Near-Circularity in Capacity and Maximally Convergent Polynomials
If f is a power series with radius R of convergence, $$R > 1$$ R > 1 , it is well-known that the method of Carathéodory–Fejér constructs polynomial approximations of f on the closed unit disk which show the typical phenomenon of near-circularity on the unit circle. Let E be compact and connect...
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| Veröffentlicht in: | Computational methods and function theory Jg. 25; H. 2; S. 279 - 300 |
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| 1. Verfasser: | |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Heidelberg
Springer Nature B.V
01.06.2025
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| Schlagworte: | |
| ISSN: | 1617-9447, 2195-3724 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | If f is a power series with radius R of convergence, $$R > 1$$ R > 1 , it is well-known that the method of Carathéodory–Fejér constructs polynomial approximations of f on the closed unit disk which show the typical phenomenon of near-circularity on the unit circle. Let E be compact and connected and let f be holomorphic on E . If $$\left\{ p_n\right\} _{n\in \mathbb {N}}$$ p n n ∈ N is a sequence of polynomials converging maximally to f on E , it is shown that the modulus of the error functions $$f-p_n$$ f - p n is asymptotically constant in capacity on level lines of the Green’s function $$g_\Omega (z,\infty )$$ g Ω ( z , ∞ ) of the complement $$\Omega $$ Ω of E in $$\overline{\mathbb {C}}$$ C ¯ with pole at infinity, thereby reflecting a type of near-circularity, but without gaining knowledge of the winding numbers of the error curves with respect to the point 0. |
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| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1617-9447 2195-3724 |
| DOI: | 10.1007/s40315-024-00528-5 |