On the rate of convergence of modified \(\alpha\)-Bernstein operators based on q-integers

Purshottam Agrawal; Dharmendra Kumar; Behar Baxhaku

doi:10.33993/jnaat511-1244

Authors

Purshottam Agrawal Indian Institute of Technology Roorkee, India
Dharmendra Kumar Indian Institute of Technology Roorkee, India
Behar Baxhaku University of Prishtina "Hasan Prishtina", Albania

DOI:

https://doi.org/10.33993/jnaat511-1244

Keywords:

Steklov mean, Peetre's K-functional, modulus of continuity, Lipschitz class, Voronovskaja type theorem, Gruss-Voronovskaja type theorem, mixed modulus of smoothness

Abstract views: 429

Abstract

In the present paper we define a q-analogue of the modified a-Bernstein operators introduced by Kajla and Acar (Ann. Funct. Anal. 10 (4) 2019, 570-582). We study uniform convergence theorem and the Voronovskaja type asymptotic theorem. We determine the estimate of error in the approximation by these operators by virtue of second order modulus of continuity via the approach of Steklov means and the technique of Peetre's \(K\)-functional. Next, we investigate the Gruss- Voronovskaya type theorem. Further, we define a bivariate tensor product of these operatos and derive the convergence estimates by utilizing the partial and total moduli of continuity. The approximation degree by means of Peetre's K- functional , the Voronovskaja and Gruss Voronovskaja type theorems are also investigated. Lastly, we construct the associated GBS (Generalized Boolean Sum) operator and examine its convergence behavior by virtue of the mixed modulus of smoothness.

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