Summability Methods Research Articles

Absolute │A; δ │<sub>k</sub> and │A; γ; δ│<sub>k</sub> Summability for n-tupled Triangle Matrices

In this study, new sequence spaces (Ak; δ) & (Ak; γ; δ) have been introduced to establish two theorems on minimal set of the sufficient conditions for a n-tupled triangle T to be a bounded operator on sequence spaces (Ank ; δ) & (Ank ; γ; δ): Generalized summability method │A; δ │k & │A; γ; δ│k have been applied for determining the sufficient conditions, where k ≥ 1; δ ≥ 0 and γ is real number. Further, a set of new and well-known applications has been deduced from the main result under suitable conditions, which shows the importance of the main result.

Degree of approximation of signals in certain Lipschitz classes by the Zweier–Euler product summability method of Fourier series

The paper introduces the notion of a new product summability method, which is obtained by superimposing the Zweier method on the Euler method. This method is applied to obtain the degree of approximation of Fourier series of signals (functions) belonging to certain Lipschitz classes.

Another proof and a generalization of a theorem of H. H. Bauschke on monotone operators

ABSTRACT Using the concept of partial-inverse of monotone operators due to Spingarn, we present a new and simple proof of a result – Theorem 2 in Bauschke [A note on the paper by Eckstein and Svaiter on “General projective splitting methods for sums of maximal monotone operators”. SIAM J Control Optim. 2009;48(4):2513–2515] – of Heinz H. Bauschke. Our proof is based on the maximal monotonicity of the partial-inverse and on the (asymptotic) closedness principle on the graph of maximal monotone operators in the topology. We also present a generalization of Bauschke's theorem to the more general setting of ε–enlargements of monotone maps.

On the Logarithmic Summability of Sequences in Intuitionistic Fuzzy Normed Spaces

We introduce logarithmic summability in intuitionistic fuzzy normed spaces($IFNS$) and give some Tauberian conditions for which logarithmic summability yields convergence in $IFNS$. Besides, we define the concept of slow oscillation with respect to logarithmic summability in $IFNS$, investigate its relation with the concept of q-boundedness and give Tauberian theorems by means of q-boundedness and slow oscillation with respect to logarithmic summability. A comparison theorem between Ces\`{a}ro summability method and logarithmic summability method in $IFNS$ is also proved in the paper.

On a Feller–Jajte strong law of large numbers

We investigate the asymptotic behavior of a large class of summability methods introduced by Jajte. Using martingale tools, we prove strong laws of large numbers for a family of random variables whose tails of distributions are subject to some restrictions. Our results complement those of Naderi et al. (Communications in Statistics - Theory and Methods 48: 5414–18, 2018) and Miao et al. (Communications in Statistics - Theory and Methods 49 (6):1435–45, 2020).

Trigonometric approximation of functions f(x,y) of generalized Lipschitz class by double Hausdorff matrix summability method

In this paper, we establish a new estimate for the degree of approximation of functions f(x,y) belonging to the generalized Lipschitz class Lip ((xi _{1}, xi _{2} );r ), r geq 1, by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from Lip ((alpha ,beta );r ) and Lip(alpha ,beta ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and (C, gamma , delta ) means.

Statistical Korovkin and Voronovskaya type theorem for the Cesaro second-order operator of fuzzy numbers

In this paper we define the Ces\'aro second-order summability method for fuzzy numbers and prove Korovkin type theorem, then as the application of it, we prove the rate of convergence. In the last section, we prove the kind of Voronovskaya type theorem and give some concluding remarks related to the obtained results.

Multiple λμ‐statistically convergence via ϕ˜ functions

The primary target of this article is to introduce a new summability sequence space In addition, we will present some of its properties and examine some interesting results by using double statistically ‐convergence, ‐ summability, and ‐summability, which are new summability methods.

Convergence in Phi-Variation and Rate of Approximation for Nonlinear Integral Operators Using Summability Process

We investigate the approximation properties of nonlinear integral operators of the convolution type. In this approximation, we use functions of bounded variation based on the appropriate functionals. To get more general results, we consider Bell-type summability methods in the approximation. Moreover, we examine the rate of approximation. Then, using summability methods, we obtain a characterization for absolute continuity. Our examples at the end of the paper clearly demonstrate why we used summability methods rather than convergence in the conventional sense.

