Weighted extremal domains and best rational approximation

作者：Baratchart Laurent; Stahl Herbert; Yattselev Maxim^*

来源：Advances in Mathematics, 2012, 229(1): 357-407.

DOI：10.1016/j.aim.2011.09.005

• Rational approximation
• Meromorphic approximation
• Extremal domains
• Weak asymptotics
• Non-Hermitian orthogonality

Let f be holomorphically continuable over the complex plane except for finitely many poles and branch points contained in the unit disk. We prove that best rational approximants to f of degree a, 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 poles and 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.

• 出版日期2012-1-15
• 单位INRIA