Lacunary Statistical Convergence of Difference Double Sequences

In this paper our purpose is to extend some results known in the literature for ordinary difference (single) to difference double sequences of real numbers.Quite recently, Esi [1] defined the statistical analogue for double difference sequences x = (xk,l) as follows: A real double sequence x = (xk,l) is said to be P-statistically ∆− convergent to L provided that for each ε > 0 P − lim m,n 1 mn {the number of (k, l) : k < m, l < n; |∆xk,l − L| ≥ ε} = 0. In this paper we introduce and study lacunary statistical convergence for difference double sequences and we shall also give some inclusion theorems. 2000 Mathematics Subject Classification: 40A05,40A35,40B05. 1.Introduction Before we go into the motivation for this paper and presentation of the main results we give some preliminaries. A double sequence x = (xk,l) has a Pringsheim limit L (denoted by P − limx = L) provided that given an ε > 0 there exists an N ∈ N such that |xk,l − L| < ε whenever k, l > N.We shall describe such an x = (xk,l) more briefly as ”P − convergent” [2].The double sequence x = (xk,l) is bounded if there exists a positive number M such that |xk,l| < M for all k and l, ‖x‖ = sup k,l |xk,l| <∞. We should note that in contrast to the case for single sequences, a convergent double sequence need not be bounded.The concept of statistical convergence was introduced by Fast [5] in 1951. A complex number sequence x = (xk) is said to be statistically convergent to the number L if for every ε > 0 lim n 1 n |{ k ≤ n : |xk − L| ≥ ε}| = 0