arXiv subjects

Aaron M. Yeager

Publications and source records attributed to Aaron M. Yeager.

4 recordsLinked to original sources

arXiv2019

The Variance of the Number of Zeros for Complex Random Polynomials Spanned by OPUC

Let $\{φ_k\}_{k=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle (OPUC) with respect to a probability measure $ μ$. We study the variance of the number of zeros of random linear combinations of the form $$ P_n(z)=\sum_{k=0}^{n}η_kφ_k(z), $$ where $\{η_k\}_{k=0}^n $ are complex-valued random variables. Under the assumption that the distribution for each $η_k$ satisfies certain uniform bounds for the fractional and logarithmic moments, for the cases when $\{φ_k\}$ are regular in the sense of Ullman-Stahl-Totik or are such that the measure of orthogonality $μ$ satisfies $dμ(θ)=w(θ)dθ$ where $w(θ)=v(θ)\prod_{j=1}^J|θ- θ_j|^{α_j}$, with $v(θ)\geq c>0$, $θ,θ_j\in [0,2π)$, and $α_j>0$, we give a quantitative estimate on the the variance of the number of zeros of $P_n$ in sectors that intersect the unit circle. When $\{φ_k\}$ are real-valued on the real-line from the Nevai class and $\{η_k\}$ are i.i.d.~complex-valued standard Gaussian, we prove a formula for the limiting value of variance of the number of zeros of $P_n$ in annuli that do not contain the unit circle.

math.CA↗

arXiv2019

Real Zeros of Random Sums with I.I.D. Coefficients

Let $\{f_k\}$ be a sequence of entire functions that are real valued on the real-line. We study the expected number of real zeros of random sums of the form $P_n(z)=\sum_{k=0}^nη_k f_k(z)$, where $\{η_k\}$ are real valued i.i.d.~random variables. We establish a formula for the density function $ρ_n$ for the expected number of real zeros of $P_n$. As a corollary, taking the random variables $\{η_k\}$ to be i.i.d.~standard Gaussian, appealing to Fourier inversion we recover the representation for the density function previously given by Vanderbei through means of a different proof. Placing the restrictions on the common characteristic function $ϕ$ of $\{η_k\}$ that $|ϕ(s)|\leq (1+as^2)^{-q}$, with $a>0$ and $q\geq 1$, as well as that $ϕ$ is three times differentiable with each the second and third derivatives being uniformly bounded, we achieve an upper bound on the density function $ρ_n$ with explicit constants that depend only on the restrictions on $ϕ$. As an application we considered the limiting value of $ρ_n$ when the spanning functions $f_k(z)=p_k(z)$, $k=0,1,\dots, n$, where $\{p_k\}$ are Bergman polynomials on the unit disk.

math.CA↗

arXiv2016

Zeros of random linear combinations of OPUC with complex Gaussian coefficients

We study zero distribution of random linear combinations of the form $$P_n(z)=\sum_{j=0}^nη_jϕ_j(z),$$ in any Jordan region $Ω\subset \mathbb C$. The basis functions $ϕ_j$ are orthogonal polynomials on the unit circle (OPUC) that are real-valued on the real line, and $η_0,\dots,η_n$ are complex-valued iid Gaussian random variables. We derive an explicit intensity function for the number of zeros of $P_n$ in $Ω$ for each fixed $n$. Using the Christoffel-Darboux formula, the intensity function takes a very simple shape. Moreover, we give the limiting value of the intensity function when the orthogonal polynomials are associated to Szegő weights.

math.PR↗

arXiv2014

Zeros of Polynomials with Random Coefficients

Zeros of many ensembles of polynomials with random coefficients are asymptotically equidistributed near the unit circumference. We give quantitative estimates for such equidistribution in terms of the expected discrepancy and expected number of roots in various sets. This is done for polynomials with coefficients that may be dependent, and need not have identical distributions. We also study random polynomials spanned by various deterministic bases.

math.PR↗