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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2506.20224 |
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| _version_ | 1866912449324646400 |
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| author | Charpentier, Stéphane Maronikolakis, Konstantinos |
| author_facet | Charpentier, Stéphane Maronikolakis, Konstantinos |
| contents | We use weighted polynomial approximation to prove the existence of a compact set K with non-empty interior and a function f is dense in the space A(K) of all continuous functions on K that are holomorphic in the interior of K, endowed with the sup norm, while the set This improves a result of Mouze. The main ideas of the proof also allows us to construct a holomorphic function while the modulus of its non-zero Taylor coecients go to $\infty$. In passing, we complement a result by Pritsker and Varga on weighted polynomial approximation by proving that, for any compact set K with connected complement, there exists a constant $α$ K > 0 such that there exists a bounded domain G containing K such that the weighted polynomials of the form z $α$n P n , with deg(P n ) $\le$ n, are dense in H(G) for the topology of locally uniform convergence if and only if $α$ < $α$ K . Explicit computations of $α$ K are given for some simple compact sets K. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20224 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New results on universal Taylor series via weighted polynomial approximation Charpentier, Stéphane Maronikolakis, Konstantinos Complex Variables We use weighted polynomial approximation to prove the existence of a compact set K with non-empty interior and a function f is dense in the space A(K) of all continuous functions on K that are holomorphic in the interior of K, endowed with the sup norm, while the set This improves a result of Mouze. The main ideas of the proof also allows us to construct a holomorphic function while the modulus of its non-zero Taylor coecients go to $\infty$. In passing, we complement a result by Pritsker and Varga on weighted polynomial approximation by proving that, for any compact set K with connected complement, there exists a constant $α$ K > 0 such that there exists a bounded domain G containing K such that the weighted polynomials of the form z $α$n P n , with deg(P n ) $\le$ n, are dense in H(G) for the topology of locally uniform convergence if and only if $α$ < $α$ K . Explicit computations of $α$ K are given for some simple compact sets K. |
| title | New results on universal Taylor series via weighted polynomial approximation |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2506.20224 |