Power-series summability methods in de Branges-Rovnyak spaces

TL;DR

This paper investigates the limitations of classical summability methods in de Branges-Rovnyak spaces, demonstrating that polynomial approximation does not guarantee convergence of various summability means, supported by a new abstract inclusion result.

Contribution

It introduces a novel abstract theorem linking scalar and Banach-space-valued summability methods, and applies it to show divergence phenomena in de Branges-Rovnyak spaces.

Findings

Existence of functions in de Branges-Rovnyak spaces with polynomial approximation but no convergence of classical means.

New abstract result on summability method inclusion for scalar and Banach-space sequences.

Demonstration that polynomial approximation does not imply convergence of Cesàro, Abel, Borel, or logarithmic means.

Abstract

We show that there exists a de Branges-Rovnyak space $\mathcal{H}(b)$ on the unit disk containing a function $f$ with the following property: even though $f$ can be approximated by polynomials in $\mathcal{H}(b)$, neither the Taylor partial sums of $f$ nor their Ces\`aro, Abel, Borel or logarithmic means converge to $f$ in $\mathcal{H}(b)$. A key tool is a new abstract result showing that, if one regular summability method includes another for scalar sequences, then it automatically does so for certain Banach-space-valued sequences too.