On Lacunary Almost Convergent Sequences

The purpose of this paper is to define and study the spaces [ĉ,M, p](∆u , q, s), [ĉ,M, p, ] θ 0(∆ m u , q, s) and [ĉ,M, p, ] θ ∞(∆ m u , q, s) of lacunary convergent sequences. We also study some inclusion relations between these spaces and some properties and theorems. 2000 Mathematics Subject Classification:40D05, 40A05. 1.Introduction and definitions Let w denote the set of all complex sequences x = (xi), and l∞, c, and c0 be the linear spaces of bounded, convergent, and null sequences with complex terms, respectively, normed by ‖ x ‖= supi | xi |, where i ∈ N, the set of positive integers. A bounded sequence x = (xi) is said to be almost convergent ( see [8] ) if all Banach limits of x coincide. In [8], it was shown that : ĉ = {x = (xi) : lim n→∞ 1 n n ∑ i=0 xi+t exists, uniformly in t}. Maddox ( [9], [10] ) defined a sequence x = (xi) to be strongly almost convergent to a number L if