is Lipα, α ∈ (0, 1), and with respect to the L1-norm under somewhat stronger differentiability conditions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 1 , January 2008 , pp. 241 - 254
Copyright
Copyright © Cambridge Philosophical Society 2008

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