Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

Authors:

D.?S.?Lubinsky author-information" > author-information__contact u-icon-before" > mailto:lubinsky@math.gatech.edu" title=" lubinsky@math.gatech.edu" itemprop=" email" data-track=" click" data-track-action=" Email author" data-track-label=" " >Email author,A.?Sidi

Affiliation:

(1) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA;(2) Department of Computer Science, Technion-Israel Institute of Technology, Haifa, 32000, Israel

Abstract:

We analyze polynomials P _n that are biorthogonal to exponentials ${e^{-sigma _{n,j}x}}_{j=1}^{n}$ , in the sense that

$int_{0}^{infty }P_{n}(x)e^{-sigma _{n,j}x}x^{alpha },dx=0,quad 1leq jleq n.$

Here α>−1. We show that the zero distribution of P _n as n→∞ is closely related to that of the associated exponent polynomial

$Q_{n}(y)=prodlimits_{j=1}^{n}(y+1/sigma _{n,j})=sum_{j=0}^{n}q_{n,j}y^{j}.$

More precisely, we show that the zero counting measures of {P _n(−4nx)}_n=1^∞ converge weakly if and only if the zero counting measures of {Q _n}_n=1^∞ converge weakly. A key step is relating the zero distribution of such a polynomial to that of the composite polynomial

$sum_{j=0}^{n}q_{n,j}Delta _{n,j}x^{j},$

under appropriate assumptions on {Δ_n,j}.

Keywords:

Biorthogonal polynomials Zero distribution Laguerre polynomials

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