Boundary behavior of optimal polynomial approximants

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 $H_{\omega }^2$ 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 $H_{\omega }^2$, the sequence $\{1-p_{n}f{\}}_{n\in \mathbb{N}}$ is uniformly bounded on the closed unit disc and, if $f$ has no zeros inside $\mathbb{D}$, the sequence $\{1-p_{n}f\}$ 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.