A Note on the Difference of Positive Operators and Numerical Aspects

Abstract

Recently, the differences between the two operators get the attention of scientists in approximation theory due to their ability to provide the approximation properties of the operator in the difference if the approximation properties of other operator in the difference are known. In other words, it gives us the ability to obtain a simultaneous approximation. On the other hand, the exponential-type operators possess better approximation properties than classical ones. Herein, the differences of the exponential-type Bernstein and Bernstein–Kantorovich operators and their differences between their higher order $$\mu $$-derivatives applied to a function with the operators applied to the same order of $$\mu $$-derivative of the function are considered. The estimates in the quantitative form are given in terms of the first modulus of continuity. Furthermore, quantitative estimates of the differences between Bernstein and Bernstein–Kantorovich operators as well as their Gruss-type difference are obtained. The numerical results obtained are in the direction of the theory, and some of them are presented.