Domain of convergence for a series of orthogonal polynomials
type: Item
instance of: scholarly article
title: Domain of convergence for a series of orthogonal polynomials
zbMATH Open document ID: 1322.42031
author: Hee Sun Jung
author: Ryozi Sakai
published in: Journal of Approximation Theory
publication date: 2015-05-13T00:00:00Z
review text: Let \(\left\{p_k\left(z\right)\right\}_{k=0}^\infty\) be a family of orthogonal polynomials with respect to an exponential weight \(w\left(x\right)=\exp\left(-Q\left(x\right)\right)\in{\mathcal F}\left(\text{Lip}\frac{1}{2}\right)\). Note that \({\mathcal F}\left(\text{Lip}\frac{1}{2}\right)\) is the Levin-Lubinsky class [\textit{E. Levin} and \textit{D. S. Lubinsky}, Orthogonal polynomials for exponential weights. New York, NY: Springer (2001; Zbl 0997.42011)]. So, \(p_n\left(x\right)=p_n\left(w^2,x\right)\) are polynomials of degree \(n\), \(n=0,1,2,\dots\), such that \[ \int_{-\infty}^\infty p_n(x)p_m(x)w^2(x)\,dx=\delta_{mn}, \] the Kronecker delta. The authors show that under certain conditions, a series of the form \(\sum_{k=0}^\infty b_k p_k(z)\) converges uniformly and absolutely on compact subsets of an open strip in the complex plane, and diverges outside the closure of this strip. To formulate the main result, we need some notations. The numbers \(a_{-t}<0<a_t\), \(t>0\), are uniquely determined by the following equations \[ t=\frac{1}{\pi}\int_{a_{-t}}^{a_t}\frac{x Q'(x)}{\sqrt{\left(x-a_{-t}\right)\left(a_t-x\right)}}\,dx; \] \[ 0=\frac{1}{\pi}\int_{a_{-t}}^{a_t}\frac{Q'(x)}{\sqrt{\left(x-a_{-t}\right)\left(a_t-x\right)}}\,dx. \] The density function for \(x\in\left[a_{-n},a_n\right]\) is defined by \[ \sigma_n\left(x\right)=\frac{1}{\pi^2}\sqrt{\left(x-a_{-n}\right)\left(a_n-x\right)} \int_{a_{-n}}^{a_n}\frac{Q'(s)-Q'(x)}{s-x}\frac{ds}{\sqrt{\left(s-a_{-n}\right)\left(a_n-s\right)}}. \] The main result of the article is given by the following statement. Theorem. Let \(w=\exp\left(-Q\right)\in{\mathcal F}\left(\text{Lip}\frac{1}{2}\right)\). For a sequence of complex numbers \(\left\{b_k\right\}\), let \[ B:=\liminf_{n\to\infty}\frac{1}{\pi\sigma_n\left(0\right)}\log\left(\frac{1}{\left|b_n\right|}\right). \] Then \(\sum b_k p_k(z)\) converges uniformly and absolutely on compact subsets of the strip \(\left|\text{Im}\,z\right|<B\) and diverges for every…
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 6435699
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: exponential weights
zbMATH Keywords: series of orthogonal polynomials
reviewed by: Alexander Tovstolis
MaRDI profile type: Publication
full work available at URL: j.jat.2015.02.003
OpenAlex ID: W2070764309
cites work: Bounds for orthogonal polynomials for exponential weights
cites work: On the domain of convergence of series in polynomials orthogonal with respect to general weight functions on the whole real line
cites work: Q4857769
DOI: 10.1016/J.JAT.2015.02.003
description: scientific article