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The central objective of this paper is to investigate the lacunary statistical convergence of order γ and strong lacunary summability of order γ 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. [ABSTRACT FROM AUTHOR]

We study the triangular θ -mean of the partial sums of f ∈ L p (T q 2) and prove the following noncommutative weak and strong type maximal inequalities: ‖ (σ n Δ , θ (f)) n ≥ 1 ‖ Λ 1 , ∞ (T q 2 , ℓ ∞) ≤ c θ ‖ f ‖ L 1 (T q 2) , p = 1 and σ n Δ , θ (f) n ≥ 1 L p (T q 2 , ℓ ∞) ≤ c p , θ ‖ f ‖ L p (T q 2) , 1 < p < ∞ , where T q 2 is a 2-dimensional quantum torus. As a consequence, we obtain the bilateral almost uniform convergence of σ n Δ , θ (f) provided f ∈ L p (T q 2). [ABSTRACT FROM AUTHOR]

The primary objective of this study is to introduce the notion of ideal convergence in quaternion-valued generalized metric spaces. We define I-convergence and I*-convergence in these spaces and establish their equivalence through the definition of property (AP). Furthermore, we introduce Z-Cauchy and I*-Cauchy sequences, adapting classically theorems to suit quaternion-valued generalized metric spaces. We also present I-statistical convergence, I-lacunary statistical convergence, strong I-Cesàro summability, strong I-lacunary summability, [V, λ] (Z)-summability, and I-λ-statistically convergencein quaternion valued generalized metric spaces, along with their associated properties. [ABSTRACT FROM AUTHOR]

The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ -Fibonacci statistical convergence, strong Δ -Fibonacci summability, and Δ -Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers. [ABSTRACT FROM AUTHOR]

The concept of summability is crucial in deriving formal solutions to partial differential equations. This paper explores the connection between the methods of statistical convergence of sequences and statistical Cesàro summability in intuitionistic fuzzy n-normed linear space (IFnNLS). While the existing literature covers Cesàro summability and its statistical variant in fuzzy, intuitionistic fuzzy, and classical normed spaces, this study stands out not only for its methodology but also for its comprehensive approach, encompassing a broader range of spaces and detailing the pathway from the statistical Cesàro summability method to statistical convergence. These results will lead us to Tauberian theorems in IFnNLS. [ABSTRACT FROM AUTHOR]

The applications of a Fibonacci sequence in mathematics extend far beyond their initial discovery and theoretical significance. The Fibonacci sequence proves to be a versatile tool with real-world implications and the practical utility of manifests in various fields, including optimization algorithms, computer science and finance. In this research paper, we introduce new versions of convergence and summability of sequences in normed spaces with the help of the Fibonacci sequence called weak Fibonacci φ-lacunary statistical convergence and weak Fibonacci φ-lacunary summability, where φ is a modulus function under certain conditions. Furthermore, we obtain some relations related to these concepts in normed spaces. [ABSTRACT FROM AUTHOR]

In this paper, we give some properties of the Dunkl–Gamma‐type operators including Appell polynomials, using into consideration the generalized power summability method. In the first section are given moments of the new defined operators. In second section are given some direct estimation related to the Dunkl–Gamma‐type operators, including Korovkin‐type theorem. In the third section, we give some results related to the weighted spaces of continuous functions, and in the last section, we give some properties in the sense of A$$ A $$‐statistically convergence, including Voronovskaya and Grüss–Voronovskaya‐type theorem. [ABSTRACT FROM AUTHOR]

