On Random Polynomials Spanned by OPUC

Abstract

We consider the behavior of zeros of random polynomials of the from \begin{equation*} P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z) \end{equation*} as ( n\to\infty ), where ( m ) is a non-negative integer (most of the work deal with the case ( m =0 ) ), ( {\eta_n}{n=0}^\infty ) is a sequence of i.i.d. Gaussian random variables, and ( {\varphi_n(z)}{n=0}^\infty ) is a sequence of orthonormal polynomials on the unit circle ( \mathbb T ) for some Borel measure ( \mu ) on ( \mathbb T ) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.