On approximation by some Bernstein–Kantorovich exponential-type polynomials

Abstract

Since the introduction of Bernstein operators, many authors defined and/or studied Bernstein type operators and their generalizations, among them are Morigi and Neamtu (Adv Comput Math 12:133–149, 2000). They proposed an analog of classical Bernstein operator and proved some convergence results for continuous functions. Herein, we introduce their integral extensions in Kantorovich sense by replacing the usual differential and integral operators with their more general analogues. By means of these operators, we are able to reconstruct the functions which are not necessarily continuous. It is shown that the operators form an approximation process in both $$C\left[ 0,1\right] $$ and $$L_{p,\mu }\left[ 0,1\right] $$ , which is an exponentially weighted space. Also, quantitative results are stated in terms of appropriate moduli of smoothness and K-functionals. Furthermore, a quantitative Voronovskaya type result is presented.