有边界黎曼曲面上 $$L^2$$ Holomorphic One-Forms 的 Faber 系列

IF 0.6 4区 数学 Q3 MATHEMATICS
Eric Schippers, Mohammad Shirazi
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引用次数: 0

摘要

考虑一个紧凑曲面(\(\mathscr {R}\)),它有区分点(z_1,\ldots ,z_n\)和从单位盘到\(\mathscr {R}\)上的非重叠准星的保角映射(f_k\),取0到(z_k\)。让 \(\Sigma \)成为从 \(\mathscr {R}\) 上移除 \(f_k\) 的图像的闭包得到的黎曼曲面。我们定义了在\(\mathscr {R}\)上只在\(z_1,\ldots ,z_n\)处有极点的非定常形式,我们称之为法布尔-铁茨形式。它们类似于球面上的法布尔多项式。我们证明了任何在\(\Sigma \)上的\(L^2\)全形一元形式都可以唯一地表达为法伯-铁茨形式的数列。这个数列既在\(L^2(\Sigma)\)中收敛,又在\(\Sigma\)的紧凑子集上均匀收敛。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Faber Series for $$L^2$$ Holomorphic One-Forms on Riemann Surfaces with Boundary

Consider a compact surface \(\mathscr {R}\) with distinguished points \(z_1,\ldots ,z_n\) and conformal maps \(f_k\) from the unit disk into non-overlapping quasidisks on \(\mathscr {R}\) taking 0 to \(z_k\). Let \(\Sigma \) be the Riemann surface obtained by removing the closures of the images of \(f_k\) from \(\mathscr {R}\). We define forms which are meromorphic on \(\mathscr {R}\) with poles only at \(z_1,\ldots ,z_n\), which we call Faber–Tietz forms. These are analogous to Faber polynomials in the sphere. We show that any \(L^2\) holomorphic one-form on \(\Sigma \) is uniquely expressible as a series of Faber–Tietz forms. This series converges both in \(L^2(\Sigma )\) and uniformly on compact subsets of \(\Sigma \).

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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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