Some Probabilistic Generalizations of the Cheney–Sharma and Bernstein Approximation Operators

Abstract

The objective of this paper is to give some probabilistic derivations of the Cheney, Sharma, and Bernstein approximation operators. Motivated by these probabilistic derivations, generalizations of the Cheney, Sharma, and Bernstein operators are defined. The convergence property of the Bernstein generalization is established. It is also shown that the Cheney–Sharma operator is the Szász–Mirakyan operator averaged by a certain probability distribution.

Some Functionals and Approximation Operators Associated with a Family of Discrete Probability Distributions

“…This enlarges the family of the investigated distributions and the area of applications involving these distributions. In our present investigation, we are motivated also by several related recent developments on approximation operators and probability distributions by (for example) Ong et al [2].…”