n-th Root Optimal Rational Approximants to Functions with Polar Singular Set
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914811781054464 |
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| author | Baratchart, L. Stahl, H. Yattselev, M. |
| author_facet | Baratchart, L. Stahl, H. Yattselev, M. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_16308 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | n-th Root Optimal Rational Approximants to Functions with Polar Singular Set Baratchart, L. Stahl, H. Yattselev, M. Complex Variables Classical Analysis and ODEs 30C15, 30E10, 31C40, 41A20, 41A25 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. |
| title | n-th Root Optimal Rational Approximants to Functions with Polar Singular Set |
| topic | Complex Variables Classical Analysis and ODEs 30C15, 30E10, 31C40, 41A20, 41A25 |
| url | https://arxiv.org/abs/2405.16308 |