Statistical Convergence of Infinite Series

Dindoš, Martin; Šalát, Tibor; Toma, Vladimír

doi:10.1023/b:cmaj.0000024535.89828.e8

Cited by 11 publications

(13 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Dindoš,Šalát and Toma [10] have studied the statistical convergence of infinite series, i.e., an infinite series ∞ n=1 a n (with real or complex terms) is called statistically convergent to an s (s ∈ R or s ∈ C) if st − lim s n = s with s n = n k=1 a k (n = 1, 2, 3, ...). By considering the unique infinite binary expansion of x ∈ (0, 1] , they have associated with the number x the infinite series (x) := ∞ k=1 e k (x)a k which after omitting the zero terms can be identified with a (infinite) subseries of the series ∞ n=1 a n .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Subsequential Results on Uniform Statistical Convergence

Yurdakadim¹,

Wieren²

2016

Sarajevo J. Math.

View full text Add to dashboard Cite

Abstract. In this paper, we present some relationships between convergence and uniform statistical convergence of a given sequence and its subsequences. The results concerning uniform statistical convergence presented here are also closely related to earlier results regarding statistical convergence and almost convergence of sequences, and are dealing with measure and in a minor case with category. Finally, we present a Cauchy type characterization of uniform statistical convergence and a result concerning uniform statistical convergence of subseries of a series.

show abstract

Section: Introductionmentioning

confidence: 99%

“…Since C( ∞ n=1 a n ) ⊂ C unif stat ( ∞ n=1 a n ) ⊂ C stat ( ∞ n=1 a n ), and considering Theorem 2.2 of [10] is useful to note the following Remark 2. Let ∞ n=1 a n be a convergent series.…”

Section: Introductionmentioning

confidence: 99%

Subsequential Results on Uniform Statistical Convergence

Yurdakadim¹,

Wieren²

2016

Sarajevo J. Math.

View full text Add to dashboard Cite

show abstract

“…It has appeared in a wide variety of topics. For example, statistical convergence has been discussed in summability of matrix, series and integral in [3,4,5,13,32,36,61,94,110,123], Fourior analysis in [70,71,84], approximation of positive operators in [39-43, 45, 46, 49, 50, 54, 86], number theory in [9], trigonometric series in [38,96,122,124], Banach space theory in [6,55,119], locally convex spaces in [24,91,92,125,126], structure of ideals of bounded continuous functions in [9], fuzzy mathematics in [80,83,101], property of continuous functions in [8,56] and making various new types of topological linear spaces in [58,61,102,113,115].…”

Section: Introductionmentioning

confidence: 99%

Measure theory of statistical convergence

Cheng

Lin

Lan

et al. 2008

Sci. China Ser. A-Math.

View full text Add to dashboard Cite

The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the studies of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper, in terms of subdifferential, first shows a representation theorem for all finitely additive probability measures defined on the σ-algebra A of all subsets of N , and proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measure μ with μ(k) = 0 for all singletons {k}). This paper also shows that classical statistical measures have many nice properties, such as: The set S of all such measures endowed with the topology of point-wise convergence on A forms a compact convex Hausdorff space; every classical statistical measure is of continuity type (hence, atomless), and every specific class of statistical measures fits a complementation minimax rule for every subset in N . Finally, this paper shows that every kind of statistical convergence can be unified in convergence of statistical measures.

show abstract

“…The concept of the statistical convergence was introduced in [5], [12] and has been developed in several directions in [2], [3], [4], [7], [8], [10], [13] and by many other authors. We will deal mainly with the generalization of the statistical convergence introduced in the [8].…”

Section: Introductionmentioning

confidence: 99%