The primary objective of this study is to introduce the concepts of T2-deferred Cesàro summability and T2-deferred statistical convergence for double sequences in fuzzy normed spaces (FNS). Furthermore, the aim is to explore the connections between these concepts and subsequently establish several theorems pertaining to the notion of T2-deferred statistical convergence in FNS for double sequences. We further define T2-deferred statistical limit points and T2-deferred statistical cluster points of a sequence within FNS and explore the relationships among these concepts. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we aim to define ℐ 2 ⁢ (풮) -summability and ℐ 2 ⁢ (풮 θ p , r ) -summability, and obtain interesting relationships among these by imposing certain conditions on p and r. Finally, we show that the space ℐ 2 ⁢ (풮 θ p , r ⁢ (G , B , Y)) ∩ l ∞ 2 ⁢ (V) is a closed subspace of l ∞ 2 ⁢ (V) . [ABSTRACT FROM AUTHOR]

In this paper, by using the notions of λ-density, (V , λ) -summability and Orlicz function ϕ, we introduce a new concept of convergence, namely λ-statistically ϕ-convergence, as a generalization of the λ-statistical convergence and statistically ϕ-convergence. Based on this concept, we introduce a new sequence space S λ - ϕ and investigate some of its properties. Also, we find its relations with statistically ϕ-convergence, [ C , 1 ] ϕ -summability and [ V , λ ] ϕ -summability. Finally, we introduce and investigate the concept of S λ - ϕ Cauchy sequences. [ABSTRACT FROM AUTHOR]

We consider fractional integral operators (I − T) d , d ∈ (− 1 , 1) acting on functions g : Z ν → R , ν ≥ 1 , where T is the transition operator of a random walk on Z ν . We obtain the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels τ (s ; d) , s ∈ Z ν of (I − T) d . The asymptotic behavior of τ (s ; d) as | s | → ∞ is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on Z ν solving the difference equation (I − T) d X = ε with white noise on the right-hand side is discussed and their scaling limits. Several examples, including fractional lattice Laplace and heat operators, are studied in detail. [ABSTRACT FROM AUTHOR]

This paper studies the summability of the transseries solution of a nonintegrable Hamiltonian system. Since our system has a resonance and is not integrable a general transseries theory does not work well as far as the author knows. In order to construct a formal transseries solution and prove its summability our main idea is to use the superintegrability of a Hamiltonian system in a class of transseries. More precisely we first show the superintegrability of a Hamiltonian system in the category of transseries via the key Lemmas 1 and 4 which follow. By virtue of the superintegrability we show the existence of a formal transseries solution. Then its summability is proved via the superintegrability. We note that the argument based on the superintegrability is elementary. [ABSTRACT FROM AUTHOR]

In the study done here regarding the theory of summability, we introduce some new concepts in fuzzy normed spaces. First, at the beginning of the original part of our study, we define the lacunary invariant statistical convergence. Then, we examine some characteristic features like uniqueness, linearity of this new notion and give its important relation with pre-given concepts. [ABSTRACT FROM AUTHOR]

Chan et al. (Integral Transforms Spec. Funct. 12 (2), 139–148, (2001)) constructed a multivariable extension of the Lagrange polynomials, popularly known as the Chan–Chyan–Srivastava polynomials. Altin and Erkuş (Integral Transforms Spec. Funct. 17, 239–244, (2006)) proposed Lagrange–Hermite polynomials in several variables. Erkuş and Srivastava (Integral Transforms Spec. Funct., 17, 267–273, (2006)) presented an unification (and generalization) of the Chan–Chyan–Srivastava polynomials and the multivariable Lagrange–Hermite polynomials, called as Erkuş–Srivastava polynomials. Duman (Taiwanese J. Math. 12 (2), 539‐543, (2008)) defined a q$$ q $$‐analogue of these generalized multivariable polynomials. Inspired by these studies, we construct a linear positive operator by means of the q$$ q $$‐Erkuş–Srivastava multivariable polynomials and study the Korovkin‐type theorems and the rate of convergence of these operators by using summability techniques of weighted A$$ A $$‐statistical convergence and the power series method. We also define a k$$ k $$th‐order Taylor generalization of the multivariable polynomials operator and investigate the approximation of k$$ k $$th‐order continuously differentiable Lipschitz class elements. Finally, we define the bivariate case of q$$ q $$‐Erkuş–Srivastava multivariable polynomials operator and study its A$$ A $$‐statistical convergence by using four‐dimensional matrix transformation. [ABSTRACT FROM AUTHOR]

