Density by Moduli and Lacunary Statistical Convergence

Abstract EN

We have introduced and studied a new concept of f-lacunary statistical convergence, where f is an unbounded modulus.

It is shown that, under certain conditions on a modulus f, the concepts of lacunary strong convergence with respect to a modulus f and f-lacunary statistical convergence are equivalent on bounded sequences.

We further characterize those θ for which Sθf=Sf, where Sθf and Sf denote the sets of all f-lacunary statistically convergent sequences and f-statistically convergent sequences, respectively.

A general description of inclusion between two arbitrary lacunary methods of f-statistical convergence is given.

Finally, we give an Sθf-analog of the Cauchy criterion for convergence and a Tauberian theorem for Sθf-convergence is also proved.