Vol. 95, No. 2, 1981

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Densities and summability

Allen Roy Freedman and John Joseph Sember

Vol. 95 (1981), No. 2, 293–305

The ordinary asymptotic density of a set A of positive integers is ν(A) = limn→∞A(n)∕n, where A(n) is the cardinality of the set A ∩{1,2,,n}. It is known that the space of bounded strongly Cesàro 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 ν(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 convergence fields of various summability methods. These include all nonnegative regular matrix methods as well as the famous nonmatrix method called almost convergence.

Mathematical Subject Classification 2000
Primary: 10L10, 10L10
Secondary: 40C05
Received: 31 July 1979
Published: 1 August 1981
Allen Roy Freedman
John Joseph Sember