This paper's main purpose is to present λ²-statistically convergent and condition of being λ²-statistically Cauchy for double sequences in fuzzy nnormed spaces. At the same time, in fuzzy n-normed space, we have introduced the concept of λ²-summability and Cesaro summability for double sequences in fuzzy n normed spaces. Then, we studied the relation between these concepts and statistical convergence in fuzzy n-normed space. [ABSTRACT FROM AUTHOR]

We first present a notion of a double lacunary sequence on product time scales. Using this notion, we define the notions of the double lacunary statistical convergence and double lacunary strongly p-Cesàro summability of 2-multiple functions on product time scales and we study some fundamental properties of both notions. We also present a theorem that connects the above-mentioned two concepts. Furthermore, we define a refinement of a double lacunary sequence on product time scales and provide some fundamental properties as well as inclusion theorems for a refined and a non-refined double lacunary sequence on product time scales. [ABSTRACT FROM AUTHOR]

Let λ = (λn) be a nondecreasing sequence of positive numbers such that λn → ∞. A sequence (ξn) is called λ-bounded if λn(ξn - α) = O(1) with the limit ...ξn = α. In this work, we obtain several Tauberian remainder theorems on λ-bounded sequences for the logarithmic summability method with help of general logarithmic control modulo of the oscillatory behavior. Tauber conditions in our main results are on the generator sequence and the general logarithmic control modulo. [ABSTRACT FROM AUTHOR]

In present paper, we aim to define a new summability method using σ-mean called σ-statistical summability in an intuitionistic fuzzy 2-normed space (briefly IF 2N S). We also define σ-statistical Cauchy sequence and Sσ-completeness in an IF 2NS and study some of their properties. We display an example that shows our method of summability is more stronger in these spaces. [ABSTRACT FROM AUTHOR]

Several new network information dimension definitions have been proposed in recent decades, expanding the scope of applicability of this seminal tool. This paper proposes a new definition based on Deng entropy and d-summability (a concept from geometric measure theory). We will prove to what extent the new formulation will be useful in the theoretical and applied points of view. [ABSTRACT FROM AUTHOR]

Statistical convergence is more extensive than the classical convergence and has recently drawn the recognition of many researchers. The Korovkin-type approximation theorems are usually based on the convergence analysis of sequences of positive linear operators. Gradually, such approximation theorems are extended over more general sequence spaces with several settings via different kinds of statistical summability techniques. In this paper, we introduce presumably a new statistical Riesz-Euler product summability technique to prove a Korovkin-type approximation theorem. Moreover, we demonstrate another result for the rate of statistical convergence under our proposed summability technique. [ABSTRACT FROM AUTHOR]

We consider a quartic O (N) -vector model. Using the loop vertex expansion, we prove the Borel summability in 1/N along the real axis of the partition function and of the connected correlations of the model. The Borel summability holds uniformly in the coupling constant, as long as the latter belongs to a cardioid like domain of the complex plane, avoiding the negative real axis. [ABSTRACT FROM AUTHOR]

The article studies the convergence of trigonometric Fourier series via a new Tauberian theorem for Cesàro summable series in abstract normed spaces. This theorem generalizes some known results of Hardy and Littlewood for number series. We find sufficient conditions for the convergence of trigonometric Fourier series in homogeneous Banach spaces over the circle. These conditions are expressed in terms of the Fourier coefficients and are weaker than Hardy's condition. We give a description of all Banach function spaces given over the circle and endowed with a norm been equivalent to a norm in a homogeneous Banach space. We study interpolation properties of such spaces and give new examples of them. We extend the classical Fejér theorem on the uniform Cesàro summability of the Fourier series on sets by means of a refined version of Cantor's theorem on the uniform continuity of a mapping between metric spaces. We also generalize the classical Hardy theorem on the uniform convergence of the Fourier series on sets. [ABSTRACT FROM AUTHOR]

