Equidistribution of zeros of some polynomials related to cyclic functions

TL;DR

This paper investigates the zero distribution of certain polynomials associated with cyclic functions in Hilbert spaces, showing they tend to distribute uniformly on the unit circle under specific conditions.

Contribution

It establishes the asymptotic equidistribution of zeros of polynomials related to cyclic functions, especially for polynomials with boundary zeros, expanding understanding of their zero behavior.

Findings

Zeros of the polynomials tend to distribute uniformly on the unit circle.

The distribution holds for polynomials without zeros inside the disk but with boundary zeros.

The results apply to a broad class of cyclic functions in reproducing kernel Hilbert spaces.

Abstract

In the study of the cyclicity of a function $f$ in reproducing kernel Hilbert spaces an important role is played by sequences of polynomials $\{p_n\}_{n\in \mathbb{N}}$ called \emph{optimal polynomial approximants} (o.p.a.). For many such spaces and when the functions $f$ generating those o.p.a. are polynomials without zeros inside the disk but with some zeros on its boundary, we find that the weakly asympotic distribution of the zeros of $1-p_nf$ is the uniform measure on the unit circle.