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Weighted Extremal Domains and Best Rational Approximation

Abstract : Let f be holomorphically continuable over the complex plane except for finitely many branch points contained in the unit disk. We prove that best rational approximants to f of degree n, in the L^2-sense on the unit circle, have poles that asymptotically distribute according to the equilibrium measure on the compact set outside of which f is single-valued and which has minimal Green capacity in the disk among all such sets. This provides us with n-th root asymptotics of the approximation error. By conformal mapping, we deduce further estimates in approximation by rational or meromorphic functions to f in the L^2-sense on more general Jordan curves encompassing the branch points. The key to these approximation-theoretic results is a characterization of extremal domains of holomorphy for f in the sense of a weighted logarithmic potential, which is the technical core of the paper.
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Contributor : Laurent Baratchart <>
Submitted on : Thursday, February 2, 2012 - 6:33:48 PM
Last modification on : Thursday, August 22, 2019 - 2:44:01 PM

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Laurent Baratchart, Herbert Stahl, Maxim Yattselev. Weighted Extremal Domains and Best Rational Approximation. Advances in Mathematics, Elsevier, 2012, 229, pp.357-407. ⟨10.1016/j.aim.2011.09.005⟩. ⟨hal-00665834⟩



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