General Optimal Polynomial Approximants, Stabilization, and Projections of Unity

TL;DR

This paper characterizes when functions in Hilbert spaces of analytic functions have optimal polynomial approximants as truncations of a single power series, introduces a generalized notion of optimal approximant, and computes orthogonal projections onto shift invariant subspaces.

Contribution

It provides a characterization of optimal polynomial approximants in Hilbert spaces and introduces a generalized concept, with explicit computations of projections.

Findings

Characterization of functions with polynomial approximants as power series truncations

Introduction of a generalized optimal approximant concept

Explicit computation of orthogonal projections onto shift invariant subspaces

Abstract

For various Hilbert spaces of analytic functions on the unit disk, we characterize when a function $f$ has optimal polynomial approximants given by truncations of a single power series. We also introduce a generalized notion of optimal approximant and use this to explicitly compute orthogonal projections of 1 onto certain shift invariant subspaces.