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Mat. Sb., 2018, Volume 209, Number 3, Pages 168–188 (Mi msb8793)  

Relative asymptotics of orthogonal polynomials for perturbed measures

E. B. Saffa, N. Stylianopoulosb

a Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN, USA
b Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus

Abstract: We survey and present some new results that are related to the behaviour of orthogonal polynomials in the plane under small perturbations of the measure of orthogonality. More precisely, we introduce the notion of a polynomially small (PS) perturbation of a measure. Namely, if $\mu_0 \ge \mu_1$ and $\{p_n(\mu_j,z)\}_{n=0}^\infty$, $j=0,1$, are the associated orthonormal polynomial sequences, then $\mu_0$ is a PS perturbation of $\mu_1$ if $\|p_n(\mu_1, \cdot )\|_{L_2(\mu_0-\mu_1)}\to 0$, as $n\to\infty$. In such a case we establish relative asymptotic results for the two sequences of orthonormal polynomials. We also provide results dealing with the behaviour of the zeros of PS perturbations of area orthogonal (Bergman) polynomials.
Bibliography: 35 titles.

Keywords: orthogonal polynomial, Christoffel function, Bergman polynomial, perturbed measure.

Funding Agency Grant Number
National Science Foundation DMS-1412428
DMS-1516400
University of Cyprus 3/311-21027
E. B. Saff's research was carried out with the support of the National Science Foundation (grant DMS-1516400). N. Stylianopoulos' research was carried out with the support of the University of Cyprus (grant 3/311-21027).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8793

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English version:
Sbornik: Mathematics, 2018, 209:3, 449–468

Bibliographic databases:

UDC: 517.538.3
MSC: 30E05, 65E05, 42C05, 30C10, 94A08, 30C40, 30C70, 41A10, 31A15
Received: 01.08.2016 and 03.06.2017

Citation: E. B. Saff, N. Stylianopoulos, “Relative asymptotics of orthogonal polynomials for perturbed measures”, Mat. Sb., 209:3 (2018), 168–188; Sb. Math., 209:3 (2018), 449–468

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