Article
Kyungpook Mathematical Journal 2022; 62(3): 467-484
Published online September 30, 2022
Copyright © Kyungpook Mathematical Journal.
Some Approximation Results by Bivariate Bernstein-Kantorovich Type Operators on a Triangular Domain
Reşat Aslan∗ and Aydin Izgi
Department of Mathematics, Faculty of Sciences and Arts, Harran University, 63100 Şanlıurfa, Turkey
e-mail : resat63@hotmail.com and a_izgi@harran.edu.tr
Received: February 3, 2021; Revised: January 17, 2022; Accepted: January 24, 2022
In this work, we define bivariate Bernstein-Kantorovich type operators on a triangular domain and obtain some approximation results for these operators. We start off by computing some moment estimates and prove a Korovkin type convergence theorem. Then, we estimate the rate of convergence using the partial and complete modulus of continuity, and derive a Voronovskaya-type asymptotic theorem. Further, we calculate the order of approximation with regard to the Peetre's K-functional and a Lipschitz type class. In addition, we construct the associated GBS type operators and compute the rate of approximation using the mixed modulus of continuity and class of the Lipschitz of Bögel continuous functions for these operators. Finally, we use the two operators to approximate example functions in order to compare their convergence.
Keywords: Bernstein-Kantorovich operators, Modulus of continuity, Voronovskaya-type asymptotic theorem, Peetre’s K-functional, GBS type operators