Optimal Polynomial Approximants in $L^p$

TL;DR

This paper extends the theory of optimal polynomial approximants (OPAs) to the $L^p$ spaces, establishing existence, uniqueness, and zero-free regions, especially focusing on the case $L^2$, and exploring their properties and coefficients.

Contribution

It introduces the concept of OPAs in $L^p$ spaces, proves existence and uniqueness for $1<p< ext{infinity}$, and analyzes zeros and orthogonality conditions, providing new tools for their study.

Findings

Existence and uniqueness of OPAs in $L^p$ for $1<p< ext{infinity}$

Zero-free disk regions for OPAs in $L^p$ when $f(0) eq 0$

Orthogonality conditions and coefficient computations for $L^p$ OPAs

Abstract

Over the past several years, optimal polynomial approximants (OPAs) have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the properties of their zeros. In this paper, we introduce the notion of optimal polynomial approximant in the space $L^p$, $1\leq p\leq\infty$. We begin our treatment by showing existence and uniqueness for $1<p<\infty$. For the extreme cases of $p=1$ and $p=\infty$, we show that uniqueness does not necessarily hold. We continue our development by elaborating on the special case of $L^2$. Here, we create a test to determine whether or not a given 1st degree OPA is zero-free in $\overline{\mathbb{D}}$. Afterward, we shed light on an orthogonality condition in $L^p$. This allows us to study OPAs in $L^p$ with the additional tools from the $L^2$ setting. Throughout this paper, we focus many of our discussions on the zeros of OPAs. In particular, we show that if $1<p<\infty$, $f\in H^p$, and $f(0)\neq 0$, then there exists a disk, centered at the origin, in which all the associated OPAs are zero-free. Toward the end of this paper, we use the orthogonality condition to compute the coefficients of some OPAs in $L^p$. To inspire further research in the general theory, we pose several open questions throughout our discussions.