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thesis
posted on 07.01.2021, 15:00 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.

History

Degree Type

Doctor of Philosophy

Department

Mathematics

Campus location

Indianapolis

Advisor/Supervisor/Committee Chair

Dr. Maxim Yattselev

Additional Committee Member 2

Dr. Pavel Bleher

Additional Committee Member 3

Dr. Evgeny Mukhin

Additional Committee Member 4

Dr. Roland Roeder