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Expected number of real zeros for random orthogonal polynomials

Published online by Cambridge University Press:  27 September 2016

DORON S. LUBINSKY ,
IGOR E. PRITSKER  and
XIAOJU XIE
DORON S. LUBINSKY
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. e-mail: lubinsky@math.gatech.edu
IGOR E. PRITSKER
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, U.S.A. e-mails: igor@math.okstate.edu; sophia.xie@okstate.edu
XIAOJU XIE
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, U.S.A. e-mails: igor@math.okstate.edu; sophia.xie@okstate.edu
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Abstract

We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only (2/π + o(1))logn expected real zeros in terms of the degree n. If the basis is given by the orthonormal polynomials associated with a compactly supported Borel measure on the real line, or associated with a Freud weight defined on the whole real line, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real zeros. We prove that the same asymptotic relation holds for all random orthogonal polynomials on the real line associated with a large class of weights, and give local results on the expected number of real zeros. We also show that the counting measures of properly scaled zeros of these random polynomials converge weakly to either the Ullman distribution or the arcsine distribution.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 1 , January 2018 , pp. 47 - 66
Copyright
Copyright © Cambridge Philosophical Society 2016 

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