S. R. Nasyrov’s Problem of Approximation by Simple Partial Fractions on an Interval

IF 0.6 4区数学 Q3 MATHEMATICS

Mathematical Notes Pub Date : 2024-07-05 DOI:10.1134/s0001434624030234

P. A. Borodin, A. M. Ershov

引用次数: 0

Abstract

In 2014, S. R. Nasyrov asked whether it is true that simple partial fractions (logarithmic derivatives of complex polynomials) with poles on the unit circle are dense in the complex space \(L_2[-1,1]\). In 2019, M. A. Komarov answered this question in the negative. The present paper contains a simple solution of Nasyrov’s problem different from Komarov’s one. Results related to the following generalizing questions are obtained: (a) of the density of simple partial fractions with poles on the unit circle in weighted Lebesgue spaces on \([-1,1]\); (b) of the density in \(L_2[-1,1]\) of simple partial fractions with poles on the boundary of a given domain for which \([-1,1]\) is an inner chord.

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S.S. R. 纳西洛夫的区间简单部分分数逼近问题

摘要 2014年，S. R. Nasyrov提出了这样一个问题：在复数空间\(L_2[-1,1]\)中，极点在单位圆上的简单部分分数（复数多项式的对数导数）是否密集？2019 年，科马洛夫（M. A. Komarov）对这个问题做出了否定的回答。本文包含了对纳西洛夫问题的不同于科马洛夫问题的简单解答。本文得到了与以下问题相关的结果：(a) \([-1,1]\)上加权 Lebesgue 空间中单位圆上有极点的简单分式的密度；(b) \(L_2[-1,1]\)中给定域边界上有极点的简单分式的密度，而 \([-1,1]\)是该域的内弦。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.