Sahand Communications in Mathematical Analysis

Korovkin-type Theorems via Statistical Derivatives of Deferred Nörlund Summability

Document Type : Research Paper

Authors

Suresh Chandra Mahapatra 1
Bidu Bhusan Jena 1
Susanta Kumar Paikray 2
M. Mursaleen 3, 4, 5

1 Faculty of Science (Mathematics), Sri Sri University, Cuttack 754006, Odisha, India.

2 Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India.

3 Department of Mathematical Sciences, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamilnadu, India.

4 Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan.

5 Department of Mathematics, Uşak University University, 64000 Uşak, Turkey.

https://doi.org/10.22130/scma.2025.2070789.2316
Abstract
This paper introduces and explores the concept of statistical derivatives within the framework of deferred N\"{o}rlund summability, complemented by illustrative examples. Leveraging this approach, we establish a new Korovkin-type theorem for a specific class of algebraic test functions, namely $1$, $x$ and $x^{2}$, within the Banach space $\mathfrak{C}[0,1]$. Our findings serve as a significant generalization of several classical and statistical Korovkin-type results in approximation theory. Furthermore, we examine the rate of convergence associated with statistical derivatives under deferred N\"{o}rlund summability, providing insights into the effectiveness of this summability method. To validate our theoretical results, we present numerical examples alongside graphical visualizations created using MATLAB, offering a clearer perspective on the convergence behavior of the proposed operators.

Keywords

Statistical derivative, Deferred Nö
rlund mean Korovkin-type theorem, Bernstein polynomials, Rate of convergence

Subjects

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Sahand Communications in Mathematical Analysis
Volume 23, Issue 1
Winter 2026
Pages 219-247

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