An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC. (arXiv:1809.04948v1 [math.CA])

Let $ \{\varphi_i\}_{i=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure $ \mu $ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say $ \mathbb E_n(\mu) $, of random polynomials \[ P_n(z) := \sum_{i=0}^n\eta_i\varphi_i(z), \] where $ \eta_0,\dots,\eta_n $ are i.i.d. standard Gaussian random variables. When $ \mu $ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that $ \mathbb E_n(|\mathrm d\xi|) $ admits an asymptotic expansion of the form \[ \mathbb E_n(|\mathrm d\xi|) \sim \frac2\pi\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where $ \mu $ is absolutely continuous with respect to arclength measure and its Radon-Nikodym derivative extends to a holomorphic non-vanishing function in some neigh 查看全文>>