Concept Of Statistical Convergence Research Articles

Statistically Convergent Sequences in Intuitionistic Fuzzy Metric Spaces

In this paper, we introduce the concepts of statistical convergence and statistical Cauchy sequences with respect to the intuitionistic fuzzy metric spaces inspired by the idea of statistical convergence in fuzzy metric spaces. Then, we give useful characterizations for statistically convergent sequences and statistically Cauchy sequences.

Generalized Cesàro summability of Fourier series and its applications

In this paper, by using generalized Cesàro means based on q-integers we study on approximating continuous and periodic functions by its Fourier series. We also discuss its connection with the concept of statistical convergence. At the end of the paper, some applications and graphical illustrations are also provided.

\mathcal{I}- almost Lacunary vector valued sequence spaces in 2- normed spaces

One of the wide-ranging applications and research areas of Summability theory is the concept of statistical convergence. This concept was studied a related concept of convergence by using lacunary sequence by Fridy and Orhan. At the last quarter of the 20th century, lacunary statistical convergence has been discussed and captured significant aspect of creating the basis of several investigations conducted in many branches of mathematics. On the other hand, in 1961 Krasnoselskii and Rutisky presented the definition of Orlicz function. Also, in 1963 G\"{a}hler introduced the notion of 2-normed spaces. The main goal of this article is to introduce $\mathcal{I}-$ almost convergence of lacunary sequences with regard to an Orlicz function in 2-normed spaces and other sequence spaces by considering the concept of ideal that was presented by Kostyrko and others. Additionally, we examine the relationship between these sequence spaces and fundamental inclusion theorems are investigated.

Statistical $$(C,1)(E,\mu )$$-summability and associated fuzzy approximation theorems with statistical fuzzy rates

The concept of statistical convergence has attracted the pervasive attention of the current researchers due basically to the fact that it is stronger than the ordinary convergence. Korovkin-type approximation theorem plays a vital role in the convergence of sequences of positive linear operators. Moreover, this type of approximation theorems has been extended through different statistical summability methods over general sequence spaces. The paper investigated statistical $$(C,1)(E,\mu )$$ product summability mean for sequences of fuzzy numbers and proved a fuzzy Korovkin-type approximation theorem. Furthermore, we have established another result for the fuzzy rate of convergence which is uniform in fuzzy Korovkin-type approximation theorem under our proposed summability mean.

On variations via statistical convergence

The paper introduces some notions of variation using the concept of statistical convergence. In particular, the ideas of variation such as regular, translationally regular, O -regular and rapid are presented in statistical context. First, some results are established connecting some statistical variations with their non-statistical counterparts. Then, certain conditions are explored under which various statistical variations can occur. Finally, the paper examined some conditions to connect two different statistical variations.

Statistical convergence of order $$\left( \beta ,\gamma \right) $$ β , γ for sequences of fuzzy numbers

The concepts of statistical convergence and strong p-Cesaro summability of sequences of real numbers were introduced in literature independently, and it was shown that if a sequence is strongly p-Cesaro summable, then it is statistically convergent and also a bounded statistically convergent sequence must be p-Cesaro summable. In the present paper, two new concepts named statistical convergence of order $$\left( \beta ,\gamma \right) $$ and strongly p-Cesaro summability of order $$\left( \beta ,\gamma \right) $$ are introduced for sequences of fuzzy numbers, where $$\alpha $$ and $$\beta $$ real numbers such that $$0<\alpha \le \beta \le 1$$ and some relations between statistical convergence of order $$\left( \beta ,\gamma \right) $$ and strongly p-Cesaro summability of order $$\left( \beta ,\gamma \right) $$ are given. Furthermore, it is shown that a bounded and statistically convergent sequence of fuzzy numbers need not strongly p-Cesaro summable of order $$\left( \beta ,\gamma \right) $$ in general for $$0<\beta \le \gamma \le 1$$ .

Statistical equi-equal convergence of positive linear operators

Many researchers have been interested in the concept of statistical convergence because of the fact that it is stronger than the classical convergence. Also, the concepts of statistical equal convergence and equi-statistical convergence are more general than the statistical uniform convergence. In this paper we define a new type of statistical convergence by using the notions of equi-statistical convergence and statistical equal convergence to prove a Korovkin type theorem. We show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems which were demonstrated by earlier authors. After, we present an example in support of our definition and result presented in this paper. Finally, we also compute the rates of statistical equi-equal convergence of sequences of positive linear operators.

Statistical Convergence of Sequences of Sets in Hyperspaces

The concept of statistical convergence in an arbitrary topological space is nothing new, it is actually a self-evident concept that comes through the structure of that space. In this paper, by considering the well known topologies on hyperspaces, we investigate the characterizations of statistical convergence of sequences of sets in the realm of these structures.

A-statistical relative modular convergence of positive linear operators

In this paper, we investigate the problem of statistical approximation to a function f by means of positive linear operators defined on a modular space. Particularly, in order to get stronger results than the classical aspects we mainly use the concept of statistical convergence. Also, a non-trivial application is presented.

