English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let $$\{\varphi _k\}_{k=0}^\infty $$ be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure $$ \mu $$. We study the variance of the number of zeros of random linear combinations of the form $$\begin{aligned} P_n(z)=\sum _{k=0}^{n}\eta _k\varphi _k(z), \end{aligned}$$where $$\{\eta _k\}_{k=0}^n $$ are complex-valued random variables. Under the assumption that the distribution for each $$\eta _k$$ satisfies certain uniform bounds for the fractional and logarithmic moments, for the cases when $$\{\varphi _k\}$$ are regular in the sense of Ullman–Stahl–Totik or are such that the measure of orthogonality $$\mu $$ satisfies $$d\mu (\theta )=w(\theta )d\theta $$ where $$w(\theta )=v(\theta )\prod _{j=1}^J|\theta - \theta _j|^{\alpha _j}$$, with $$v(\theta )\ge c>0$$, $$\theta ,\theta _j\in [0,2\pi )$$, and $$\alpha _j>0$$, we give a quantitative estimate on the variance of the number of zeros of $$P_n$$ in sectors that intersect the unit circle. When $$\{\varphi _k\}$$ are real-valued on the real-line from the Nevai class and $$\{\eta _k\}$$ are i.i.d. complex-valued standard Gaussian, we obtain 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.

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