Expected number of real zeros for random Freud orthogonal polynomials

Pritsker, Igor E.; Xie, Xiaoju

doi:10.48550/arxiv.1505.04762

Cited by 1 publication

(2 citation statements)

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“…The authors recently showed [29] that if the basis is given by the orthonormal polynomials associated to a finite Borel measure with compact support on the real line, then random linear combinations have n/ √ 3 + o(n) expected real zeros under mild conditions on the weight. The second and the third authors [33] extended this asymptotic to random orthogonal polynomials associated with the Freud weights W (x) = e −c|x| λ , c > 0, λ > 1. The results of this paper provide detailed information on the expected number of real zeros for random polynomials associated with a large class of weights defined on the whole real line.…”

Section: Random Orthogonal Polynomialsmentioning

confidence: 97%

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Expected number of real zeros for random orthogonal polynomials

Lubinsky¹,

Pritsker²,

Xie³

2016

Math. Proc. Camb. Phil. Soc.

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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)) log n 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/ √ 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.

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Section: Random Orthogonal Polynomialsmentioning

confidence: 97%

“…The results of this paper provide detailed information on the expected number of real zeros for random polynomials associated with a large class of weights defined on the whole real line. In particular, they cover the case of random Freud polynomials considered in [33].…”

Section: Random Orthogonal Polynomialsmentioning

confidence: 99%