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

TL;DR

This paper establishes a universal asymptotic formula for the expected number of real zeros in random linear combinations of orthogonal polynomials, extending known results from specific bases like Legendre to a broad class.

Contribution

It proves a universal asymptotic relation for the expected number of real zeros in random orthogonal polynomial combinations, generalizing previous specific cases.

Findings

Expected zeros grow linearly with degree for orthogonal polynomials

Universal asymptotic relation holds for a broad class of orthogonal polynomials

Local results on expected number of real zeros are provided

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.