scholarly journals Comparison of strong and statistical convergences in some families of summability methods

Filomat ◽  
2014 ◽  
Vol 28 (6) ◽  
pp. 1225-1236
Author(s):  
Anna Seletski ◽  
Anne Tali
Keyword(s):  
Strong Convergence ◽  
Statistical Convergence ◽  
Strong Summability ◽  
Convexity Theorem ◽  
Summability Methods

The paper deals with certain families {A?}(?>?0) of summability methods. Strong and statistical convergences in Ces?ro- and Euler-Knopp-type families {A?} are investigated. Convergence of a sequence x = (xn) with respect to the different strong summability methods [A?+1]t (with positive exponents t = (tn)) in a family {A?} is compared, and characterized with the help of statistical convergence. A convexity theorem for comparison of three strong summability methods [A?+1]t, [A?+1]t and [A?+1]t (? > ? > ? > ?0) in a Ces?ro-type family {A?} is proved. This theorem can be seen as a generalization of some convexity theorems known earlier. Interrelations between strong convergence and certain statistical convergence are also studied and described with the help of theorems proved here. All the results can be applied to the families of generalized N?rlund methods (N, p?n, qn).

scholarly journals I-lacunary summability function of order α

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 69-76
Author(s):  
Ekrem Savaş
Keyword(s):  
Measurable Function ◽  
Statistical Convergence ◽  
Open Problems ◽  
Summability Methods

In this paper, we further generalize recently introduced summability methods in [23](where ideals of N were used to extend certain important summability methods) and introduce new notions, namely, I-statistical convergence of order ?, where 0 < ? < 1 by taking nonnegative real-valued Lebesque measurable function in the interval (1,?). We mainly investigate their relationship and also make some observations about these classes. The study leaves a lot of interesting open problems

scholarly journals Orlicz–Pettis Theorem through Summability Methods

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 895 ◽  
Author(s):  
Fernando León-Saavedra ◽  
María del Pilar Romero de la Rosa ◽  
Antonio Sala
Keyword(s):  
Statistical Convergence ◽  
Summability Method ◽  
Matrix Summability ◽  
Summability Methods

This paper unifies several versions of the Orlicz–Pettis theorem that incorporate summability methods. We show that a series is unconditionally convergent if and only if the series is weakly subseries convergent with respect to a regular linear summability method. This includes results using matrix summability, statistical convergence with respect to an ideal, and other variations of summability methods.

scholarly journals Statistical Summability through de la Vallée-Poussin Mean in Probabilistic Normed Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ayhan Esi
Keyword(s):  
Statistical Convergence ◽  
The Other ◽  
Normed Spaces ◽  
Summability Methods ◽  
Probabilistic Normed Spaces ◽  
New Type ◽  
De La Vallée Poussin

Two concepts—one of statistical convergence and the other of de la Vallée-Poussin mean—play an important role in recent research on summability theory. In this work we define a new type of summability methods and statistical completeness involving the ideas of de la Vallée-Poussin mean and statistical convergence in the framework of probabilistic normed spaces.

scholarly journals A bounded consistency theorem for strong summabilities

1989 ◽  
Vol 12 (1) ◽  
pp. 39-46 ◽  
Author(s):  
C. S. Chun ◽  
A. R. Freedman
Keyword(s):  
Strong Summability ◽  
Regular Matrix ◽  
Matrix Methods ◽  
Summability Methods ◽  
Summability Field

The study of R-type summability methods is continued in this paper by showing that two such methods are identical on the bounded portion of the strong summability field associated with the methods. It is shown that this “bounded consistency” applies for many non-matrix methods as well as for regular matrix methods.

scholarly journals A Link between Approximation Theory and Summability Methods via Four-Dimensional Infinite Matrices

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1895
Author(s):  
Hari M. Srivastava ◽  
Khursheed J. Ansari ◽  
Faruk Özger ◽  
Zeynep Ödemiş Özger
Keyword(s):  
Computer Graphics ◽  
Power Series ◽  
Approximation Theory ◽  
Statistical Convergence ◽  
Series Method ◽  
Power Series Method ◽  
Summability Methods ◽  
Korovkin Type Theorems ◽  
Convergence Type ◽  
Theoretical Results

In this study, we present a link between approximation theory and summability methods by constructing bivariate Bernstein-Kantorovich type operators on an extended domain with reparametrized knots. We use a statistical convergence type and power series method to obtain certain Korovkin type theorems, and we study certain rates of convergences related to these summability methods. Furthermore, we numerically analyze the theoretical results and provide some computer graphics to emphasize the importance of this study.

On I-lacunary statistical convergence of weight g of sequences of sets

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5315-5322 ◽  
Author(s):  
Ekrem Savaş
Keyword(s):  
Statistical Convergence ◽  
Open Problems ◽  
New Approach ◽  
Summability Methods ◽  
Lacunary Statistical Convergence

In this paper, following a very recent and new approach of [1], we further generalize recently introduced summability methods in [13] and introduce new notions, namely, I-statistical convergence of weight 1 and I-lacunary statistical convergence of weight g, where g : N ? [0,?) is a function satisfying lim n?? g(n) = ? and n/g(n) ?? 0 as n ? ?, for sequences of sets. We mainly investigate their relationship and also make some observations about these classes. The study leaves a lot of interesting open problems.

Statistical convergence in a locally convex space

1988 ◽  
Vol 104 (1) ◽  
pp. 141-145 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Convex Space ◽  
Statistical Convergence ◽  
Present Note ◽  
Locally Convex Space ◽  
Strong Summability ◽  
Modulus Function ◽  
Local Convexity ◽  
Complex Sequences ◽  
Topological Linear ◽  
Locally Convex

The notion of statistical convergence was introduced by Fast[1] and has been investigated in a number of papers[2, 5, 6]. Recently, Fridy [2] has shown that k(xk–xk+l) = O(1) is a Tauberian condition for the statistical convergence of (xk). Existing work on statistical convergence appears to have been restricted to real or complex sequences, but in the present note we extend the idea to apply to sequences in any locally convex Hausdorif topological linear space. Also we obtain a representation of statistical convergence in terms of strong summability given by a modulus function, an idea recently introduced in Maddox [3, 4]. Moreover Fridy's Tauberian result is extended so as to apply to sequences of slow oscillation in a locally convex space, and we also examine the local convexity of w(f) spaces.

Some properties of modified Szász–Mirakyan operators in polynomial spaces via the power summability method

2020 ◽  
Vol 26 (1) ◽  
pp. 79-90 ◽  
Author(s):  
Naim L. Braha
Keyword(s):  
Statistical Convergence ◽  
Type Theorem ◽  
Summability Method ◽  
Korovkin Type Theorem ◽  
Summability Methods ◽  
Voronovskaya Type Theorem

AbstractIn this paper we will prove the Korovkin type theorem for modified Szász–Mirakyan operators via A-statistical convergence and the power summability method. Also we give the rate of the convergence related to the above summability methods, and in the last section, we give a kind of Voronovskaya type theorem for A-statistical convergence and Grüss–Voronovskaya type theorem.

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