A Matrix Characterization of Statistical Convergence

Abstract

The sequence χ is statistically convergent to L if for each e > 0, lim η -1 {the number of k < τι :It is known that this method of summability cannot be included by any matrix method, but for bounded sequences it is included by the Cesáro matrix method C\. In this paper these results eure extended by comparing statistical convergence with the intersection of a collection Τ of summability matrices, each of which is somewhat like C\. It is shown that a bounded sequence is statistically convergent if and only if it is summable by every matrix in T. On the other hand, no countable collection of matrices can include statistical convergence for unbounded sequences. Also, the class Τ is studied to determine which classical summability matrices belong to T.

Statistical Limit Points

“…In [2,3,5,6,9] this concept was studied as a nonmatrix summability method. In the present paper we return to the view of statistical convergence as a sequential limit concept, and we extend this concept in a natural way to define a statistical analogue of the set of limit points or cluster points of a number sequence.…”

I-convergence theorems for a class of k-positive linear operators

“…Let A be the Taylor coefficients defined in (7) and assume that (4) is satisfied. Then for all ∈ D ω K we have…”

“…A sequence = ( ) is said to be A−statistically convergent to L if δ A { ∈ N : | − L| ≥ ε} = 0 and it is denoted by st A − lim = L (see [5,7,13]). In the case A = C 1 the Cesaro matrix of order one, C 1 -statistical convergence is equivalent to the statistical convergence ( [6,8]).…”

I: Fréchet-Urysohn spaces

“…But in general, s-convergent sequences satisfy many of the properties of ordinary convergent sequences in metric spaces. It has been discussed and developed by many authors [3,5,6,9,10,11,21,22,25,26].…”