More properties of optimal polynomial approximants in Hardy spaces

Raymond Cheng and Christopher Felder

Vol. 327 (2023), No. 2, 267–295

DOI: 10.2140/pjm.2023.327.267

Abstract

We study optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, $H^p$ ($1<p<\infty$). For fixed $f\in H^p$ and $n\in \mathbb{N}$, the OPA of degree $n$ associated to $f$ is the polynomial which minimizes the quantity $\parallel qf-1{\parallel }_p$ over all complex polynomials $q$ of degree less than or equal to $n$. We begin with some examples which illustrate, when $p\ne 2$, how the Banach space geometry makes the above minimization problem interesting. We then weave through various results concerning limits and roots of these polynomials, including results which show that OPAs can be witnessed as solutions of certain fixed-point problems. Finally, using duality arguments, we provide several bounds concerning the error incurred in the OPA approximation.

Keywords

optimal polynomial approximant, Pythagorean inequality, duality, fixed point

Mathematical Subject Classification

Primary: 30E10

Secondary: 46E30

Milestones

Received: 25 November 2023

Revised: 4 February 2024

Accepted: 4 February 2024

Published: 12 March 2024