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.

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.

On Strong Matrix Summability with Respect to a Modulus and Statistical Convergence

1989 ◽  
Vol 32 (2) ◽  
pp. 194-198 ◽  
Author(s):  
Jeff Connor
Keyword(s):  
Statistical Convergence ◽  
Summability Method ◽  
Regular Matrix ◽  
Cesàro Summability ◽  
Matrix Summability ◽  
Cesaro Summability ◽  
Definition Of ◽  
Bounded Sequences

AbstractThe definition of strong Cesaro summability with respect to a modulus is extended to a definition of strong A -summability with respect to a modulus when A is a nonnegative regular matrix summability method. It is shown that if a sequence is strongly A-summable with respect to an arbitrary modulus then it is A-statistically convergent and that Astatistical convergence and strong A-summability with respect to a modulus are equivalent on the bounded sequences.

A new factor theorem on absolute matrix summability method

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Şebnem Yıldız
Keyword(s):  
Infinite Series ◽  
Summability Method ◽  
Summability Factor ◽  
Matrix Summability ◽  
Absolute Matrix Summability ◽  
Matrix Generalization ◽  
Summability Methods ◽  
Power Increasing Sequences ◽  
Factor Theorem ◽  
Increasing Sequences

Abstract In this paper, we have a new matrix generalization with absolute matrix summability factor of an infinite series by using quasi-β-power increasing sequences. That theorem also includes some new and known results dealing with some basic summability methods

scholarly journals Inclusion results for convolution submethods

2004 ◽  
Vol 2004 (2) ◽  
pp. 55-64
Author(s):  
Jeffrey A. Osikiewicz ◽  
Mohammad K. Khan
Keyword(s):  
Summability Method ◽  
Borel Summability ◽  
Matrix Summability ◽  
Summability Methods ◽  
The Matrix ◽  
Summability Matrix ◽  
Equivalence Result ◽  
Inclusion Results

IfBis a summability matrix, then the submethodBλis the matrix obtained by deleting a set of rows from the matrixB. Comparisons between Euler-Knopp submethods and the Borel summability method are made. Also, an equivalence result for convolution submethods is established. This result will necessarily apply to the submethods of the Euler-Knopp, Taylor, Meyer-König, and Borel matrix summability methods.

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 Absolute matrix summability methods

2011 ◽  
Vol 24 (12) ◽  
pp. 2102-2106 ◽  
Author(s):  
H.S. Özarslan ◽  
T. Ari
Keyword(s):  
Matrix Summability ◽  
Absolute Matrix Summability ◽  
Summability Methods

Summability of Matrix Transforms of Subsequences

1981 ◽  
Vol 24 (3) ◽  
pp. 359-364
Author(s):  
Thomas A. Keagy
Keyword(s):  
Limit Point ◽  
Regularity Condition ◽  
Summability Method ◽  
Finite Limit ◽  
Regular Matrix ◽  
Matrix Summability ◽  
Matrix Transforms

AbstractD. F. Dawson has considered several questions of the following nature. Suppose T is a regular matrix summability method. If A is a regular matrix and x is a sequence having a finite limit point, then there exists a subsequence y of x such that each finite limit point of x is a T-limit point of Ay. In the present paper, we show the regularity condition for A may be replaced by the requirement that A be a limit preserving bv to c map. This leads to summability characterizations for several classes of sequences.

Submethods Of Regular Matrix Summability Methods

1956 ◽  
Vol 8 ◽  
pp. 40-46 ◽  
Author(s):  
Casper Goffman ◽  
G. M. Petersen
Keyword(s):  
Matrix Method ◽  
Regular Matrix ◽  
Matrix Summability ◽  
Summability Methods ◽  
First Category ◽  
The Matrix

By a submethod of a regular matrix method A we mean a method (see 1 or 3) whose matrix is obtained by deleting a set of rows from the matrix A. We establish a one-one correspondence between the submethods of A and the points of the interval 0 < ξ ≤We designate the submethod which corresponds to 𝞷 by A (𝞷) and are accordingly able to speak of sets of submethods of measure 0, of the first category, etc.

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