Linear polynomial approximation schemes in Banach holomorphic function spaces

TL;DR

This paper characterizes Banach holomorphic function spaces on the unit disk that allow for linear polynomial approximation schemes, revealing that polynomial density alone is not sufficient for such schemes to exist.

Contribution

It provides a complete characterization of spaces admitting linear polynomial approximation schemes, extending understanding beyond polynomial density.

Findings

Spaces with polynomial density do not necessarily admit approximation schemes

Complete characterization of spaces with linear polynomial approximation schemes

Conditions for approximation schemes depend on more than polynomial density

Abstract

Let $X$ be a Banach holomorphic function space on the unit disk. A linear polynomial approximation scheme for $X$ is a sequence of bounded linear operators $T_n:X\to X$ with the property that, for each $f\in X$, the functions $T_n(f)$ are polynomials converging to $f$ in the norm of the space. We completely characterize those spaces $X$ that admit a linear polynomial approximation scheme. In particular, we show that it is NOT sufficient merely that polynomials be dense in $X$.