On Some Lacunary Generalized Difference Sequence Spaces Defined by a Modulus Function in a Locally Convex Space

In this paper, we introduce some new lacunary generalized difference sequence spaces defined by a modulus function in a locally convex Hausdorff topological linear space whose topology is determined by a finite set Q of seminorms. Various algebraic and topological properties of these spaces, and some inclusion relations between these spaces have been discussed. We also characterize those lacunary sequences theta for which N-theta [Delta(m)(v),f, p, Q] = w(Delta(m)(v), f, p, Q). Finally, a new concept of (Delta(m)(v), Q)-lacunary statistical convergence is introduced. It is shown that if a sequence is (Delta(m)(v), Q)-lacunary strongly summable then it is (Delta(m)(v), Q)-lacunary statistically convergent and the concepts of (Delta(m)(v), Q)-lacunary strong summability and (Delta(m)(v), Q)-lacunary statistical convergence are equivalent on (Delta(m)(v), Q) -bounded sequences. Our results generalize and unify the corresponding earlier results of A.R. Freedman et al., I.J. Maddox, Y. Altin and M. Et and others.