A-Statistical Core of a Sequence

Abstract

Abstract. In this paper we extend the concepts of statistical limit superior and inferior (as introduced by FVidy and Orhan) to A-statistical limit superior and inferior and give some .¿-statistical analogue of properties of statistical limit superior and inferior for a sequence of real numbers. Also we extend the concept of statistical core to A-statistical core and get necessary and sufficient conditions on a matrix T so that the Knopp core of Tx is contained in the Astatistical core of a bounded complex number sequence x. IntroductionIf A" is a set of positive integers, K n will denote the set {k € K : k < n} and \K n \ will denote the cardinality of K n . The natural density of K [16] is given by 6{K) := lim" if it exists. The number sequence x = (x*) is statistically convergent to L provided that for every e > 0 the set K := K(e) := {k € N : |x* -L\ > «} has natural density zero [6], [8]. In this case, we write st -limx = L. Hence x is statistically convergent to L iff Statistical convergence can be generalized by using a regular nonnegative summability matrix A in place of Ci.Following Freedman and Sember [7], we say that a set K Ç N has. A-density if 6A(K) := lim"(Axjr)n = limn 52K€K ^fc exists where A is a nonnegative regular matrix.The number sequence x = (x*) is A-statistically convergent to L provided that for every e > 0 the set K(e) has A-density zero [3]. In this case we write SÎA -limx = L. This idea was studied by Freedman and Sember [7]

Statistical approximation by positive linear operators on modular spaces

“…A-statistical analogs of these concepts have been examined by Connor and Kline [5], and Demirci [6] as follows. The A-statistical limit superior of a number sequence x = {x n }, denoted by…”

Inclusion results on statistical cluster points

“…If for every ε > 0, δ A (K ε ) = 0, γ is called the A-statistical cluster point of x, where K ε = {k ∈ N : |x k − γ| < ε}. We denote the set of all A-statistical cluster points of x by Γ A (x), ( [4]). …”

“…Then in [10] they studied the statistical core for complex number sequences. In [4] Demirci extended these concepts by taking a nonnegative regular matrix A instead of Cesàro matrix. On the other hand Goffman and Petersen [11] introduced the submethod by deleting some rows from a matrix method.…”

Banach and Statistical Cores of Bounded Sequences

“…In [10] the sequence x is defined to be statistically bounded if x has a bounded subsequence of density one; and the statistical core of such an x of real values is the closed interval [st-lim inf x, st-lim sup x], where st-lim inf x and st-lim sup x are the least and greatest statistical cluster points of x (see [6], [10], [11], [16]). Recall [10] that, for a sequence x the number β is the st-lim sup x if and only if for every ε > 0, δ{k : x k > β − ε} = 0 and δ{k : x k > β + ε} = 0.…”