Orthogonal polynomials in weighted Bergman spaces

IF 0.9 3区数学 Q2 MATHEMATICS Journal of Approximation Theory Pub Date : 2023-09-24 DOI:10.1016/j.jat.2023.105972

Erwin Miña-Díaz

引用次数: 0

Abstract

Let $w$ be a weight on the unit disk $D$ having the form $w (z) = | v (z) |^{2} \prod_{k = 1}^{s} {(\frac{z - a_{k}}{1 - z {\bar{a}}_{k}})}^{m_{k}}, m_{k} > - 2, | a_{k} | < 1,$ where $v$ is analytic and free of zeros in $\bar{D}$ , and let ${(p_{n})}_{n = 0}^{\infty}$ be the sequence of polynomials ( $p_{n}$ of degree $n$ ) orthonormal over $D$ with respect to $w$ . We give an integral representation for $p_{n}$ from which it is in principle possible to derive its asymptotic behavior as $n \to \infty$ at every point $z$ of the complex plane, the asymptotic analysis of the integral being primarily dependent on the nature of the first singularities encountered by the function $v {(z)}^{- 1} \prod_{k = 1}^{s} {(1 - z {\bar{a}}_{k})}^{- 1}$ .

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加权Bergman空间中的正交多项式

设w是单位圆盘D上的权重，其形式为w（z）=|v（z）|2πk=1sz−ak1−za’kmk，mk>；−2，|ak|<；1，其中v是解析的，并且在D’中没有零，并且设（pn）n=0∞是D上关于w的正交多项式序列（n阶pn）。我们给出了pn的积分表示，从中原则上可以导出其渐近行为为n→∞ 在复平面的每个点z，积分的渐近分析主要取决于函数v（z）−1πk=1s（1−za’k）−1遇到的第一个奇点的性质。

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.

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