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

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)) 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/ √ 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.

Real Zeroes of Random Analytic Functions Associated with Geometries of Constant Curvature

“…Similar approach has already been applied to real zeroes of random trigonometric polynomials in one and many variables [15,1], as well as to random Taylor series [16]. It should be possible to treat random linear combinations of orthogonal polynomials by similar methods, thus proving the universality of the limiting mean density function computed in [33,32] in the case of Gaussian coefficients. We believe that the second moment assumption on ξ 0 is nearly optimal.…”

Expected Number of Real Roots for Random Linear Combinations of Orthogonal Polynomials Associated with Radial Weights

“…Recently, Lubinsky, Pritsker & Xie [LPX15,PX15,LPX16] generalized this result to the random orthogonal polynomials with suitable weights supported on the real line. In this note we study real roots of random polynomials which are random linear combinations of orthogonal polynomials induced by a non-negative radially symmetric (i.e.…”

“…where P n j are entire functions that take real values on the real line (see also [Van15,LPX15] for recent treatment of this problem). In particular, Edelman and Kostlan [EK95,§3] proved that…”

“…Gaussian case. It follows from [EK95] (see also [Van15,LPX15]) that for every measurable set E ⊂ R the expected number of real zeros of f n in E is given by…”

Real zeros of random sums with i.i.d. coefficients

“…Asymptotics for the density function ρ n (x) in the case when the random variables {η k } are i.i.d. standard Gaussian has been well studied when the spanning functions are trigonometric functions [36], polynomials orthogonal on the real line ( [6], [7], [2], [28], [29], [31], [39]), and polynomials orthogonal on the unit circle ( [39], [1], [38]). As an application we consider the case…”