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arXiv · 2405.16308

n-th Root Optimal Rational Approximants to Functions with Polar Singular Set

L. Baratchart·H. Stahl·M. Yattselev
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Abstract

Let $ D $ be a bounded Jordan domain and $ A $ be its complement on the Riemann sphere. We investigate the $ n $-th root asymptotic behavior in $ D $ of best rational approximants, in the uniform norm on $ A $, to functions holomorphic on $ A $ having a multi-valued continuation to quasi every point of $ D $ with finitely many branches. More precisely, we study weak$^*$ convergence of the normalized counting measures of the poles of such approximants as well as their convergence in capacity. We place best rational approximants into a larger class of $ n $-th root optimal meromorphic approximants, whose behavior we investigate using potential-theory on certain compact bordered Riemann surfaces.

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L. Baratchart, H. Stahl, M. Yattselev. 2024-05-25. n-th Root Optimal Rational Approximants to Functions with Polar Singular Set. https://arxiv.org/abs/2405.16308

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