Remarks on Inner Functions and Optimal Approximants

TL;DR

This paper explores the relationship between inner functions and optimal polynomial approximants in reproducing kernel Hilbert spaces, revisiting classical examples and proposing modifications to existing constructions.

Contribution

It provides new insights into the connection between inner functions and optimal approximants, including a modified construction based on Shapiro and Shields' method.

Findings

Inner functions relate closely to optimal polynomial approximants.

Classical examples are revisited with a new perspective.

A modified construction for producing inner functions is proposed.

Abstract

We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions.