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2018 Zeros of random orthogonal polynomials with complex Gaussian coefficients
Aaron Yeager
Rocky Mountain J. Math. 48(7): 2385-2403 (2018). DOI: 10.1216/RMJ-2018-48-7-2385

Abstract

Let $\{f_j\}_{j=0}^n$ be a sequence of orthonormal polynomials where the orthogonality relation is satisfied on either the real line or on the unit circle. We study zero distribution of random linear combinations of the form $$P_n(z)=\sum _{j=0}^n\eta _jf_j(z),$$ where $\eta _0,\ldots ,\eta _n$ are complex-valued iid standard Gaussian random variables. Using the Christoffel-Darboux formula, the density function for the expected number of zeros of $P_n$ in these cases takes a very simple shape. From these expressions, under the mere assumption that the orthogonal polynomials are from the Nevai class, we give the limiting value of the density function away from their respective sets where the orthogonality holds. In the case when $\{f_j\}$ are orthogonal polynomials on the unit circle, the density function shows that the expected number of zeros of $P_n$ are clustering near the unit circle. To quantify this phenomenon, we give a result that estimates the expected number of complex zeros of $P_n$ in shrinking neighborhoods of compact subsets of the unit circle.

Citation

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Aaron Yeager. "Zeros of random orthogonal polynomials with complex Gaussian coefficients." Rocky Mountain J. Math. 48 (7) 2385 - 2403, 2018. https://doi.org/10.1216/RMJ-2018-48-7-2385

Information

Published: 2018
First available in Project Euclid: 14 December 2018

zbMATH: 06999267
MathSciNet: MR3892137
Digital Object Identifier: 10.1216/RMJ-2018-48-7-2385

Subjects:
Primary: 26C10 , 30B20 , 30C15 , 60B99

Keywords: Christoffel-Darboux formula , Nevai class , orthogonal polynomials , random polynomials

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 7 • 2018
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