Article

On the Generalized Weighted Statistical Convergence

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

Statistical convergence and summability represent a significant generalization of traditional convergence for sequences of real or complex values, allowing for a broader interpretation of convergence phenomena. This concept has been extensively examined by numerous researchers using various mathematical tools and applied to different mathematical structures over time, revealing its relevance across multiple disciplines. In the present study, a generalized definition of the concepts of statistical convergence and summability, termed (△_v^m )_u-generalized weighted statistical convergence and (△_v^m )_u-generalized weighted by [¯N_t ]-summability for real sequences, is introduced using the weighted density and generalized difference operator. Based on this definition, several fundamental properties and inclusion results, obtained by differentiating the components used in the definitions, are provided.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
In this study, we introduce the concepts of φλ,μ-double statistically convergence of order β in fuzzy sequences and strongly λ- double Cesaro summable of order β for sequences of fuzzy numbers. Also we give some inclusion theorems.
Article
Full-text available
In this work, we first define a difference operator of natural order m with respect to -integers. We then introduce the concepts of -statistical convergence, statistical -summability and strong -summability of order γ by the weighted method. Furthermore, based on the definition of statistical -summability, we prove a Korovkin type approximation theorem for functions of two variables. By using -analogue of Bernstein operator of two variables we give an example which shows that proposed method successfully works. Finally, some Voronovskaja type approximation results are obtained.
Article
Full-text available
The definition of weighted statistical convergence was first introduced by Karakaya and Chishti (2009) [1]. After that the definition was modified by Mursaleen et al. (2012) [2]. But some problems are still there; so it will be further modified in this paper. Using it some newly developed definitions of the convergence of a sequence of random variables in probability have been introduced and their interrelations also have been investigated, and in this way a partial answer to an open problem posed by Das and Savas (2014) [3] has been given.
Article
Full-text available
We define the sequence spaces ℓ ∞ (Δ r m ), c(Δ r m ) and c 0 (Δ r m ) (m∈ℕ and r∈ℝ), where for instance ℓ ∞ (Δ r m )={x=(x k ):(k r Δ m x k )∈ℓ ∞ }, give some topological properties, inclusion relations of these spaces and compute their continuous and α, β and γ-duals. Furthermore, we characterize some matrix classes related to these sequence spaces. This study generalizes some results of M. Et and R. Çolak [Soochow J. Math. 21, 377–386 (1995; Zbl 0841.46006) and Hokkaido Math. J. 26, 483–492 (1997; Zbl 0888.40002)] and M. A. Sarıgöl [J. Karadeniz Techn. Univ., Fac. Arts Sci., Ser. Math.-Phys. 10, 63–71 (1987; Zbl 0679.46005)] in special cases.
Article
Full-text available
In this paper, we define the sequence spaces ) , c f " and 0 c be the linear spaces of bounded, convergent, and null sequences ) ( k x x with complex terms, respectively, normed by k k x x sup f
Conference Paper
In this study, we introduce and examine the concepts of Δm−weighted statistical convergence and Δm−weighted (N¯,pn)−summability. Also some relations between Δm−weighted statistical convergence and Δm−weighted (N¯,pn)−summability are given.
Article
In this paper, we propose to investigate a new weighted statistical convergence by applying the Nörlund–Cesáro summability method. Based upon this definition, we prove some properties of statistically convergent sequences and a kind of the Korovkin type theorems. We also study the rate of the convergence for this kind of weighted statistical convergence and a Voronovskaya type theorem.
Article
In this paper, the notion of (N̄, pn)- summability to generalize the concept of statistical convergence is used. We call this new method weighted statistically convergence. We also establish its relationship with statistical convergence, (C,1)-summability and strong (N̄, pn)-summability.
Article
In this paper define the spaces l ∞ (Δ), c(Δ), and c 0 (Δ), where for instance l ∞ (Δ) = {x=(x k ):sup k |x k -x k + l |< ∞} , and compute their duals (continuous dual, α-dual, β-dual and γ-dual). We also determine necessary and sufficient conditions for a matrix A to map l ∞ (Δ) or c(Δ) into l ∞ or c , and investigate related questions.
Article
The sequence X is statistically convergent to L if for each ∊ > 0, limn n-1 [the number of k ≤ n : | xk - L | ≥ ∊] = 0; x is a statistically Cauchy sequence if for each ∊ > 0 there is a positive integer N = N(e) such that [formula omitted][the number of k ≤ n : | xn - xk | ≥ ∊] = 0. n These concepts are shown to be equivalent. Also, statistical convergence is studied as a regular summability method, and it is shown that it cannot be included by any matrix method. There are two Tauberian theorems proved: one uses the Tauberian condition Δxk = 0(1/k), which is shown to be “best possible,” and the other is concerned with gap sequences.
Article
The notion of the statistical convergence of sequences of real numbers was introduced in papers [1] and [5]. In the present paper we shall show that the set of all bounded statistically convergent sequences of real numbers is a nowhere dense subset of the linear normed space m (with the sup-norm) of all bounded sequences of real numbers and the set of all statistically convergent sequences of real numbers is a dense subset of the first Baire category in the Frechet space s.
Article
The object of this paper is to introduce the concepts of weighted lambda-statistical convergence and statistical summability ((N) over bar (lambda), p). We also establish some inclusion relations and some related results for these new summability methods. Further, we determine a Korovkin type approximation theorem through statistical summability ((N) over bar (lambda), p) and we show that our approximation theorem is stronger than classical Korovkin theorem by using classical Bernstein polynomials.
Article
It is shown that if a sequence is strongly p-Cesàro summable or w p convergent for 0<p<∞ then the sequence must be statistically convergent and that a bounded statistically convergent sequence must be w p convergent for any p, 0<p<∞. It is also shown that the statistically convergent sequences do not form a locally convex FK space. A characterization of conservative matrices which map the bounded statistically convergent sequences into convergent sequences is given and applied to Nörlund and Nörlund-type means.
Article
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 (N, p_n)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^-x; e^-2x.We apply the classical Baskakov operator to construct an example in support of our result.
Article
The purpose of this paper is to introduce the space of sequences that are strongly (Vσ,λ,q)-summable with respect to an Orlicz function. We give some relations related to these sequence spaces. We also show that the spaces [Vσ,λ,M,p,q]1∩ℓ∞(q) may be represented as a space.
△^m-Statistical Convergence
  • M Et
  • F Nuray
Et M, Nuray F. 2001. △ m −Statistical Convergence. Indian J Pure Appl Math, 32(6): 961-969.
Weighted statistical convergence of order α
  • H Ş Kandemir
  • M Et
  • H Çakallı
Kandemir HŞ, Et M, Çakallı H. 2023. Weighted statistical convergence of order α. Facta Univ Series: Math Info, 38(2): 317-327.