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.

Keywords

• generalized Laguerre polynomials
• Korovkin theorem
• noncentral negative binomial
• probabilistic derivation
• Weierstrass approximation theorem
• Szász–Mirakyan operator