Applications of relative statistical convergence and associated approximation theorem

Lian-Ta Su; Kuldip Raj; Sonali Sharma; Qing-Bo Cai; Lian-Ta Su; Kuldip Raj; Sonali Sharma; Qing-Bo Cai

doi:10.3934/math.20221142

AIMS Mathematics

2022, Volume 7, Issue 12: 20838-20849. doi: 10.3934/math.20221142

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Applications of relative statistical convergence and associated approximation theorem

1.
Fujian Provincial Key Laboratory of Data-Intensive Computing, Fujian University Laboratory of Intelligent Computing and Information Processing, School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, Fujian, China
2.
School of Mathematics Shri Mata Vaishno Devi University, Katra 182320, J & K, India

Received: 13 July 2022 Revised: 14 October 2022 Accepted: 24 October 2022 Published: 02 November 2022
MSC : 40A05, 40A30

In this work, we investigate a new type of convergence known as relative statistical convergence through the use of the deferred Nörlund and deferred Riesz means. We demonstrate that the idea of deferred Nörlund and deferred Riesz statistically relative uniform convergence is significantly stronger than deferred Nörlund and deferred Riesz statistically uniform convergence. We provide some interesting examples which explain the validity of the theoretical results and effectiveness of constructed sequence spaces. Furthermore, as an application point of view we prove the Korovkin-type approximation theorem in the context of relative equi-statistical convergence for real valued functions and demonstrate that our theorem effectively extends and most of the earlier existing results. Finally, we present an example involving the Meyer-König-Zeller operator of real sequences proving that our theorem is a stronger approach than its classical and statistical version.
- statistical convergence,
- deferred Nörlund,
- deferred Riesz,
- relative uniform convergence,
- pointwise convergence
Citation: Lian-Ta Su, Kuldip Raj, Sonali Sharma, Qing-Bo Cai. Applications of relative statistical convergence and associated approximation theorem[J]. AIMS Mathematics, 2022, 7(12): 20838-20849. doi: 10.3934/math.20221142

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Abstract

In this work, we investigate a new type of convergence known as relative statistical convergence through the use of the deferred Nörlund and deferred Riesz means. We demonstrate that the idea of deferred Nörlund and deferred Riesz statistically relative uniform convergence is significantly stronger than deferred Nörlund and deferred Riesz statistically uniform convergence. We provide some interesting examples which explain the validity of the theoretical results and effectiveness of constructed sequence spaces. Furthermore, as an application point of view we prove the Korovkin-type approximation theorem in the context of relative equi-statistical convergence for real valued functions and demonstrate that our theorem effectively extends and most of the earlier existing results. Finally, we present an example involving the Meyer-König-Zeller operator of real sequences proving that our theorem is a stronger approach than its classical and statistical version.

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