This paper aims to explore the concepts of statistical convergence sequence and statistical summability in non-Archimedean fields (NAF). Statistical convergence has been studied in various mathematical fields such as measure theory, probability theory, and number theory, and plays a significant role in summability theory and functional analysis. The goal of this study is to provide characterizations for triple sequences using the (Mλt,u,v) method of summability over NAF and to prove inclusion relations between statistical convergence and statistical (Mλt,u,v) summability for triple sequences over such fields. Furthermore, statistics At,u,v-summability for triple sequence has been examined in non-Archimedean fields. The non-trivially valued, complete, non-Archimedean fields are denoted by K throughout the article. [ABSTRACT FROM AUTHOR]

This article explores a Chlodowsky-type extension of Szász operators using the general-Appell polynomials. The convergence properties of these operators are established by employing the universal Korovkin-type property, and the order of approximation is determined using the classical modulus of continuity. Additionally, the weighted B -statistical convergence and statistically weighted B -summability properties of the operators are derived. Theoretical results are supported by numerical and graphical examples. [ABSTRACT FROM AUTHOR]

This paper gives an introduction to the resurgence theory developed by French mathematician Jean Écalle and its important applications in mathematics and physics. The theory of simple resurgent functions and mould calculus are systematically presented with briefly mentioning general singularities. Euler series, Stirling series, Airy function and quantum dilogarithm are taken as examples to explain the subtlety of the theory. Applications in quantum mechanics（exact WKB theory, semi-classical trace formula）, gauge theory（especially complex Chern-Simons theory）and topological string theory are also mentioned. [ABSTRACT FROM AUTHOR]

Let p = (pj) and q = (qk) be real sequences of nonnegative numbers with the property that P m = ∑ j = 0 m p j ≠ 0 and Q m = ∑ k = 0 n q k ≠ 0 for all m and n. Also let (Pm) and (Qn) be regularly varying positive indices. Assume that (umn) is a double sequence of complex (real) numbers, which is ( N ¯ , p, q; α, β)-summable and has a finite limit, where (α, β) = (1, 1), (1, 0), or (0, 1). We present some conditions imposed on the weights under which (umn) converges in Pringsheim's sense. These results generalize and extend the results obtained by the authors in [Comput. Math. Appl., 62, No. 6, 2609–2615 (2011)]. [ABSTRACT FROM AUTHOR]

This paper contains a power series summability method based on Korovkin type approximation theorem for a sequence of fuzzy positive linear operators in two variables. Besides that, a theorem related to fuzzy rate of convergence has been derived with the help of fuzzy modulus of continuity, which gives clarity to understanding the analogous observations between fuzzy set theory and classical theory. Additionally, an example is illustrated which gives a better idea to understand that summability in two variable-based Korovkin type theorem has less drawback over fuzzy set theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new version of the definition of a quasinorm (in particular, a norm) in Lebesgue spaces with variable order of summability. Using it, we prove an analogue of Hölder's inequality for such spaces, which is more general and more precise than those known earlier. [ABSTRACT FROM AUTHOR]

This paper interprets the relationship between the statistical Cesàro summability method and statistical convergence for sequences in intuitionistic fuzzy normed spaces. There are several studies in the literature on the Cesàro summability method and its statistical version, in classical, fuzzy, and intuitionistic fuzzy normed spaces. However, what distinguishes this study from others, besides the techniques used, is that it is a more comprehensive research in terms of both the space being studied and the path to the statistical convergence from the statistical Cesàro summability method. Our primary goal in this work is to investigate whether conditions designated as O-type like q-boundedness and statistical slow oscillation require statistical Cesàro summable sequences be statistical convergent with respect to the intuitionistic fuzzy norm (μ , ν). Moreover, we inquire about the existence of a more inclusive condition than these conditions, from which we can derive a similar inference. Our second aim is to present a series of auxiliary results designed to be utilized in the proofs of our main results. To go into a little detail, we discuss some algebraic properties of statistical limits, including additivity and homogeneity, as well as the existence of a statistical limit of subsequences and moving averages of a sequence. Finally, we give some examples of the application of the intuitionistic fuzzy normed space in engineering, in order to extend the area of influence of the relevant space and to support the studies carried out in the context of this space. [ABSTRACT FROM AUTHOR]

