Densities and summability

Abstract

The ordinary asymptotic density of a set A of positive integers is where A(n) is the cardinality of the set A⋂(1, 2, n). It is known that the space of bounded strongly Cesáro summable sequences are just those bounded sequences that converge (in the ordinary sense) after the removal of a suitable collection of terms, the indices of which form a set A for which v(A)=0. In this paper we introduce a general concept of density and then examine the relationship, suggested by the above observation, between these densities and the strong convengence fields of various summability methods. These include all nonnegative regular matrix methods as well as the famous nonmatrix method called almost convergence. © 1981, University of California, Berkeley. All Rights Reserved.