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Statistical convergence was extended to weighted statistical convergence in [24], by using a sequence of real numbers sk, satisfying some conditions. Later, weighted statistical convergence was considered in [35] and [19] with modified conditions on sk. Weighted statistical convergence is an extension of statistical convergence in the sense that, for sk = 1, for all k, it reduces to statistical convergence. A definition of weighted ??-statistical convergence of order ?, considered in [25] does not have this property. To remove this extension problem the definition given in [25] needs some modifications. In this paper, we introduced the modified version of weighted ??-statistical convergence of order ?, which is an extension of ??-statistical convergence of order ?. Our definition, with sk = 1, for all k, reduces to ??-statistical convergence of order ?. Moreover, we use this definition of weighted ??-statistical convergence of order ?, to prove Korovkin type approximation theorems via, weighted ??-equistatistical convergence of order ? and weighted ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Also we prove Korovkin type approximation theorems via ??-equistatistical convergence of order ? and ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Some examples of positive linear operators are constructed to show that, our approximation results works, but its classical and statistical cases do not work. Finally, rates of weighted ??-equistatistical convergence of order ? is introduced and discussed.

Section: Weighted αβ-Statistical Convergence Of Order γmentioning

confidence: 99%

Korovkin type approximation theorems proved viaweighted αβ-equistatistical convergence for bivariate functions

Aktuğlu¹,

Gezer²

2018

“…In [17] the relation between lacunary statistical convergence and statistical convergence was established among the other thinks. In the year 2017 weighted lacunary statistical convergence is a generalization of lacunary statistical convergence was introduce by Ghosal et al [21]. Definition 1.3.…”

Section: Introductionmentioning

confidence: 99%

“…Definition 1.3. [21]. Let θ = {k r } r∈N∪{0} be a lacunary sequence and {t n } n∈N be a sequence of real numbers such that t n > α, ∀ n ∈ N (where α is a positive real number) and T n = n k=1 t k (where n ∈ N and T 0 = 0).…”

Section: Introductionmentioning

confidence: 99%

“…A sequence x = {x n } n∈N in a normed linear space ||.||, is said to be rough I-convergent to x * w.r.t. the roughness of degreer (or shortly:r-I-convergent to x * ) if for every ε > 0, the set On further progress, we combine the approaches of I-lacunary statistical convergence [10], rough Iconvergence [13,26], statistical cluster point [3,27] and weighted lacunary statistical convergence [21] and introduce new and more advance summability methods namely, rough weighted I-lacunary statistical limit set and weighted I-lacunary statistical cluster points set of a sequence in a metric space. Some new examples are constructed to ensure the deviation from basic results such as: Theorem 1.7 [13,26].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Effects on rough I-lacunary statistical convergence to induce the weighted sequence

Ghosal¹,

Banerjee²

2018

Self Cite

Two classes of sets are introduced: rough weighted I-lacunary statistical limit set and weighted I-lacunary statistical cluster points set which are natural generalizations of rough I-limit set and I-cluster points set respectively. To highlight the variation from basic results we place into some new examples. So our aim is to analyze the different behaviors of the new convergences and characterize both the sets with topological approach like closedness, boundedness, compactness etc.

“…The notion of a modulus function was introduced by Nakano [17]. The concept of weighted statistical convergence was initially introduced by Karakaya and Chishti [13] in the year 2009 and gradually developed by Mursaleen et al [16], Ghosal [10] and Das et al [6] by taking the weighted sequence {t n } n∈N of real numbers such that lim inf n→∞ t n > 0 (this was also done to some extent in [9,11]). However an important question remains unanswered: If the weighted sequence is properly divergent to +∞ then lim inf n→∞ t n does not exist.…”

Section: Introductionmentioning

confidence: 99%

Different behaviors of rough weighted statistical limit set under unbounded moduli

Ghosal

Som

2018

Self Cite

In this paper we introduce f-rough weighted statistical limit set and f-weighted statistical cluster points set which are natural generalizations of rough statistical limit set and f-statistical cluster points set of sequence respectively. Some new examples are constructed to ensure the deviation of basic results. So both the sets don?t follow the nature of usual extension properties which will be discussed here.