Kernel-Summability Methods and the Silverman-Toeplitz Theorem

TL;DR

This paper develops kernel-summability methods in Banach spaces, extending the Silverman-Toeplitz theorem to vector-valued functions and applying these results to the summability of Taylor series in holomorphic function spaces.

Contribution

It introduces kernel-summability methods in Banach spaces, proves an analogue of the Silverman-Toeplitz theorem for these methods, and extends inclusion results from scalar to vector-valued functions.

Findings

Established an analogue of the Silverman-Toeplitz theorem for kernel-summability methods.

Proved that inclusion of scalar kernel-summability methods implies inclusion for Banach space-valued functions.

Applied the abstract results to the summability of Taylor series in Banach spaces of holomorphic functions.

Abstract

We introduce kernel-summability methods in Banach spaces using the vector-valued integrals and prove an analogue of the Silverman-Toeplitz Theorem for regular kernel-summability methods. We also show that if $X$ is a Banach space and one kernel-summability method is included in another kernel-summability method for scalar-valued functions, then the first method is included in the second method, for $X$-valued functions. This extends a previous result from Javad Mashreghi, Thomas Ransford and the author. We then apply these abstract results to the summability of Taylor series of functions in a Banach space of holomorphic functions on the unit disk.