Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems

TL;DR

This paper investigates the zeros of optimal polynomial approximants in Hilbert spaces of analytic functions, revealing their distribution, extremal properties, and connections to Jacobi matrices and Jentzsch-type theorems.

Contribution

It introduces a nonlinear extremal problem linked to Jacobi matrices to determine zero moduli and establishes new Jentzsch-type theorems for zero accumulation points.

Findings

Identifies minimal zero moduli via extremal problems.

Proves zero accumulation points follow Jentzsch-type distributions.

Provides detailed zero behavior of reproducing kernels in weighted spaces.

Abstract

We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzsch-type theorems describing where they accumulate. As a consequence, we obtain detailed information regarding zeros of reproducing kernels in weighted spaces of analytic functions.