Lupaş Bernstein-Kantorovich operators using Jackson and Riemann type (p,q)-integrals

Mohammad Iliyas, Rameez A. Bhatt, Asif Khan, M. Mursaleen


In this paper, Lupaş Bernstein-Kantorovich operators have been studied using Jackson and Riemann type (p,q)-integrals. It has been shown that (p,q)-integrals as well as Riemann type (p,q)-integrals are not well defined for 0<q<p<1 and thus further analysis is needed. Throughout the paper, the case 1≤q<p<∞ has been used. Advantages of using Riemann type (p,q)-integrals are discussed over general (p,q)-integrals. Lupaş Bernstein-Kantorovich operators constructed via Jackson integral need not be positive for every f≥0. So to make these operators based on general (p,q)-integral positive, one need to consider strictly monotonically increasing functions, and to handle this situation Lupaş Bernstein-Kantorovich operators are constructed using Riemann type (p,q)-integrals. However Lupaş (p,q)-Bernstein-Kantorovich operators based on Riemann type (p,q)-integrals are always positive linear operators. Approximation properties for these operators based on Korovkin's type approximation theorem are investigated. The rate of convergence via modulus of continuity and function f belonging to the Lipschitz class is computed.


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