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AIMS Mathematics

2020, Volume 5, Issue 1: 650-672. doi: 10.3934/math.2020044
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Relatively equi-statistical convergence via deferred Nörlund mean based on difference operator of fractional-order and related approximation theorems

  • 1.
    Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India
  • 2.
    Department of General Required Courses, Mathematics, Faculty of Applied Studies, King A bdulaziz University, Jeddah 21589, Saudi Arabia
  • 3.
    Operator Theory and Applications Research Group, Department of Mathematics, King A bdulaziz University, Jeddah 21589, Saudi Arabia
  • 4.
    Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484 887, India
  • Received: 23 September 2019 Accepted: 29 November 2019 Published: 17 December 2019

  • MSC : 40A05, 41A36, 40G15

  • In the proposed paper, we have introduced the notion of point-wise relatively statistical convergence, relatively equi-statistical convergence and relatively uniform statistical convergence of sequences of functions based on the difference operator of fractional order including (p, q)-gamma function via the deferred Nörlund mean. As an application point of view, we have proved a Korovkin type approximation theorem by using the relatively deferred Nörlund equi-statistical convergence of difference sequences of functions and intimated that our theorem is a generalization of some well-established approximation theorems of Korovkin type which was presented in earlier works. Moreover, we estimate the rate of the relatively deferred Nörlund equi-statistical convergence involving a non-zero scale function. Furthermore, we use the modulus of continuity to estimate the rate of convergence of approximating positive linear operators. Finally, we set up various fascinating examples in connection with our results and definitions presented in this paper.

    Citation: B. B. Jena, S. K. Paikray, S. A. Mohiuddine, Vishnu Narayan Mishra. Relatively equi-statistical convergence via deferred Nörlund mean based on difference operator of fractional-order and related approximation theorems[J]. AIMS Mathematics, 2020, 5(1): 650-672. doi: 10.3934/math.2020044

    Related Papers:

  • In the proposed paper, we have introduced the notion of point-wise relatively statistical convergence, relatively equi-statistical convergence and relatively uniform statistical convergence of sequences of functions based on the difference operator of fractional order including (p, q)-gamma function via the deferred Nörlund mean. As an application point of view, we have proved a Korovkin type approximation theorem by using the relatively deferred Nörlund equi-statistical convergence of difference sequences of functions and intimated that our theorem is a generalization of some well-established approximation theorems of Korovkin type which was presented in earlier works. Moreover, we estimate the rate of the relatively deferred Nörlund equi-statistical convergence involving a non-zero scale function. Furthermore, we use the modulus of continuity to estimate the rate of convergence of approximating positive linear operators. Finally, we set up various fascinating examples in connection with our results and definitions presented in this paper.


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