On Cesàro and Abel Summability Factors for Integrals

Abstract

Many results have been obtained about factors transforming integrals summable by ordinary and absolute Cesàro methods of non-negative orders into integrals summable by such methods (see [4], [2], [6], [3]) and also into integrals summable by the ordinary and absolute Abel methods (see [7]). Since the Cesàro summability methods (C, α) and |C, α| for integrals are defined for α ≦ –1, it is natural to try to extend the above mentioned results for α ≦ 0 to the case –1 ≧ α < 0. In this paper we restrict attention to the simplest case α = –1, and classify the summability factors from (C, –1) and |C, –1| to (C, –1) , |C, –1|, (C, λ), |C, λ|, A and |A|, where λ ≦ 0 and A denotes Abel summability.