Classical theorems through convergence methods

TL;DR

This paper surveys various classical theorems reinterpreted through different summability methods, analyzing the necessary properties for these methods and deriving sharp conditions that lead to new research questions.

Contribution

It introduces a unified framework for understanding classical theorems via summability methods, highlighting conditions needed and revealing new problems.

Findings

Sharp conditions for summability methods to adapt classical results

Identification of properties necessary for summability methods

Emergence of new research problems from the analysis

Abstract

We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods. Sometimes very sharp conditions are obtained, giving a focused view of the subject and from which new problems emerge.