Approximation by Modified Meyer–König and Zeller Operators via Power Series Summability Method

In this paper, we study the Korovkin-type theorem for modified Meyer–Konig and Zeller operators via A-statistical convergence and power series summability method. The rate of convergence for this new summability methods is also obtained with the help of the modulus of continuity. Further, we establish Voronovskaya-type and Gruss–Voronovskaya-type theorems for A-statistical convergence.

TAUBERIAN THEOREMS FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY OF INTEGRALS

Let $q$ be a positive weight function on $\mathbf{R}_{+}:=[0, \infty)$ which is integrable in Lebesgue's sense over every finite interval $(0,x)$ for $0<x<\infty$, in symbol: $q \in L^{1}_{loc} (\mathbf{R}_{+})$ such that $Q(x):=\int_{0}^{x} q(t) dt\neq 0$ for each $x>0$, $Q(0)=0$ and $Q(x) \rightarrow \infty $ as $x \to \infty $.Given a real or complex-valued function $f \in L^{1}_{loc} (\mathbf{R}_{+})$, we define $s(x):=\int_{0}^{x}f(t)dt$ and$$\tau^{(0)}_q(x):=s(x), \tau^{(m)}_q(x):=\frac{1}{Q(x)}\int_0^x \tau^{(m-1)}_q(t) q(t)dt\,\,\, (x>0, m=1,2,...),$$provided that $Q(x)>0$. We say that $\int_{0}^{\infty}f(x)dx$ is summable to $L$ by the $m$-th iteration of weighted mean method determined by the function $q(x)$, or for short, $(\overline{N},q,m)$ integrable to a finite number $L$ if$$\lim_{x\to \infty}\tau^{(m)}_q(x)=L.$$In this case, we write $s(x)\rightarrow L(\overline{N},q,m)$. It is known thatif the limit $\lim _{x \to \infty} s(x)=L$ exists, then $\lim _{x \to \infty} \tau^{(m)}_q(x)=L$ also exists. However, the converse of this implicationis not always true. Some suitable conditions together with the existence of the limit $\lim _{x \to \infty} \tau^{(m)}_q(x)$, which is so called Tauberian conditions, may imply convergence of $\lim _{x \to \infty} s(x)$. In this paper, one- and two-sided Tauberian conditions in terms of the generating function and its generalizations for $(\overline{N},q,m)$ summable integrals of real- or complex-valued functions have been obtained. Some classical type Tauberian theorems given for Ces\`{a}ro summability $(C,1)$ and weighted mean method of summability $(\overline{N},q)$ have been extended and generalized.

The Role of Intelligent Organisations in Creating Favourable Conditions for the Development of Entrepreneurship

Purpose: The theoretical purpose of this study was to determine the essence of smart specialisation in local administrative units, together with the identification of its impact on the level of entrepreneurship and socio-economic development of spatial units. The empirical purpose was to indicate the relationship between the level of development of intelligent organisations (IO) in counties and the degree of entrepreneurship intensity. Design/Methodology/Approach: The paper uses the method of literature studies, the standardised sums method, the weighted correlation method and cartographic analysis. Findings: Intelligent specialisation in a local administrative unit (LAU) can contribute to the development of entrepreneurship and accelerate the socio-economic development of the area. Investment attractiveness is more important for stimulating the development of entrepreneurship than intelligent organisation in local administrative units at the county level in Poland. Practical Implications: Results of the research can be used by local administrative units to shape the development strategy of an intelligent organization as well as of a smart specialisation. They can provide valuable practical guidance and synergies. Originality/Value: The paper identifies key features of the smart specialization strategy. The paper defines the intelligent organizations development level index in local administrative units of the county level in Poland in 2018 based on a set of variables. A method of constructing typologies of local administrative units is based on indices determining the level of intelligent organization, investment attractiveness and entrepreneurship. The methodological concept can be used to assess the level of development of IOs in taxonomic units of different levels in different countries.

Korovkin type theorem for Bernstein–Kantorovich operators via power summability method

In this paper we will prove the Korovkin type theorem for Bernstein–Kantorovich type operators via $$A-$$ statistical convergence and power summability method. Also we give the rate of the convergence related to the above summability methods and in the last sections we give a kind of Voronovskaya type theorem for $$A-$$ statistical convergence and Gruss–Voronovskaya type theorem.

