Density by Moduli and Lacunary Statistical Convergence

Abstract

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 $\theta$ for which $S_{\theta }^f=S^f$, where $S_{\theta }^f$ and $S^f$ 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 $S_{\theta }^f$-analog of the Cauchy criterion for convergence and a Tauberian theorem for $S_{\theta }^f$-convergence is also proved.