2015/03/22 by D. S. Lubinsky, Lubinsky, D. S., Igor E. Pritsker +5 · 1 citation
Mathematics · #60B10 #Analytic Number Theory Research #FOS: Mathematics #Geometry and complex manifolds #Mathematical functions and polynomials #Primary: 30C15 #Probability (math.PR) #Secondary: 30B20 #math.PR #msc:30B20 #msc:30C15 #msc:60B10
paper · pdf · doi:10.48550/arxiv.1503.06376
13 pages
arxiv created 2015/03/22 · openalex publication_date 2015/03/22 · arxiv updated 2015/03/24 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
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. 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.