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On estimates for the ratio of errors in best rational approximation of analytic functions


Authors: S. Kouchekian and V. A. Prokhorov
Journal: Trans. Amer. Math. Soc. 361 (2009), 2649-2663
MSC (2000): Primary 41A20, 30E10; Secondary 47B35
Published electronically: December 4, 2008
MathSciNet review: 2471933
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Abstract: Let $ E$ be an arbitrary compact subset of the extended complex plane $ \overline {\mathbb{C}}$ with nonempty interior. For a function $ f$ continuous on $ E$ and analytic in the interior of $ E$ denote by $ \rho_n(f; E)$ the least uniform deviation of $ f$ on $ E$ from the class of all rational functions of order at most $ n$. In this paper we show that if $ f$ is not a rational function and if $ K$ is an arbitrary compact subset of the interior of $ E,$ then $ \prod_{k=0}^n (\rho_k(f; K) /\rho_k(f; E) ),$ the ratio of the errors in best rational approximation, converges to zero geometrically as $ n \to \infty$ and the rate of convergence is determined by the capacity of the condenser $ (\partial E, K)$. In addition, we obtain results regarding meromorphic approximation and sharp estimates of the Hadamard type determinants.


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Additional Information

S. Kouchekian
Affiliation: Department of Mathematics & Statistics, University of South Florida, Tampa, Florida 33620–5700
Email: skouchek@cas.usf.edu

V. A. Prokhorov
Affiliation: Department of Mathematics & Statistics, ILB 325, University of South Alabama, Mobile, Alabama 36668
Email: prokhoro@jaguar1.usouthal.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04628-X
Keywords: Rational approximation, singular number, meromorphic approximation, Hadamard type determinants
Received by editor(s): October 2, 2005
Received by editor(s) in revised form: August 3, 2007
Published electronically: December 4, 2008
Additional Notes: The first author was supported in part by the National Science Foundation grant DMS–0500916
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.