scholarly journals Statistical σ-convergence of double sequences with application

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2783-2792
Author(s):  
M. Mursaleen ◽  
Cemal Belen ◽  
Syed Rizvi
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Invariant Mean ◽  
Double Sequences ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Functions Of Two Variables

The concepts of ?-statistical convergence, statistical ?-convergence and strong ?q-convergence of single (ordinary) sequences have been introduced and studied in [M. Mursaleen, O.H.H. Edely, On the invariant mean and statistical convergence, App. Math. Lett. 22, (2011), 1700-1704] which were obtained by unifying the notions of density and invariant mean. In this paper, we extend these ideas from single to double sequences. We also use the concept of statistical ?-convergence of double sequences to prove a Korovkin-type approximation theorem for functions of two variables and give an example to show that our Korovkin-type approximation theorem is stronger than those proved earlier by other authors.

scholarly journals Generalized Weighted Statistical Convergence for Double Sequences of Fuzzy Numbers and Associated Korovkin-Type Approximation Theorem

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Abdullah Alotaibi
Keyword(s):  
Statistical Convergence ◽  
Fuzzy Numbers ◽  
Approximation Theorem ◽  
Bernstein Operators ◽  
Double Sequences ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Functions Of Two Variables ◽  
Sequences Of Fuzzy Numbers ◽  
Bivariate Functions

We define the notions of weighted λ,μ-statistical convergence of order γ1,γ2 and strongly weighted λ,μ-summability of γ1,γ2 for fuzzy double sequences, where 0<γ1,γ2≤1. We establish an inclusion result and a theorem presenting a connection between these concepts. Moreover, we apply our new concept of weighted λ,μ-statistical convergence of order γ1,γ2 to prove Korovkin-type approximation theorem for functions of two variables in a fuzzy sense. Finally, an illustrative example is provided with the help of q-analogue of fuzzy Bernstein operators for bivariate functions which shows the significance of our approximation theorem.

scholarly journals Korovkin-Type Theorems for ModularΨ-A-Statistical Convergence

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Carlo Bardaro ◽  
Antonio Boccuto ◽  
Kamil Demirci ◽  
Ilaria Mantellini ◽  
Sevda Orhan
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Orlicz Spaces ◽  
Double Sequences ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Modular Spaces ◽  
New Type ◽  
Korovkin Type Theorems

We deal with a new type of statistical convergence for double sequences, calledΨ-A-statistical convergence, and we prove a Korovkin-type approximation theorem with respect to this type of convergence in modular spaces. Finally, we give some application to moment-type operators in Orlicz spaces.

scholarly journals Approximation Theorems for Functions of Two Variables viaσ-Convergence

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Mohammed A. Alghamdi
Keyword(s):  
Approximation Theorem ◽  
Bernstein Polynomials ◽  
Test Functions ◽  
Double Sequences ◽  
Korovkin Type Approximation Theorem ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Functions Of Two Variables

Çakan et al. (2006) introduced the concept ofσ-convergence for double sequences. In this work, we use this notion to prove the Korovkin-type approximation theorem for functions of two variables by using the test functions 1,x,y, andx2+y2and construct an example by considering the Bernstein polynomials of two variables in support of our main result.

scholarly journals Statistical Approximation for Periodic Functions of Two Variables

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Abdullah Alotaibi ◽  
M. Mursaleen ◽  
S. A. Mohiuddine
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Periodic Functions ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Functions Of Two Variables ◽  
Function Of Two Variables

We prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical summability(C,1,1). We also study the rate of statistical summability(C,1,1)of positive linear operators. Finally we construct an example to show that our result is stronger than those previously proved for Pringsheim's convergence and statistical convergence.

On αβ-statistical convergence for sequences of fuzzy mappings and Korovkin type approximation theorem

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3749-3760 ◽  
Author(s):  
Ali Karaisa ◽  
Uğur Kadak
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Bernstein Operator ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Inclusion Relations ◽  
Prior Investigation

Upon prior investigation on statistical convergence of fuzzy sequences, we study the notion of pointwise ??-statistical convergence of fuzzy mappings of order ?. Also, we establish the concept of strongly ??-summable sequences of fuzzy mappings and investigate some inclusion relations. Further, we get an analogue of Korovkin-type approximation theorem for fuzzy positive linear operators with respect to ??-statistical convergence. Lastly, we apply fuzzy Bernstein operator to construct an example in support of our result.

scholarly journals Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 636 ◽  
Author(s):  
Hari Mohan Srivastava ◽  
Bidu Bhusan Jena ◽  
Susanta Kumar Paikray
Keyword(s):  
Banach Space ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Real Sequence ◽  
Test Functions ◽  
Real Numbers ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Algebraic Test

The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund statistical convergence and the statistical deferred Nörlund summability for sequences of real numbers defined over a Banach space. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of real numbers on a Banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results (in classical and statistical versions). Finally, we have presented an example involving the generalized Meyer–König and Zeller operators of a real sequence demonstrating that our theorem is a stronger approach than its classical and statistical versions.

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