Asymptotic Error Estimates for L² Best Rational Approximants to Markov Functions

Authors: Baratchart L.¹; Stahl H.²; Wielonsky F.³

Source: Journal of Approximation Theory, Volume 108, Number 1, January 2001 , pp. 53-96(44)

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Abstract:

Let f(z)= int (t-z)^-1 d(t) be a Markov function, where is a positive measure with compact support in Ropf . We assume that supp() sub (-1, 1), and investigate the best rational approximants to f in the Hardy space H⁰₂(V), where V colone {z isin ¯ Copf mid |z|>1} and H⁰₂(V) is the subset of functions f isin H₂(V) with f( infin )=0. The central topic of the paper is to obtain asymptotic error estimates for these approximants. The results are presented in three groups. In the first one no specific assumptions are made with respect to the defining measure of the function f. In the second group it is assumed that the measure is not too thin anywhere on its support so that the polynomials p_n, orthonormal with respect to the measure , have a regular nth root asymptotic behavior. In the third group the defining measure is assumed to belong to the Szeg odblac class. For each of the three groups, asymptotic error estimates are proved in the L²-norm on the unit circle and in a pointwise fashion. Also the asymptotic distribution of poles, zeros, and interpolation points of the best L² approximants are studied. Copyright 2001 Academic Press.

Keywords: best rational approximation in the L²-norm on the unit circle; asymptotic error estimates; Markov'; s theorem

Language: English

Document Type: Research article

Affiliations: 1: INRIA, 2004, route des Lucioles, Sophia Antipolis Cedex, 06902, France 2: TFH-Berlin/FB II, Luxemburger Strasse 10, Berlin, 13353, Germany 3: INRIA, 2004, route des Lucioles, Sophia Antipolis Cedex, 06902, France

Publication date: 2001-01-01