scholarly journals Statistical Riemann and Lebesgue Integrable Sequence of Functions with Korovkin-Type Approximation Theorems

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 229
Author(s):  
Hari Mohan Srivastava ◽  
Bidu Bhusan Jena ◽  
Susanta Kumar Paikray
Keyword(s):  
Limit Theorems ◽  
Bernstein Polynomials ◽  
Linear Operators ◽  
Test Functions ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Korovkin Type Approximation Theorems ◽  
Algebraic Test ◽  
Lebesgue Summability ◽  
Lebesgue Integrability

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based upon our proposed techniques, we establish the Korovkin-type approximation theorems with algebraic test functions. Finally, we present two illustrative examples under the consideration of positive linear operators in association with the Bernstein polynomials to exhibit the effectiveness of our findings.

Deferred weighted 𝒜-statistical convergence based upon the (p,q)-Lagrange polynomials and its applications to approximation theorems

2018 ◽  
Vol 24 (1) ◽  
pp. 1-16 ◽  
Author(s):  
H. M. Srivastava ◽  
Bidu Bhusan Jena ◽  
Susanta Kumar Paikray ◽  
U. K. Misra
Keyword(s):  
Statistical Convergence ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Lagrange Polynomials ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Special Cases ◽  
Korovkin Type Approximation Theorems ◽  
Appl Math

AbstractRecently, the notion of positive linear operators by means of basic (orq-) Lagrange polynomials and{\mathcal{A}}-statistical convergence was introduced and studied in [M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means ofq-Lagrange polynomials andA-statistical approximation, Appl. Math. Comput. 219 2013, 12, 6911–6918]. In our present investigation, we introduce a certain deferred weighted{\mathcal{A}}-statistical convergence in order to establish some Korovkin-type approximation theorems associated with the functions 1,tand{t^{2}}defined on a Banach space{C[0,1]}for a sequence of (presumably new) positive linear operators based upon{(p,q)}-Lagrange polynomials. Furthermore, we investigate the deferred weighted{\mathcal{A}}-statistical rates for the same set of functions with the help of the modulus of continuity and the elements of the Lipschitz class. We also consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.

scholarly journals Korovkin Second Theorem via -Statistical -Summability

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
M. Mursaleen ◽  
A. Kiliçman
Keyword(s):  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Real Interval ◽  
Test Functions ◽  
Approximation Process ◽  
The Real ◽  
Approximation Theorems ◽  
Whole Space ◽  
Korovkin Type Approximation Theorems

Korovkin type approximation theorems are useful tools to check whether a given sequence of positive linear operators on of all continuous functions on the real interval is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, , and in the space as well as for the functions 1, cos, and sin in the space of all continuous 2-periodic functions on the real line. In this paper, we use the notion of -statistical -summability to prove the Korovkin second approximation theorem. We also study the rate of -statistical -summability of a sequence of positive linear operators defined from into .

scholarly journals Korovkin type approximation theorems via lacunary equistatistical convergence

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3641-3647 ◽  
Author(s):  
Abdullah Alotaibi ◽  
M. Mursaleen
Keyword(s):  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Approximation Theorem ◽  
Test Functions ◽  
Central European ◽  
Korovkin Type Approximation Theorem ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Korovkin Type Approximation Theorems

Aktu?lu and H. Gezer [Central European J. Math. 7 (2009), 558-567] introduced the concepts of lacunary equistatistical convergence, lacunary statistical pointwise convergence and lacunary statistical uniform convergence for sequences of functions. In this paper, we apply the notion of lacunary equistatistical convergence to prove a Korovkin type approximation theorem by using test functions 1, x/1-x,(x/1-x)2.

scholarly journals Korovkin type approximation theorems proved viaweighted αβ-equistatistical convergence for bivariate functions

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6253-6266
Author(s):  
Hüseyin Aktuğlu ◽  
Halil Gezer
Keyword(s):  
Uniform Convergence ◽  
Statistical Convergence ◽  
Linear Operators ◽  
Extension Problem ◽  
Real Numbers ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Definition Of ◽  
Korovkin Type Approximation Theorems ◽  
Bivariate Functions

Statistical convergence was extended to weighted statistical convergence in [24], by using a sequence of real numbers sk, satisfying some conditions. Later, weighted statistical convergence was considered in [35] and [19] with modified conditions on sk. Weighted statistical convergence is an extension of statistical convergence in the sense that, for sk = 1, for all k, it reduces to statistical convergence. A definition of weighted ??-statistical convergence of order ?, considered in [25] does not have this property. To remove this extension problem the definition given in [25] needs some modifications. In this paper, we introduced the modified version of weighted ??-statistical convergence of order ?, which is an extension of ??-statistical convergence of order ?. Our definition, with sk = 1, for all k, reduces to ??-statistical convergence of order ?. Moreover, we use this definition of weighted ??-statistical convergence of order ?, to prove Korovkin type approximation theorems via, weighted ??-equistatistical convergence of order ? and weighted ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Also we prove Korovkin type approximation theorems via ??-equistatistical convergence of order ? and ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Some examples of positive linear operators are constructed to show that, our approximation results works, but its classical and statistical cases do not work. Finally, rates of weighted ??-equistatistical convergence of order ? is introduced and discussed.

scholarly journals ABSTRACT KOROVKIN THEOREMS VIA RELATIVE MODULAR CONVERGENCE FOR DOUBLE SEQUENCES OF LINEAR OPERATORS

2020 ◽  
pp. 561
Author(s):  
Sevda Yıldız ◽  
Kamil Demirci
Keyword(s):  
Positive Operators ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Double Sequences ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Modular Spaces ◽  
Abstract Version ◽  
Modular Convergence ◽  
Korovkin Type Approximation Theorems

We will obtain an abstract version of the Korovkin type approximation theorems with respect to the concept of statistical relative convergence in modular spaces for double sequences of positive linear operators. We will give an application showing that our results are stronger than classical ones. We will also study an extension to non-positive operators.

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 Approximation by Parametric Extension of Szász-Mirakjan-Kantorovich Operators Involving the Appell Polynomials

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Md. Nasiruzzaman ◽  
A. F. Aljohani
Keyword(s):  
Rate Of Convergence ◽  
Weighted Space ◽  
Modulus Of Smoothness ◽  
Linear Operators ◽  
Appell Polynomials ◽  
Global Approximation ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Dunkl Analogue ◽  
Kantorovich Operators

The purpose of this article is to introduce a Kantorovich variant of Szász-Mirakjan operators by including the Dunkl analogue involving the Appell polynomials, namely, the Szász-Mirakjan-Jakimovski-Leviatan-type positive linear operators. We study the global approximation in terms of uniform modulus of smoothness and calculate the local direct theorems of the rate of convergence with the help of Lipschitz-type maximal functions in weighted space. Furthermore, the Voronovskaja-type approximation theorems of this new operator are also presented.

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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