容量和最大收敛多项式的近圆性

IF 0.6 4区 数学 Q3 MATHEMATICS
Hans-Peter Blatt
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引用次数: 0

摘要

如果 f 是收敛半径为 R 的幂级数,即 \(R>1\),众所周知,Carathéodory-Fejér 方法可以在封闭的单位圆盘上构造 f 的多项式近似值,这些近似值在单位圆上显示出近似圆的典型现象。让 E 紧凑且连通,让 f 在 E 上是全态的。如果 \(\left\{ p_n\right\} _{n\in \mathbb {N}}\) 是在 E 上最大程度收敛于 f 的多项式序列,那么可以证明误差函数 \(f-p_n\) 的模量在格林函数 \(g_\Omega (z.) 的水平线上的容量中是渐近恒定的、\E 在 \(\overline{\mathbb {C}}\) 中的补集 \(\Omega \),极点位于无穷大,从而反映了一种近似圆周性,但并没有获得关于点 0 的误差曲线的缠绕数的知识。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Near-Circularity in Capacity and Maximally Convergent Polynomials

If f is a power series with radius R of convergence, \(R > 1\), 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 \(\left\{ p_n\right\} _{n\in \mathbb {N}}\) is a sequence of polynomials converging maximally to f on E, it is shown that the modulus of the error functions \(f-p_n\) is asymptotically constant in capacity on level lines of the Green’s function \(g_\Omega (z,\infty )\) of the complement \(\Omega \) of E in \(\overline{\mathbb {C}}\) 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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来源期刊
Computational Methods and Function Theory
Computational Methods and Function Theory MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.20
自引率
0.00%
发文量
44
审稿时长
>12 weeks
期刊介绍: CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.
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