Rainwater-Simons-type convergence theorems for generalized convergence methods

TL;DR

This paper extends Rainwater-Simons convergence theorems to generalized convergence methods like strong matrix summability, statistical convergence, and almost convergence, including (I)-generating sets.

Contribution

It introduces new convergence theorems applicable to broader classes of convergence methods and sets, generalizing existing results.

Findings

Theorems for strong matrix summability convergence.

Results for statistical convergence and almost convergence.

Extension to (I)-generating sets.

Abstract

We extend the well-known Rainwater-Simons convergence theorem to various generalized convergence methods such as strong matrix summability, statistical convergence and almost convergence. In fact we prove these theorems not only for boundaries but for the more general notion of (I)-generating sets introduced by Fonf and Lindenstrauss.