Δm – Weighted Statistical Convergence

In this paper we will define the new weighted statistically summbaility method, known as the weighted Norlund-Euler statistical convergence. We will show some properties of this method and we have proved Korovkin type theorem.

International Journal of Research -GRANTHAALAYAH, 2017

In this paper, we have established some new theorems on double weighted mean statistical convergence of double sequences, which gives some new results and generalizes the some previous known results of Karakaya.

Proyecciones (Antofagasta), 2021

In this paper we investigate the notion of I-statistical ϕ-convergence and introduce IS-ϕ limit points and IS-ϕ cluster points of real number sequence and also studied some of its basic properties.

Mathematical and Computer Modelling, 2009

A real-valued finitely additive measure µ on N is said to be a measure of statistical type provided µ(k) = 0 for all singletons {k}. Applying the classical representation theorem of finitely additive measures with totally bounded variation, we first present a short proof of the representation theorem of statistical measures. As its application, we show that every kind of statistical convergence is just a type of measure convergence with respect to a specific class of statistical measures.

Applied Mathematics and Computation, 2012

The concept of weighted statistical convergence was introduced and studied by Karakaya and Chishti (2009) [7]. In this paper, we modify the definition of weighted statistical convergence and find its relationship with the concept of statistical summability ðN; p n Þ due to Moricz and Orhan (2004) [10]. We apply this new summability method to prove a Korovkin type approximation theorem by using the test functions 1; e Àx ; e À2x. We apply the classical Baskakov operator to construct an example in support of our result.

2019

The Korovkin theory has effective role in approximation theory. This theory is connected with the approximation to continuous functions by means of positive linear operators. Many mathematicians have investigated the Korovkin-type theorems by for a sequence of positive linear operators defined on different spaces by using various types of convergence. Firstly, A.D. Gadjiev has proved the weighted Korovkin type theorems, (Math. Zamet., 20 (1976) 781-786 (in Russian)). Later, these theorems are studied by many authors by means of different convergence methods. Recently, The definition of equal convergence for real functions was introduced by Császár and Laczkovich and they improved their investigations on this * convergence. Later Das et. al. introduced the ideas of I and I-equal convergence with the help of ideals by extending the equal convergence (Mat. Vesnik, vol:66, 2 (2014),165-177). In our work, we introduce a new type of statistical convergence on weighted spaces by using the notions of the equal convergence. We study its use in the Korovkin-type approximation theory. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work.

2013

The object of this present paper is to dene and study generalised statistical convergence for the sequences in any locally convex Hausdorff space X whose topology is determined by a set Q of continuous seminorms q and their relation with the nearly convergent sequence space using a bounded modulus function along with regular and almost positive method.

Science in China Series A: Mathematics, 2008

The purpose of this paper is to unify various kinds of statistical convergence by statistical measure convergence and to present Jordan decomposition of finitely additive measures. It is done through dealing with the most generalized statistical convergence-ideal convergence by applying geometric functional analysis and Banach space theory. We first show that for each type of ideal I(⊂ 2 N ) convergence, there exists a set S of statistical measures such that the measure S-convergence is equivalent to the statistical convergence. To search for Jordan decomposition of measures of statistical type, we show that the subspace X I ≡ span{χ A : A ∈ I} is an ideal of the space ℓ ∞ in the sense of Banach lattice, hence the quotient space ℓ ∞ /X I is isometric to a C (K ) space. We then prove that a statistical measure has a Jordan decomposition if and only if its corresponding functional is norm-attaining on ℓ ∞ , and which in turn induces an approximate null-ideal preserved Jordan decomposition theorem of finitely additive measures. Finally, we show this characterization and the approximate decomposition theorem are true for finitely additive measures defined on a general measurable space.  n j=1 χ S (j) = 0 is said to be a statistically null set, or simply, a null set if there is no confusion arise, where χ A denotes the characteristic function of a set A. On one hand, properties of statistical convergence has been studied in many pure and applied mathematical fields (see, for example, ). On the other hand, the notion of statistical convergence has been generalized in different ways. The original notion was introduced for X = R, and there are dozens of its generalizations. Generally speaking, this notion was extended in two directions: One is to discuss statistical convergence in more general spaces, for example, locally convex spaces , including Banach spaces with the weak topologies , and general topological spaces . The other is to consider generalized notions defined by various limit processes, for example, A-statistical convergence [6], lacunary statistical convergence . The most general notion of statistical convergence is ideal (or filter) convergence .

Journal of Mathematical Analysis and Applications, 1996

Ž . This article extends the concept of a statistical limit cluster point of a sequence Ž .

Kragujevac journal of mathematics, 2018

A sequence of real numbers {x n } n∈N is said to be αβ-statistically convergent of order γ (where 0 < γ ≤ 1) to a real number x [1] if for every δ > 0, lim n→∞ 1 (β n − α n + 1) γ |{k ∈ [α n , β n ] : |x k − x| ≥ δ}| = 0, where {α n } n∈N and {β n } n∈N are two sequences of positive real numbers such that {α n } n∈N and {β n } n∈N are both non-decreasing, β n ≥ α n for all n ∈ N, (β n − α n) → ∞ as n → ∞. In this paper we study a related concept of convergences in which the value x k is replaced by P (|X k − X| ≥ ε) and E(|X k − X| r) respectively (where X, X k are random variables for each k ∈ N, ε > 0, P denotes the probability, and E denotes the expectation) and we call them αβ-statistical convergence of order γ in probability and αβ-statistical convergence of order γ in r th expectation respectively. The results are applied to build the probability distribution for αβ-strong p-Cesàro summability of order γ in probability and αβ-statistical convergence of order γ in distribution. So our main objective is to interpret a relational behaviour of above mentioned four convergences. We give a condition under which a sequence of random variables will converge to a unique limit under two different (α, β) sequences and this is also use to prove that if this condition violates then the limit value of αβstatistical convergence of order γ in probability of a sequence of random variables for two different (α, β) sequences may not be equal. Key words and phrases. αβ-statistical convergence, αβ-statistical convergence of order γ in probability, αβ-strong p-Cesàro summability of order γ in probability, αβ-statistical convergence of order γ in r th expectation, αβ-statistical convergence of order γ in distribution.