A-Statistical Convergence with a Rate and Applications to Approximation

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.