scholarly journals Density by Moduli and Lacunary Statistical Convergence

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Vinod K. Bhardwaj ◽  
Shweta Dhawan
Keyword(s):  
Strong Convergence ◽  
Tauberian Theorem ◽  
Statistical Convergence ◽  
General Description ◽  
Lacunary Statistical Convergence ◽  
Bounded Sequences ◽  
Convergent Sequences

We have introduced and studied a new concept off-lacunary statistical convergence, wherefis an unbounded modulus. It is shown that, under certain conditions on a modulusf, the concepts of lacunary strong convergence with respect to a modulusfandf-lacunary statistical convergence are equivalent on bounded sequences. We further characterize thoseθfor whichSθf=Sf, whereSθfandSfdenote the sets of allf-lacunary statistically convergent sequences andf-statistically convergent sequences, respectively. A general description of inclusion between two arbitrary lacunary methods off-statistical convergence is given. Finally, we give anSθf-analog of the Cauchy criterion for convergence and a Tauberian theorem forSθf-convergence is also proved.

scholarly journals On statistical convergence and strong Cesàro convergence by moduli

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Fernando León-Saavedra ◽  
M. del Carmen Listán-García ◽  
Francisco Javier Pérez Fernández ◽  
María Pilar Romero de la Rosa
Keyword(s):  
Statistical Convergence ◽  
Convergent Sequence ◽  
Modulus Function ◽  
Cesaro Convergence ◽  
Bounded Sequences ◽  
Convergent Sequences

AbstractIn this paper we will establish a result by Connor, Khan and Orhan (Analysis 8:47–63, 1988; Publ. Math. (Debr.) 76:77–88, 2010) in the framework of the statistical convergence and the strong Cesàro convergence defined by a modulus function f. Namely, for every modulus function f, we will prove that a f-strongly Cesàro convergent sequence is always f-statistically convergent and uniformly integrable. The converse of this result is not true even for bounded sequences. We will characterize analytically the modulus functions f for which the converse is true. We will prove that these modulus functions are those for which the statistically convergent sequences are f-statistically convergent, that is, we show that Connor–Khan–Orhan’s result is sharp in this sense.

scholarly journals Lacunary statistical convergence and inclusion properties between lacunary methods

2000 ◽  
Vol 23 (3) ◽  
pp. 175-180 ◽  
Author(s):  
Jinlu Li
Keyword(s):  
Statistical Convergence ◽  
Lacunary Sequence ◽  
General Description ◽  
Lacunary Sequences ◽  
Inclusion Properties ◽  
Integer Sequence ◽  
Lacunary Statistical Convergence

A lacunary sequence is an increasing integer sequenceθ={kr}such thatkr−kr−1→∞asr→∞. A sequencexis calledsθ-convergent toLprovided that for eachϵ>0,limr(1/(kr−kr−1)){the number of   kr−1<k≤kr:|xk−L|≥ϵ}=0. In this paper, we study the general description of inclusion between two arbitrary lacunary sequences convergent.

scholarly journals On Some Further Generalizations of Strong Convergence in Probabilistic Metric Spaces Using Ideals

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Pratulananda Das ◽  
Kaustubh Dutta ◽  
Vatan Karakaya ◽  
Sanjoy Ghosal
Keyword(s):  
Strong Convergence ◽  
Metric Space ◽  
Statistical Convergence ◽  
Metric Spaces ◽  
Strong Topology ◽  
Probabilistic Metric Space ◽  
Probabilistic Metric Spaces ◽  
New Approach ◽  
Lacunary Statistical Convergence ◽  
Probabilistic Metric

Following the line of (Das et al., 2011, Savas and Das, 2011), we make a new approach in this paper to extend the notion of strong convergence and more general strong statistical convergence (Şençimen and Pehlivan, 2008) using ideals and introduce the notion of strongℐ- andℐ*-statistical convergence and two related concepts, namely, strongℐ-lacunary statistical convergence and strongℐ-λ-statistical convergence in a probabilistic metric space endowed with strong topology. We mainly investigate their interrelationship and study some of their important properties.

Lacunary statistical convergence of order (α, β) in topological groups

2017 ◽  
Vol 26 (3) ◽  
pp. 339-344
Author(s):  
HACER SENGUL ◽  
◽  
MIKAIL ET ◽  
Keyword(s):  
Statistical Convergence ◽  
Topological Groups ◽  
Lacunary Statistical Convergence ◽  
Inclusion Relations ◽  
Convergent Sequences

In this paper, the concept of lacunary statistical convergence of order (α, β) is generalized to topological groups, and some inclusion relations between the set of all statistically convergent sequences of order (α, β) and the set of all lacunary statistically convergent sequences of order (α, β) are given.

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