Statistical Summability Research Articles

A New Notion of Convergence Defined by The Fibonacci Sequence: A Novel Framework and Its Tauberian Conditions

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.

Steinhaus Type Theorem for Nörlund-(M, λn) and Nörlund-Euler-(M, λn) Methods of Summability in Non-Archimedean Fields

In the present research paper, an investigation is undertaken of Steinhaus type theorems for Nörlund-(M, λn) and Nörlund-Euler(M, λn) method of summability in K, a complete non-trivially valued Non-Archimedean field. The conditions for statistical summability for those matrices are discussed in such fields K. The consistency of Nörlund-(M, λn) method of summability is investigated when different sequences are used in the summation process. Further, the relation between Nörlund-Euler(M, λn) summable and its statistical summability is also established.

Tauberian conditions under which statistical convergence follows from statistical summability by weighted mean method in two-normed spaces

Tauberian conditions under which statistical convergence follows from statistical summability by weighted mean method in two-normed spaces

Some Results of Generalized Weighted Norlund-Euler-<img src=image/13428701_01.gif> Statistical Convergence in Non-Archimedean Fields

Non-Archimedean analysis is the study of fields that satisfy the stronger triangular inequality, also known as ultrametric property. The theory of summability has many uses throughout analysis and applied mathematics. The origin of summability methods developed with the study of convergent and divergent series by Euler, Gauss, Cauchy and Abel. There is a good number of special methods of summability such as Abel, Borel, Euler, Taylor, Norlund, Hausdroff in classical Analysis. Norlund, Euler, Taylor and weighted mean methods in Non-Archimedan Analysis have been investigated in detail by Natarajan and Srinivasan. Schoenberg developed some basic properties of statistical convergence and also studied the concept as a summability method. The relationship between the summability theory and statistical convergence has been introduced by Schoenberg. The concept of weighted statistical convergence and its relations of statistical summability were developed by Karakaya and Chishti. Srinivasan introduced some summability methods namely y-method, Norlund method and Weighted mean method in p-adic Fields. The main objective of this work is to explore some important results on statistical convergence and its related concepts in Non-Archimedean fields using summability methods. In this article, Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence, generalized weighted summability using Norlund-Euler-<img src=image/13428701_02.gif> method in an Ultrametric field are defined. The relation between Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence and Statistical Norlund-Euler-<img src=image/13428701_02.gif> summability has been extended to non-Archidemean fields. The notion of Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence and inclusion results of Norlund-Euler statistical convergent sequence has been characterized. Further the relation between Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence of order α & β has been established.

LACUNARY STATISTICAL HARMONIC SUMMABILITY

In this paper, the concepts of lacunary (H,1) summability, lacunary strongly harmonically summability, lacunary statistical (H,1) summability, lacunary statistical logarithmic convergence of sequences of real numbers are introduced and relations between these concepts are investigated.

Tauberian theorems for weighted mean statistical summability of double sequences of fuzzy numbers

We discuss Tauberian conditions under which the statistical convergence of double sequences of fuzzy numbers follows from the statistical convergence of their weighted means. We also prove some other results which are necessary to establish the main results.

Multiplier Sequence Spaces Defined by Statistical Summability and Orlicz-Pettis Theorem

In this paper, we introduce some new multiplier spaces related to a series in a normed space X through statistical summability and give some characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in X and respectively. We also obtain a new version of the Orlicz-Pettis theorem within the frame of the statistical convergence.

Statistical Voronoi mean and applications to approximation theorems

In this paper, we give statistical Voronoi mean which is a new statistical summability method, is not need to be regular and positive. We prove a Korovkin type approximation theorem via this method that covers many important summability methods scattered in the literature. Also, we demonstrate that our theorem is stronger than proven by earlier authors with an interesting application. Finally, we establish the rate of convergence.

Nörlund statistical convergence and Tauberian conditions for statistical convergence from statistical summability using Nörlund means in non-Archimedean fields

Nörlund statistical convergence and Tauberian conditions for statistical convergence from statistical summability using Nörlund means in non-Archimedean fields

Weighted (λ,μ)-statistical convergence and statistical summability methods of double sequences of fuzzy numbers with application to Korovkin-type fuzzy approximation theorem

The paper aims to investigate different types of weighted statistical convergence and weighted statistical summability of double sequences of fuzzy numbers. Relations connecting statistical convergence and statistical summability have been investigated in the environment of the newly defined classes of double sequences of fuzzy numbers. Relevant inclusion relations and related results concerning the new fuzzy summability methods have also been examined in detail. Finally, a Korovkin-type approximation theorem for double sequences of fuzzy positive linear operators has been established using the most generalized version of the summability method and showed its efficiency by means of a concrete example.