Hyperbolic rules of the cooperative organization of eukaryotic and prokaryotic genomes

The author's method of oligomer sums for analysis of oligomer compositions of eukaryotic and prokaryotic genomes is described. The use of this method revealed the existence of general rules for the cooperative oligomeric organization of a wide list of genomes. These rules are called hyperbolic because they are associated with hyperbolic sequences including the harmonic progression 1, 1/2, 1/3, .., 1/n. These rules are demonstrated by examples of quantitative analysis of many genomes from the human genome to the genomes of archaea and bacteria. The hyperbolic (harmonic) rules, speaking about the existence of algebraic invariants in full genomic sequences, are considered as candidates for the role of universal rules for the cooperative organization of genomes. The results concerns additionally the problem of the origin of life. The described phenomenological results were obtained as consequences of the previously published author's quantum-information model of long DNA sequences. The oligomer sums method was also applied to the analysis of long genes and viruses including the COVID-19 virus; this revealed, in characteristics of many of them, the phenomenon of such rhythmically repeating deviations from model hyperbolic sequences, which are associated with DNA triplets. In addition, an application of the oligomer sums method is shown to the analysis of amino acid sequences in long proteins like the protein Titin. The topics of the algebraic harmony in living bodies and of the quantum-information approach in biology are discussed.

Schur Lemma and Uniform Convergence of Series through Convergence Methods

In this note, we prove a Schur-type lemma for bounded multiplier series. This result allows us to obtain a unified vision of several previous results, focusing on the underlying structure and the properties that a summability method must satisfy in order to establish a result of Schur’s lemma type.

Characterization of Absolute and Uniform Continuity

In the present paper, by considering nonlinear integral operators and using their approximations via regular summability methods, we obtain characterizations for some function spaces including the space of absolutely continuous functions, the space of uniformly continuous functions, and their other variants. We observe that Bell-type summability methods are quite effective to generalize and improve some related results in the literature. At the end of the paper, we discuss some special cases and applications.

Matrix Application of Power Increasing Sequences to Infinite Series and Fourier Series

UDC 517.54 The aim of the paper is a generalization, under weaker conditions, of the main theorem on quasi-$\sigma$-power increasing sequences applied to $|A,\theta_{n}|_{k}$ summability factors of infinite series and Fourier series. We obtain some new and known results related to basic summability methods.

Regular methods of summability for direct sums of Banach spaces

Using Ramsey sets and the Galvin-Prikry partition theorem we show that the Banach-Saks property with respect to the regular positive matrix methods of summability, such as Cesàro, Nørlund or Hölder methods, is inherited by the countable direct sums of Banach spaces from the spaces forming this sum.

قابلية جمع المتسلسلات المتعامدة بالطريقة المصفوفية المطلقة

ظهرت في مطلع القرن العشرين نظرية المتسلسلات المتعامدة، التي تتعامل مع جمل من الدوال المتعامدة كتعميم طبيعي لنظرية المتسلسلات. وقد تمت دراسة قابلية جمع المتسلسلات المتعامدة بالعديد من الطرائق مثل طرائق سيزارو وهولدر وريمان وبوريل وهاوسدورف ونيورلند ونيورلند المعممة والطريقة المصفوفية. كما تم في هذا البحث دراسة قابلية جمع المتسلسلات المتعامدة البسيطة بطرائق مختلفة نوعاً ما تسمى الطرائق المطلقة والتي تعتمد على الفروق بشكل عام. وذلك من خلال الاعتماد على بعض الأبحاث والمقالات والكتب العلمية في مجال قابلية جمع المتسلسلات. إن الطريقة المصفوفية المطلقة تعد من أهم طرائق قابلية الجمع المطلقة، حيث إن معظم طرائق قابلية الجمع المطلقة، ما هي إلا حالات خاصة من الطريقة المصفوفية المطلقة. ونوصي أخيراً بدراسة الحالة المضاعفة.

Necessary and sufficient Tauberian conditions for weighted mean methods of summability in two-normed spaces

We first define the concept of weighted mean method of summability and then present necessary and sufficient Tauberian conditions for the weighted mean summability of sequences in two-normed spaces. As corollaries, we establish two-normed analogues of two classical Tauberian theorems.