In the present paper, we aim to define Sθ-summability in neutrosophic 2-normed spaces and study some of its properties. We provide examples that shows our method of summability is stronger in these spaces. Finally we introduce Sθ-Cauchy and Sθ-completeness and prove that every neutrosophic-2-normed spaces is Sθ-complete. [ABSTRACT FROM AUTHOR]

In the present paper, we aim to define λ-statistical summability, Vλ-statistical summability in neutrosophic-2-normed spaces (briefly called N - 2 - NS) and study some relationships among these notions. We give an example that shows λ-statistical summability is stronger method in neutrosophic-2-normed spaces. Finally, we define λ-statistically Cauchy sequence and λ-statistically completeness in neutrosophic-2-normed spaces and obtain the Cauchy convergence criteria in these spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new summability methods for sequences of sets named Riesz summability of order α and weighted statistical convergence of order α, where α = (0; 1]. Also, we examine some properties of these methods and give some inclusion theorems. [ABSTRACT FROM AUTHOR]

In this paper, a theorem dealing with the |N, pn|k summability factors of infinite series has been generalized to |A, pn, β; γ|k summability method by using (Φ, δ)- monotone sequences. This new theorem also includes some new results. [ABSTRACT FROM AUTHOR]

Let s ∈ be an open bounded set. In this work we study the existence of solutions to problems (E±) Lu ± g(u) = μ and u = 0 a.e. in ℝN \ Ω, where g ∈ C(ℝ) is a nondecreasing function, μ is a bounded Radon measure on and Ω is an integro-differential operator with order of differentiability s ∈ (0, 1) and summability ... More precisely, L is a fractional p-Laplace type operator. We establish sufficient conditions for the solvability of problems (E±). In the particular case ..., these conditions are expressed in terms of Bessel capacities. [ABSTRACT FROM AUTHOR]

This paper presents a new perspective on the relationship between the ( N̅, p, q, r) method and P-convergence for triple sequences.Our primary goal is to derive Tauberian conditions that dictate the behavior of the weighted generator sequence (zlmn) in relation to the sequences (Pl), (Qm), and (Rn), with the intention of establishing a fresh interpretation. These conditions control the OL- and O-oscillatory properties and establish a connection from ( N̅, p, q, r) summability to P-convergence, subject to certain restrictions on the weight sequences. Furthermore, we demonstrate that specific cases, such as the OL-condition of Landau type and the O-condition of Hardy type with respect to (Pl), (Qm) and (Rn), serve as Tauberian conditions for ( N̅, p, q, r) summability under additional conditions. Consequently, our findings encompass all the classical Tauberian theorems, including conditions related to slow decrease and slow oscillation in specific contexts. [ABSTRACT FROM AUTHOR]

The usual convergence of sequences has many generalizations with the aim of providing deeper insights into the summability theory. In this paper, following a very recent and new approach, we introduce the notion of I3 and I*3-convergence of triple sequences in neutrosophic normed spaces mainly as a generalization of statistical convergence of triple sequences. We investigate a few fundamental properties and study the relationship between the two notions. We also introduce and investigate the concept of I3 and I*3-Cauchy sequence of triple sequences and show that the condition (AP3) plays a crucial role to study the interrelationship between them. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce and study the strong Cesaro summability of difference sequence spaces through fusion of modulus function. On the newly established sequence space, linear structure is imposed and a paranorm is established. Apart from various inclusion relations, a new variant of statistical convergence is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of a generalized summability, called Cesaro summability in neutrosophic normed spaces (briefly NNS). We obtain conditions under which ordinary summability follows from Cesaro summability. Later, we define a concept of slowly oscillating sequences in NNS and establish related Tauberian Theorems in neutrosophic normed spaces. [ABSTRACT FROM AUTHOR]

