# Boundary behavior of optimal polynomial approximants

## TL;DR

This paper develops an efficient method to analyze the boundary behavior of optimal polynomial approximants in Hilbert spaces of analytic functions, revealing convergence properties and estimates related to zeros of the polynomial $f$.

## Contribution

It introduces a new computational approach for Taylor coefficients of $1-p_nf$, extending understanding of boundary behavior and convergence of optimal polynomial approximants, including cases with multiple zeros.

## Key findings

- Sequence $\\{1-p_nf\}\$ is uniformly bounded on the closed unit disc.
- If $f$ has no zeros inside $\mathbb{D}$, $\\{1-p_nf\}\$ converges uniformly to 0 outside the zeros.
- Provides precise convergence rate estimates on compacta.

## Abstract

In this paper, we provide an efficient method for computing the Taylor coefficients of $1-p_n f$, where $p_n$ denotes the optimal polynomial approximant of degree $n$ to $1/f$ in a Hilbert space $H^2_\omega$ of analytic functions over the unit disc $\mathbb{D}$, and $f$ is a polynomial of degree $d$ with $d$ simple zeros. As a consequence, we show that in many of the spaces $H^2_\omega$, the sequence $\{1-p_nf\}_{n\in \mathbb{N}}$ is uniformly bounded on the closed unit disc and, if $f$ has no zeros inside $\mathbb{D}$, the sequence $\{1-p_nf \}$ converges uniformly to 0 on compact subsets of the complement of the zeros of $f$ in $\bar{\mathbb{D}}, $ and we obtain precise estimates on the rate of convergence on compacta. We also treat the previously unknown case of a single zero with higher multiplicity.