Distribution of the zeros of polynomials near the unit circle

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory Pub Date : 2024-08-08 DOI:10.1016/j.jat.2024.106087

Mithun Kumar Das

引用次数: 0

Abstract

We estimate the number of zeros of a polynomial in $ℂ [z]$ within any small circular disk centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erdélyi, and Littmann in 2008. Furthermore, by combining this result with Euclidean geometry, we derive an upper bound on the number of zeros of such a polynomial within a region resembling a gear wheel. Additionally, we obtain a sharp upper bound on the annular discrepancy of such zeros near the unit circle. Our approach builds upon a modified version of the method described in Borwein et al. (2008), combined with the refined version of the best-known upper bound for angular discrepancy of zeros of polynomials.

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单位圆附近多项式零点的分布

我们估算了ℂ[z]多项式在以单位圆为中心的任何小圆盘内的零点数，这改进并全面扩展了博尔文、埃尔德利和利特曼在 2008 年建立的一个结果。此外，通过将这一结果与欧几里得几何相结合，我们推导出了在类似齿轮的区域内该多项式的零点个数上限。此外，我们还获得了单位圆附近此类零点环差的尖锐上界。我们的方法建立在 Borwein 等人（2008 年）所描述方法的改进版基础之上，并结合了多项式零点角度差异的最著名上界的改进版。

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

Journal of Approximation Theory 数学-数学

CiteScore

1.90

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.