A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2022; you can also visit the original URL.
The file type is
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 holdsdoi:10.1090/proc/12836 fatcat:b6vs3gorknfj5exputj3vb7qem