Some Spaces of Lacunary Convergent Sequences Defined by Orlicz Functions

We introduce the sequence spaces [ ˆ w(M)] and [ ˆ w(M)] θ of strong almost convergence and lacunary strong almost convergence respectively defined by the Orlicz function M. We establish certain inclusion relations and show that the above spaces are same for any bounded sequences. 1. Definitions and notations A sequence x ∈ l ∞ , the space of bounded sequences x = (x k), is said to be almost convergent [5] to s if lim k→∞ t km (x) = s uniformly in m, where t km (x) = 1 k + 1 k i=0 x m+i. Recently, Das and Sahoo [2] introduced the following sequence spaces using the concept of almost convergence. ˆ w = x : lim n 1 n + 1 n k=0 t km (x − s) = 0, uniformly in m, for some s , [ ˆ w] = x : lim n 1 n + 1 n k=0 |t km (x − s)| = 0, uniformly in m, for some s. By a lacunary sequence θ = (k r), r = 0, 1, 2, · · · , where k 0 = 0, we shall mean an increasing sequence of non-negative integers h r = (k r − k r−1) → ∞, (r → ∞). The intervals determined by θ are denoted by I r = (k r−1 − k r ] and the ratio k r kr−1 will be denoted by q r .