Abel summability in topological spaces

Ünver, Mehmet

doi:10.1007/s00605-014-0717-0

Cited by 17 publications

(8 citation statements)

References 16 publications

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“…Therefore, many authors have restricted the scope by assuming either the topological space to have a group structure or a linear structure. Recently, some authors have studied some summability methods that directly can be defined in arbitrary Hausdorff spaces such as A-statistical convergence and A-distributional convergence (see e.g., [2,15,[17][18][19]).…”

Section: Preliminariesmentioning

confidence: 99%

Lacunary distributional convergence in topological spaces

Uluçay

Ünver

2019

Filomat

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Most of the summability methods cannot be defined in an arbitrary Hausdorff topological space unless one introduces a linear or a group structure. In the present paper, using distribution functions over the Borel σ-field of the topology and lacunary sequences we define a new type of convergence method in an arbitrary Hausdorff topological space and we study some inclusion theorems with respect to the resulting summability method. We also investigate the inclusion relation between lacunary sequence and lacunary refinement of it.

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Section: Preliminariesmentioning

confidence: 99%

Lacunary distributional convergence in topological spaces

Uluçay

Ünver

2019

Filomat

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“…A and ∆A will denote the set of Abel convergent sequences and the set of Abel quasi Cauchy sequences, respectively. Recently the concept of Abel statistical convergence of a sequence is investigated in [34] (see also [20]) in the sense that a sequence (α k ) is called Abel statistically convergent to a real number L if…”

Section: Abel Statistical Ward Continuitymentioning

confidence: 99%

Abel statistical quasi Cauchy sequences

Cakalli

2019

Filomat

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In this paper, we investigate the concept of Abel statistical quasi Cauchy sequences. A real function f is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchy sequences, where a sequence (α k) of point in R is called Abel statistically quasi Cauchy if lim x→1 − (1 − x) k:|∆α k | ε x k = 0 for every ε > 0, where ∆α k = α k+1 − α k for every k ∈ N. Some other types of continuities are also studied and interesting results are obtained. It turns out that the set of Abel statistical ward continuous functions is a closed subset of the space of continuous functions.

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“…Recently the concept of Abel statistical convergence of a sequence is introduced and investigated in [58] (see also [55]). Although the definitions and the most of the results are also valid in a topological Hausdorff group, which allows countable local base at the origin, we investigate the notion in the real setting: a sequence p = (p k ) is called Abel statistically convergent to a real number L if Abel density of the set {k ∈ N 0 :…”

Section: Abel Statistical Continuitymentioning

confidence: 99%

On Abel statistical convergence

Taylan¹,

Cakalli²

2017

Preprint

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In this paper, we introduce and investigate a concept of Abel statistical continuity. A real valued function f is Abel statistically continuous on a subset E of R, the set of real numbers, if it preserves Abel statistical convergent sequences, i.e. (f (p k )) is Abel statistically convergent whenever (p k ) is an Abel statistical convergent sequence of points in E, where a sequence (p k ) of point in R is called Abel statistically convergent to a real number L if Abel density of the set {k ∈ N 0 : |p k − L| ≥ ε} is 0 for every ε > 0. Some other types of continuities are also studied and interesting results are obtained.

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Abel summability in topological spaces

Cited by 17 publications

References 16 publications

Lacunary distributional convergence in topological spaces

Lacunary distributional convergence in topological spaces

Abel statistical quasi Cauchy sequences

On Abel statistical convergence

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