Expected number of real zeros for random linear combinations of orthogonal polynomials

Lubinsky, D.; Pritsker, I.; Xie, X.

doi:10.1090/proc/12836

Expected number of real zeros for random linear combinations of orthogonal polynomials
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by D. S. Lubinsky, I. E. Pritsker and X. Xie PDF

Proc. Amer. Math. Soc. 144 (2016), 1631-1642 Request permission

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 $(2/\pi + o(1))\log {n}$ expected real zeros in terms of the degree $n$. On the other hand, if the basis is given by Legendre (or more generally by Jacobi) polynomials, then random linear combinations have $n/\sqrt {3} + o(n)$ expected real zeros. We prove that the latter asymptotic relation holds universally for a large class of random orthogonal polynomials on the real line, and also give more general local results on the expected number of real zeros.

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