On Lacunary Statistical Limit Points

Abstract

Abstract. In this paper we study the concepts of lacunary statistical limit points and lacunary statistical cluster points as well as the concept of lacunary statistical core for a bounded complex number sequence. Introduction and NotationsThe idea of statistical convergence was first introduced by Steinhaus [20] in 1949 and studied by Fast and several authors [7], [10], [2].Recall that a number sequence x = (xfc) is said to be statistically convergent to the number L if for every e > 0, lim -|{fc <n:\x k -L\> e}| = 0, n n where the vertical bars indicate the number of elements in the enclosed set.If if is a subset of the natural numbers N, K n will denote the set {k < n : k € K}, and \K n \ will denote the cardinality of K n . The natural density of K [19] is given by 6(K) :-lim-|A" n |, if it exsits. The number sequence n n x -(%k) is statistically convergent to L provided that for every c > 0 the set K(e) := {fc € N : |x fc -L\ > e} has natural density zero [7], [10]. Hence x = (xfc) is statistically convergent to L iff {C\XK(t))n -> 0, (as ra -» oo, for every e > 0), where C\ is the Cesaro mean of order one and XK(<.) is the chsiracteristic function of the set K(e).Statistical convergence can be generalized by using a nonnegative regular matrix A in place of C\: we say that a subset K of N has yl-density This research was supported by Kirikkale University, Turkey. 1991 Mathematics Subject Classification: Primary 40A05, Secondary 26A03, 11B05. Key words and phrases-, lacunary statistical limit point, lacunary statistical cluster point, lacunary statistical core of a sequence.

Inclusion results on statistical cluster points

“…In this note, by taking a nonnegative regular matrix A instead of the Cesàro matrix, we give extentions of the results given in [5] under weaker conditions. In Section 2 of this paper we refine [5, Theorem 2.1] by omitting a condition.…”

“…Osikiewicz and Khan [16] extended this concept to a more general matrix method A. Later Demirci [5] studied the concepts of statistical cluster point, statistical core of a sequence for C λ methods.…”

Lacunary Statistical Limit and Cluster Points of Generalized Difference Sequences of Fuzzy Numbers

“…In 1993, Fridy and Orhan [7] presented an interesting generalization of statistical convergence with the help of a lacunary sequence and called it lacunary statistical convergence or S θ -convergence. Demirci [8] defined S θ -limit and cluster points of number sequences and obtained some interesting results analogous to [4]. In past years, statistical convergence has also become an interesting area of research for sequences of fuzzy numbers.…”

Deferred statistical cluster points of double sequences

“…Later on some generalizations and applications of these notions were investigated by many authors (see [5,7,8,12,14,19,23,24]). Some equivalence results for Cesàro submethods were studied by Goffman and Petersen [10], Armitage and Maddox [2] and Osikiewicz [17].…”