Best Meromorphic Approximation of Markov Functions on the Unit Circle |
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Authors: | L Baratchart V A Prokhorov E B Saff |
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Institution: | (1) INRIA 2004 Route des Lucioles B.P. 93 06902 Sophia Antipolis Cedex, France baratcha@sophia.inria.fr, FR;(2) Department of Mathematics and Statistics University of South Alabama Mobile, AL 36688-0002, USA prokhorov@mathstat.usouthal.edu, US;(3) Institute for Constructive Mathematics Department of Mathematics University of South Florida Tampa, FL 33620, USA esaff@math.usf.edu, US |
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Abstract: | Let E \subset(-1,1) be a compact set, let μ be a positive Borel measure with support \supp μ =E , and let H
p
(G),
1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z \colon |z|=1} . Let Δ
n,p
be the error in best approximation of the Markov function \frac{1}{2π i} ∈t_E \frac{d μ(x)}{z-x} in the space L
p
(Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H
p
(G),
Q is a polynomial of degree at most n , Q\not \equiv 0 . We investigate the rate of decrease of Δ
n,p
,
1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants
are described in the case when 1<p≤∈fty and the measure μ with support E=a,b] satisfies the Szegő condition ∈t_a^b \frac{\log(d μ/ d x)}{\sqrt{(x-a)(b-x)}} dx >- ∈fty.
July 27, 2000. Final version received: May 19, 2001. |
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Keywords: | , Meromorphic approximation, Markov functions, Best approximation, AMS Classification, 41A20, 30E10, 47B35, |
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