Comparison of Two-Parameter Bernstein Operator and Bernstein–Durrmeyer Variants

Abstract

The quantum calculus and the post-quantum calculus have recently gained broad popularity in computational science and engineering due to their applications to diverse areas such as solution of differential equations, approximation theory and computer-aided geometric design. Herein, we consider two parameters but two different modified Bernstein–Durrmeyer operators along with two-parameter Bernstein operator. We obtain estimates to the differences between the Bernstein operator and each modified Bernstein–Durrmeyer operator using classical modulus of continuity. In addition, similar estimates are obtained for Chebyshev functional of these operators. Main purpose of using two-parameter operators is to allow us more flexible approximations compared to their classical versions, namely depending on values of parameters, the approximation can be speeded up. Numerical results presented approves the theoretical results.