Chebyshev polynomials corresponding to a vanishing weight

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory Pub Date : 2024-05-02 DOI:10.1016/j.jat.2024.106048

Alex Bergman, Olof Rubin

引用次数: 0

Abstract

We consider weighted Chebyshev polynomials on the unit circle corresponding to a weight of the form ${(z - 1)}^{s}$ where $s > 0$ . For integer values of $s$ this corresponds to prescribing a zero of the polynomial on the boundary. As such, we extend findings of Lachance et al. (1979), to non-integer $s$ . Using this generalisation, we are able to relate Chebyshev polynomials on lemniscates and other, more established, categories of Chebyshev polynomials. An essential part of our proof involves the broadening of the Erdős–Lax inequality to encompass powers of polynomials. We believe that this particular result holds significance in its own right.

查看原文本刊更多论文

与消失权重相对应的切比雪夫多项式

我们考虑的是单位圆上的加权切比雪夫多项式，对应于 s>0 的 (z-1)s 形式的权值。对于 s 的整数值，这相当于在边界上规定多项式的零点。因此，我们将 Lachance 等人（1979 年）的发现扩展到了非整数 s。利用这一概括，我们就能将 lemniscates 上的切比雪夫多项式与其他更成熟的切比雪夫多项式类别联系起来。我们证明的一个重要部分是扩大厄尔多斯-拉克斯不等式的范围，使其包括多项式的幂。我们相信，这一特殊结果本身就具有重要意义。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

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.