Optimal Polynomial Approximants in $H^p$

TL;DR

This paper investigates optimal polynomial approximants in Hardy spaces, providing estimates for low-degree approximants and analyzing root bounds for certain functions, using novel inequalities from Banach space theory.

Contribution

It introduces new bounds and root location results for OPAs in $H^p$, especially for inner functions and specific $p$, via innovative Banach space inequalities.

Findings

Root bounds depend only on $p$ for inner functions

Estimates for degree zero and one OPAs

Novel inequalities derived from Banach space Pythagorean theorem

Abstract

This work studies optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, $H^p$ ($1 < p < \infty$). In particular, we uncover some estimates concerning the OPAs of degree zero and one. It is also shown that if $f \in H^p$ is an inner function, or if $p>2$ is an even integer, then the roots of the nontrivial OPA for $1/f$ are bounded from the origin by a distance depending only on $p$. For $p\neq 2$, these results are made possible by the novel use of a family of inequalities which are derived from a Banach space analogue of the Pythagorean theorem.