Abstract
In this paper, Lupas, 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 < oo has been used. Advantages of using Riemann type (p, q)-integrals are discussed over general (p, q)-integrals. Lupas, 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 con -sider strictly monotonically increasing functions, and to handle this situation Lupas, Bernstein-Kantorovich operators are constructed using Riemann type (p, q)-integrals. However Lupas, (p, q)-Bernstein-Kantorovich operators based on Riemann type (p, q)-integrals are always positive linear operators. Approximation prop-erties 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.