A-statistical convergence with a rate and applications to approximation

Abstract

A = (ank) be a regular summability matrix. In the present paper we deal with subspaces of the space of A?statistically convergent sequences obtained by the rate at which the A?statistical limit tends to zero. We prove that a sequence is the A?strongly convergent if and only if it is the A?statistically convergent and the A?uniformly integrable with the rate of o (an) where a = (an) is a positive nonincreasing sequence. We also make a link between the A?strong convergence and the A?distributional convergence with the rate of o (an). Finally, as an application we present an approximation theorem of Korovkin type.