STATISTICAL CONVERGENCE IN FUZZY $n$-NORMED SPACES

In this paper we introduce the concept of statistical convergence of a sequence in fuzzy n-normed space and defined limit point, statistical limit point and statistical cluster point of a sequence in fuzzy n-normed space. Also we establish the relationship between limit point, statistical limit point and statistical cluster point of a sequence in fuzzy n-normed space.

Statistical uniform convergence in 2- normed spaces

The concept of statistical convergence in 2-normed spaces for double sequence was introduced in [17].In the rst , we introduce concept strongly statistical convergence in 2- normed spaces and generalize some results. moreover, we dene the concept of statistical uniform convergence in 2

$\mathcal{I}$ and $\mathcal{I}^{\ast}$ convergence of multiple sequences of fuzzy numbers

Recently, the concept of statistical convergence for multiple sequences of fuzzy numbers has been studied by Kumar et al. This motivate us to extend the idea of $\mathcal{I}$-convergence to sequences of fuzzy numbers of multiplicity greater than two.

On Some New Sequence Spaces and Statistical Convergence Methods for Double Sequences

Abstract In this paper, we introduce some new double sequence spaces with respect to an Orlicz function and define two new convergence methods related to the concepts of statistical convergence and lacunary statistical convergence for double sequences. We also present some inclusion theorems for our newly defined sequence spaces and statistical convergence methods

Statistical approximation by double sequences of positive linear operators on modular spaces

In this paper, using the concept of statistical convergence, which is stronger than the Pringsheim convergence, we study the problem of approximation to a function by means of double sequences of positive linear operators defined on a modular space. Also, a non-trivial application is presented.

On some spaces of almost lacunary convergent sequences derived by Riesz mean and weighted almost lacunary statistical convergence in a real n-normed space

In this paper, we introduce some new spaces of almost convergent sequences derived by Riesz mean and the lacunary sequence in a real n-normed space. By combining the definitions of lacunary sequence and Riesz mean, we obtain a new concept of statistical convergence which will be called weighted almost lacunary statistical convergence in a real n-normed space. We examine some connections between this notion with the concept of almost lacunary statistical convergence and weighted almost statistical convergence, where the base space is a real n-normed space.MSC:40C05, 40A35, 46A45, 40A05, 40F05.

Korovkin type approximation theorems proved via [formula omitted]-statistical convergence

The concept of statistical convergence was introduced by H. Fast, and studied by various authors. Recently, by using the idea of statistical convergence, M. Balcerzak, K. Dems and A. Komisarski introduced a new type of convergence for sequences of functions called equistatistical convergence. In the present paper we introduce the concepts of αβ-statistical convergence and αβ-statistical convergence of order γ. We show that αβ-statistical convergence is a non-trivial extension of ordinary and statistical convergences. Moreover we show that αβ-statistical convergence includes statistical convergence, λ-statistical convergence, and lacunary statistical convergence. We also introduce the concept of αβ-equistatistical convergence which is a non-trivial extension of equistatistical convergence. Moreover, we prove that αβ-equistatistical convergence lies between αβ-statistical pointwise convergence and αβ-statistical uniform convergence. Finally we prove Korovkin type approximation theorems via αβ-statistical uniform convergence of order γ and αβ-equistatistical convergence of order γ.

STATISTICAL AND LACUNARY CONVERGENCE-ITS LATEST DEVELOPMENT FROM THE POINT OF VIEW OF SEQUENCE SPACES

The development of the theory of sequence spaces is nowadays effected by the introduction of new convergence methods and theories in the process. Some of them are statistical convergence, lacunary convergence, lacunary statistical convergence etc (see [1], [2], [3], [8], [11]). Here, I have basically concentrated on these concepts. Statistical convergence while introduced over nearly fifty years ago has only recently become an area of active research in sequence spaces. Fast [5] extended the concept of sequential limit which he called statistical convergence. Schoenberg (1959) gave some basic properties of statistical convergence and studied the concept as summability method. The basic concept of statistical convergence is based on the notion of natural density of sets A ⊆ N = {1, 2, . . . , n, . . .}, refer to [5], [6]. If A ⊆ N and A(n)=|A ∩ {1, 2, . . . , n}|, where the vertical bar denotes the cardinality of the

On lacunary double statistical convergence in locally solid Riesz spaces

The concept of statistical convergence is one of the most active areas of research in the field of summability. Most of the new summability methods have relation with this popular method. In this paper, we introduce the concept of double I θ -statistical-τ-convergence which is a more general idea of statistical convergence. We also investigate the ideas of double I θ -statistical-τ-boundedness and double I θ -statistical-τ-Cauchy condition of sequences in the framework of locally solid Riesz space endowed with a topology τ and investigate some of their consequences.MSC:40G15, 40A35, 46A40.

On Generalized Statistical Convergence of Order of Difference Sequences

We introduce the concept of statistical convergence of order of difference sequences, and we give some relations between the set of statistical convergence of order of difference sequences and strong -summability of order . Furthermore some relations between the spaces and are examined.

On generalized double statistical convergence in a random 2-normed space

Recently, the concept of statistical convergence has been studied in 2-normed and random 2-normed spaces by various authors. In this paper, we shall introduce the concept of λ-double statistical convergence and λ-double statistical Cauchy in a random 2-normed space. We also shall prove some new results. MSC: 40A05; 40B50; 46A19; 46A45