Statistical product convergence of martingale sequences and its applications to Korovkin‐type approximation theorems

2021 ◽  
Author(s):  
Hari M. Srivastava ◽  
Bidu Bhusan Jena ◽  
Susanta Kumar Paikray
Keyword(s):  
Approximation Theorems ◽  
Type Approximation ◽  
Korovkin Type Approximation Theorems

scholarly journals Korovkin type approximation theorems on the disk algebra

2000 ◽  
Vol 29 (1) ◽  
pp. 103-117
Author(s):  
Go HIRASAWA ◽  
Keiji IZUCHI ◽  
Kazuhiro KASUGA
Keyword(s):  
Disk Algebra ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Korovkin Type Approximation Theorems

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 Dunkl-type generalization of the second kind beta operators via $(p,q)$-calculus

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Md. Nasiruzzaman ◽  
Abdullah Alotaibi ◽  
M. Mursaleen
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Lipschitz Class ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Research Article ◽  
Local Approximations ◽  
Korovkin Type Approximation Theorems

AbstractThe main purpose of this research article is to construct a Dunkl extension of $(p,q)$ ( p , q ) -variant of Szász–Beta operators of the second kind by applying a new parameter. We obtain Korovkin-type approximation theorems, local approximations, and weighted approximations. Further, we study the rate of convergence by using the modulus of continuity, Lipschitz class and Peetre’s K-functionals.

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.

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.

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