Complete proofs of the main theorems on summability factors

Abstract

In this article the shortest proofs up to the present of the main theorems on summability factors for Cesàro methods are given. The necessity part of these theorems is proved for more general methods. Especially short is the proof of the sufficiency part of the theorem for summability factors of types (C₀^α,|C^β|) and (C^α,|C^β|).