Advances in almost convergence

Abstract

‎In this paper‎, ‎we first give the concept of properly distributed‎ ‎sequence‎, ‎and prove that it is almost convergent with F-limit‎ ‎expressed as a formal integral‎. ‎Basing on these‎, ‎we review the work‎ ‎of Feng and Li‎, ‎which is shown to be a special case of our‎ ‎generalized theory‎. ‎Then we generalize Banach limit to Banach limit‎ ‎functional‎, ‎which is the minimum requirement to characterize strong‎ ‎almost convergence for bounded sequences in normed vector space‎. ‎With this machinery‎, ‎we show that Hajdukovi\'{c}'s almost‎ ‎convergence and quasi-almost convergence are both equivalent to our‎ ‎strong almost convergence‎.