In this work, vector-valued continuous functions are approximated uniformly on the unit hypercube by Shepard operators. If λ denotes the usual parameter of the Shepard operators and m is the dimension of the hypercube, then our results show that it is possible to obtain a uniform approximation of a continuous vector-valued function by these operators when λ ≥ m + 1 . By using three-dimensional parametric plots, we illustrate this uniform approximation for some vector-valued functions. Finally, the influence in approximation by regular summability processes is studied, and their motivation is shown. [ABSTRACT FROM AUTHOR]

In this paper, we obtain necessary and sufficient conditions for the equivalence of two matrix summability methods. Some known results are also presented. [ABSTRACT FROM AUTHOR]

We define Cesaro summability in a Neutrosophic n -Normed Linear Space in this article. In a Neutrosophic n-Normed Linear Space, we show the Cesaro summability method to be regular, albeit this does not imply typical convergence in general. We also look for more circumstances in which the opposite is true. [ABSTRACT FROM AUTHOR]

Over the last few years, numerous researchers have contributed significantly to summability theory by connecting various notions of convergence concepts of sequences. In this paper, we introduce the concepts of ℐ -statistical supremum and ℐ -statistical infimum of a real-valued sequence and study some fundamental features of the newly introduced notions. We also introduce the concept of ℐ -statistical monotonicity and establish the condition under which an ℐ -statistical monotonic sequence is ℐ -statistical convergent. We end up giving a necessary and a sufficient condition for the ℐ -statistical convergence of a real-valued sequence. [ABSTRACT FROM AUTHOR]

In the study conducted here, we have given some new concepts in summability theory. In this sense, firstly, using the lacunary sequence we have given the concept of strongly Iθ2*-convergence and we have examined the relations between Iθ2*-convergence and strongly Iθ2*-convergence and also between strongly Iθ2-convergence and strongly Iθ2*-convergence. Also, using the lacunary sequence we have given the concept of strongly Iθ2*-Cauchy sequence and examined the relations between strongly Iθ2-Cauchy sequence and strongly Iθ2*-Cauchy sequence. [ABSTRACT FROM AUTHOR]

With this work, we present the asymptotical strongly p-deferred invariant and asymptotical deferred invariant statistical equivalence of order a (0 < a ≤ 1) for sequences of sets in the Wijsman sense. Furthermore, we investigate the connections between these concepts and conduct their properties. [ABSTRACT FROM AUTHOR]

This paper aims to introduce the notions of deferred Nörlund statistical Riemann integrability and statistical deferred Nörlund Riemann summability for sequence of real-valued functions and to apply them in Korovkin-type new approximations. First, we present an inclusion theorem to understand the connection between these new notions. Then, based on these potential notions we establish new versions of Korovkin-type theorems with three algebraic test functions. Finally, we compute an example, under the consideration of a positive linear operator in association with the Bernstein polynomials to exhibit the effectiveness of our findings. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to make a novel interpretation of relation between (N̅,p,q) method and P-convergence for double sequences of fuzzy numbers. In accordance with this aim, we derive some Tauberian conditions, controlling O-oscillatory behavior of a double sequence of fuzzy numbers in certain senses, from (N̅,p,q) summability to P- convergence with some restrictions on the weight sequences. As special cases, we indicate that O-condition of Hardy type with respect to (Pm) and Qn are Tauberian conditions for (N̅,p,q) summability under some additional conditions. In the sequel, we prove a fuzzy Korovkin-type approximation theorem by using the (N̅,p,q) summability method for fuzzy positive linear operators. [ABSTRACT FROM AUTHOR]