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# ON RANDOM POLYNOMIALS SPANNED BY OPUC

thesis
posted on 2021-01-07, 15:00 authored by Hanan AljubranHanan Aljubran

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.

## Degree Type

Doctor of Philosophy

Mathematics

## Campus location

Indianapolis

Dr. Maxim Yattselev

Dr. Pavel Bleher

Dr. Evgeny Mukhin