Summability Of Order Research Articles

Lacunary Statistically Convergence via Modulus Function Sequences

Modifying the definition of density functions is one method used to generalise statistical convergence. In the present study, we use sequences of modulus functions and order $\alpha \in \left( 0,1\right] $ to introduce a new density. Based on this density framework, we define strong $(f_k)$-lacunary summability of order $\alpha $ and $(f_k)$-lacunary statistical convergence of order $\alpha $ for a sequence of modulus functions $(f_k)$. This concept holds an intermediate position between the usual convergence and the statistical convergence for lacunary sequences. We also establish inclusion theorems and relations between these two concepts in the study.

Modulated Lacunary 𝒫-Statistical Convergence of Order γ of Sequences of Fuzzy Functions

The central objective of this paper is to investigate the lacunary statistical convergence of order [Formula: see text] and strong lacunary summability of order [Formula: see text] for sequences of fuzzy functions by using the modulus function and a regular matrix. We examine the relevant inclusion relations between these spaces under specific conditions. Additionally, we study properties related to these spaces and establish several interesting results.

On statistical convergence of order α of sequences in gradual normed linear spaces

Abstract In the current paper, we introduce the notion of statistical convergence of order α and strongly p-Cesàro summability of order α of sequences in the gradual normed linear spaces. We investigate several properties and a few inclusion relations of the newly introduced notions.

ON ρ-STATISTICAL CONVERGENCE OF ORDER (α,β) FOR SEQUENCES OF FUZZY NUMBERS

In this study, we first give the definition of ρ-statistical convergence of order (α,β) for sequences of fuzzy numbers. We also define the strongly w(ρ,F,q)-summable of order (α,β) and the strongly w(ρ,F,q,f)-summable of order (α,β), defined by a modulus function f for sequences of fuzzy numbers. Later we give some coverage theorems between these sets and the set S_α^β (ρ,F).

LACUNARY STATISTICAL CONVERGENCE OF ORDER α IN PARTIAL METRIC SPACES

The present study introduces the notions of statistical convergence of order $\alpha$ and strongly $q-$ summability of order $\alpha$ in partial metric spaces. We examinethe inclusion relations linked to these these concepts.

Neutrosophic Y-Cesàro summability of a sequence of order α, of neutrosophic random variables in probability

In this paper, we define the notions of neutrosophic $ \mathfrak{Y} $-Ces\`aro summability of a sequence of order $ \alpha $, neutrosophic $ \mathfrak{Y} $-lacunary statistical convergence of order $ \alpha $, neutrosophic strongly $ \mathfrak{Y} $-lacunary statistical convergence of order $ \alpha $ and neutrosophic strongly $ \mathfrak{Y} $-Ces\`aro summability of order $ \alpha $ in neutrosophic probability. Besides, we prove some relations among them.

On harmonic summability of order α

In this study, the concepts of lacunary [Formula: see text]summability of order [Formula: see text], lacunary strongly harmonic summability of order [Formula: see text], lacunary statistical logarithmic convergence of order [Formula: see text] of sequences of real numbers are introduced and relations between these concepts are investigated.

Scaling a theorem of Harald Bohr

A outstanding result of H. Bohr shows that under a certain condition on the frequency λ=(λn) (preventing the λn's from getting too close too fast) every λ-Dirichlet series D=∑ane−λns converges uniformly on all half-planes [Re>σ] with σ>0, provided D is pointwise convergent on some half-plane and has a limit function extending to a bounded holomorphic function f on [Re>0]. Riesz summability plays an important role within the classical literature of general Dirichlet series - recall that a Dirichlet series D=∑ane−λns is said to be Riesz summable of order ℓ≥0 in s∈C, whenever the sequence of all Riesz means ∑λn<xan(1−λnx)ℓe−λns converges as x tends to infinity. Extending classical work of M. Riesz, we for a given ℓ≥0 isolate and study the class of those frequencies λ, which have the following property: If the Dirichlet series D=∑ane−λns is pointwise Riesz summable for some order on some half-plane and has a limit function extending to a holomorphic function f on [Re>0] with growth of order O((1+s)ℓ), then uniformly on all half-planes [Re>σ],σ>0, we have f(s)(1+s)ℓ=1(1+s)ℓlimx→∞⁡∑λn<xan(1−λnx)ℓe−λns. Among others, we show that this holds true if λ satisfies a well-known condition isolated by E. Landau, which recovers the classical bounded case ℓ=0.

∆λ−statistical boundedness of order α on time scales

In this study, we introduce the notions ∆λ−statistical convergence of order α (for αϵ(0,1]) and λp−summable of order α (for αϵ(0,1]) on an arbitrary time scale. Moreover, some relations about these notions are obtained. We define ∆λ− statistically boundedness of order α (for αϵ(0,1]) on a time scale. Furthermore, We give connections between S (λ,α) T (b) , S (β,θ) T (b) and ST (b) for various sequences µ∆λ(t) and µ∆β(t) are determined in class Λ.