Statistical matrix summability of difference sequences with speed and characterization of matrix classes

The paper introduces the notion of A-statistical convergence of difference sequences with speed λ = (λk), where 0 < λk↗ ∞. The primary aim is to characterize several matrix classes involving this class of sequences along with others including arbitrary sequence spaces that satisfy certain conditions such as subsequence closed and section closed.

Some inequalities for statistical (C; 1) (H; 1)-summability

The concept of statistical summability C, 1)(H, 1) has recently been introduced by [8]. In this paper we establish some inequalities related to concept of statistical summability (C, 1)(H, 1).

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.

Weighted λ-statistical convergence connecting a statistical summability of sequences of fuzzy numbers and Korovkin-type approximation theorems

In this paper, we investigated the notion of generalized weighted λ-statistical convergence connecting $$ \left( {\bar{N}_{\lambda } , p_{n} } \right)^{F} $$ -statistical summability of sequences of fuzzy numbers. Some inclusion relations and other related results for the new summability methods are also given. Furthermore, we give a Korovkin-type approximation theorem for fuzzy positive linear operators using the notion of generalized weighted $$ \left( {\bar{N}_{\lambda } , p_{n} } \right)^{F} $$ -statistical summability.

Tauberian theorems for statistically (C, 1, 1) summable double sequences

In this paper, we obtain some Tauberian conditions in terms of slow oscillation and slow decreasing in certain senses, under which convergence of a double sequence in Pringsheim’s sense follows from its statistical (C, 1, 1) summability.

Tauberian theorems for statistical summability methods of sequences of fuzzy numbers

We give Tauberian conditions of slow decrease type under which statistical Cesaro summability and statistical logarithmic summability of sequences of fuzzy numbers imply convergence in fuzzy number space. Besides, we obtain a sufficient condition for statistically convergent sequences of fuzzy numbers to converge in the ordinary sense.

Weighted Statistical Convergence of Double Sequences in Non-Archimedean Fields

Let K be a complete, non-trivially valued non-archimedean field. In this paper, weighted statistical convergence and statistical summability of double sequences are studied. Also the notion of (, p, q) - summability and inclusion relations have been discussed in such fields.

Tauberian conditions under which statistical convergence follows from statistical summability $(EC)_{n}^1$

Let $(x_k)$, for $k\in \mathbb{N}\cup \{0\}$ be a sequence of real or complex numbers and set $(EC)_{n}^{1}=\frac{1}{2^n}\sum_{j=0}^{n}{\binom{n}{j}\frac{1}{j+1}\sum_{v=0}^{j}{x_v}},$ $n\in \mathbb{N}\cup \{0\}.$ We present necessary and sufficient conditions, under which $st-\lim_{}{x_k}= L$ follows from $st-\lim_{}{(EC)_{n}^{1}} = L,$ where L is a finite number. If $(x_k)$ is a sequence of real numbers, then these are one-sided Tauberian conditions. If $(x_k)$ is a sequence of complex numbers, then these are two-sided Tauberian conditions.

Statistical deferred Cesàro summability and its applications to approximation theorems

Statistical (C,1) summability and a Korovkin type approximation theorem has been proved by Mohiuddine et al. [20] (see [S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical summability (C,1) and a Korovkin type approximation theorem, J. Inequal. Appl. 2012 (2012), Article ID 172, 1-8). In this paper, we apply statistical deferred Ces?ro summability method to prove a Korovkin type approximation theorem for the set of functions 1, e-x and e-2x defined on a Banach space C[0;1) and demonstrate that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. We also establish a result for the rate of statistical deferred Ces?ro summability method. Some interesting examples are also discussed here in support of our definitions and results.

Tauberian theorems for the statistical convergence and the statistical (C,1,1) summability

Every P-convergent double sequence is statistically convergent and every bounded statistically convergent sequence is statistical (C,1,1) summable. The converse of these implications are not always true. Theorems on which conditioned converses are searched are known as Tauberian theorems. Inspired by the convergence to zero of the difference sequence between a sequence and its arithmetic means in the single sequence case, we obtain Tauberian theorems for the statistical convergence and statistical (C,1,1) summability method by imposing some conditions on the difference sequence between a double sequence and its different arithmetic means.