Approximation properties of generalized Baskakov operators

Purshottam Narain Agrawal; Behar Baxhaku; Abhishek Kumar; Purshottam Narain Agrawal; Behar Baxhaku; Abhishek Kumar

doi:10.3934/math.2021410

AIMS Mathematics

2021, Volume 6, Issue 7: 6986-7016. doi: 10.3934/math.2021410

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Research article

Approximation properties of generalized Baskakov operators

1.
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
2.
Department of Mathematics, University of Prishtina "Hasan Prishtina", Prishtina, Kosovo

Received: 20 January 2021 Accepted: 08 April 2021 Published: 26 April 2021
MSC : 41A25, 41A36

The present article is a continuation of the work done by Aral and Erbay ^[1]. We discuss the rate of convergence of the generalized Baskakov operators considered in the above paper with the aid of the second order modulus of continuity and the unified Ditzian Totik modulus of smoothness. A bivariate case of these operators is also defined and the degree of approximation by means of the partial and total moduli of continuity and the Peetre's K-functional is studied. A Voronovskaya type asymptotic result is also established. Further, we construct the associated Generalized Boolean Sum (GBS) operators and investigate the order of convergence with the help of mixed modulus of smoothness for the Bögel continuous and Bögel differentiable functions. Some numerical results to illustrate the convergence of the above generalized Baskakov operators and its comparison with the GBS operators are also given using Matlab algorithm.
- Voronovskaya theorem,
- Ditzian-Totik modulus of smoothness,
- Peetre's K-functional,
- Bögel continuous function,
- Bögel differentiable function
Citation: Purshottam Narain Agrawal, Behar Baxhaku, Abhishek Kumar. Approximation properties of generalized Baskakov operators[J]. AIMS Mathematics, 2021, 6(7): 6986-7016. doi: 10.3934/math.2021410

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Abstract

The present article is a continuation of the work done by Aral and Erbay ^[1]. We discuss the rate of convergence of the generalized Baskakov operators considered in the above paper with the aid of the second order modulus of continuity and the unified Ditzian Totik modulus of smoothness. A bivariate case of these operators is also defined and the degree of approximation by means of the partial and total moduli of continuity and the Peetre's K-functional is studied. A Voronovskaya type asymptotic result is also established. Further, we construct the associated Generalized Boolean Sum (GBS) operators and investigate the order of convergence with the help of mixed modulus of smoothness for the Bögel continuous and Bögel differentiable functions. Some numerical results to illustrate the convergence of the above generalized Baskakov operators and its comparison with the GBS operators are also given using Matlab algorithm.

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