On the negative order Cesáro summability of double series with respect to block-orthonormal systems

Abstract Some statements connected with the ( C , - 1 &lt; α &lt; 0 , - 1 &lt; β &lt; 0 ) {(C,-1&lt;\alpha&lt;0,-1&lt;\beta&lt;0)} summability almost everywhere of double series with respect to block-orthonormal systems are given. In particular, using the obtained results, we can determine Weyl multipliers for establishing negative order summability of double series with respect to Δ p , q {\Delta_{p,q}} -ONS.

ρ-STATISTICAL CONVERGENCE DEFINED BY A MODULUS FUNCTION OF ORDER (α,β)

The concept of strong w [ρ, f, q] −summability of order (α, β) for sequences of complex (or real) numbers is introduced in this work. We also give some inclusion relations between the sets of ρ-statistical convergence of order (α, β), strong wαβ [ρ, f, q] −summability and strong wαβ (ρ, q) −summability.

On (f, ρ)−Statistical convergence and strong (f, ρ)−summability of order α

Abstract The main object of this article is to introduce the concepts of (f, ρ)− statistical convergence of order α and strong (f, ρ)− summability of order α of sequences of real numbers and give some inclusion relations between these spaces.

Applications of deferred Cesàro statistical convergence of sequences of fuzzy numbers of order (ξ, ω)

The purpose of this article is to study deferred Cesrào statistical convergence of order (ξ, ω) associated with a modulus function involving the concept of difference sequences of fuzzy numbers. The study reveals that the statistical convergence of these newly formed sequence spaces behave well for ξ ≤ ω and convergence is not possible for ξ &gt; ω. We also define p-deferred Cesàro summability and establish several interesting results. In addition, we provide some examples which explain the validity of the theoretical results and the effectiveness of constructed sequence spaces. Finally, with the help of MATLAB software, we examine that if the sequence of fuzzy numbers is bounded and deferred Cesàro statistical convergent of order (ξ, ω) in (Δ, F, f), then it need not be strongly p-deferred Cesàro summable of order (ξ, ω) in general for 0 &lt; ξ ≤ ω ≤ 1.

On strong lacunary summability of order α with respect to modulus functions

This research paper focuses on defining the relationships between the sets of strongly lacunary summable and lacunary statistically convergent sequences by using different modulus functions f and g under certain conditions and different orders. Furthermore, for some special modulus functions, we establish the relations between the sets of strongly f-lacunary summable sequences and strongly f-lacunary summable sequences of order alpha.

A Note on Probability Convergence Defined by Unbounded Modulus Function and $alphabeta$-Statistical Convergence

In this paper we define f − αβ-statistical convergence of order γ in probability and f − αβ-strong p-Ces`aro summability of order γ in probability for a sequence of random variables under unbounded modulus function and examine the relation between these two concepts. We show by an example that this notion of f − αβ-statistical convergence of order γ in probability is stronger than αβ-statistical convergence of order γ in probability.

Tauberian theorems for statistical Cesàro and statistical logarithmic summability of sequences in intuitionistic fuzzy normed spaces

We define statistical Cesàro and statistical logarithmic summability methods of sequences in intuitionistic fuzzy normed spaces(IFNS) and give slowly oscillating type and Hardy type Tauberian conditions under which statistical Cesàro summability and statistical logarithmic summability imply convergence in IFNS. Besides, we obtain analogous results for the higher order summability methods as corollaries. Also, two theorems concerning the convergence of statistically convergent sequences in IFNS are proved in the paper.

SOME STATISTICAL CONVERGENCE TYPES OF ORDER $\alpha$ FOR DOUBLE SET SEQUENCES

In this study, we have introduced the concepts of Wijsman statistical convergence of order $\alpha$, Hausdorff statistical convergence of order $\alpha$ and Wijsman strongly $p$-Ces\`{a}ro summability of order $\alpha$ for double set sequences. Also, we have investigated some properties of these concepts and examined the relationships among them.

Applications of statistical convergence of order (η, δ + γ) in difference sequence spaces of fuzzy numbers

In this article, we introduce and study some difference sequence spaces of fuzzy numbers by making use of λ-statistical convergence of order (η, δ + γ) . With the aid of MATLAB software, it appears that the statistical convergence of order (η, δ + γ) is well defined every time when (δ + γ) &gt; η and this convergence fails when (δ + γ) &lt; η. Moreover, we try to set up relations between (Δv, λ)-statistical convergence of order (η, δ + γ) and strongly (Δv, p, λ)-Cesàro summability of order (η, δ + γ) and give some compelling instances to show that the converse of these relations is not valid. In addition to the above results, we also graphically exhibits that if a sequence of fuzzy numbers is bounded and statistically convergent of order (η, δ + γ) in (Δv, λ), then it need not be strongly (Δv, p, λ)-Cesàro summable of order (η, δ + γ).

Statistical Convergence of order β for (λ, μ) double sequences of fuzzy numbers

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.

Deferred statistical convergence of order &lt;i&gt;α&lt;/i&gt; in metric spaces

In this paper, the concepts of deferred statistical convergence of order α and deferred strong Cesaro summability are generalized to general metric spaces and some relations between deferred strong Cesaro summability of order α and deferred statistical convergence of order α are given in general metric spaces.