Pointwise Approximation Theorems for Combinations and Derivatives of Bernstein Polynomials

We establish the pointwise approximation theorems for the combinations of Bernstein polynomials by the rth Ditzian–Totik modulus of smoothness \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \omega ^{r}_{\Phi } (f,t) $$\end{document}where Φ is an admissible step–weight function. An equivalence relation between the derivatives of these polynomials and the smoothness of functions is also obtained.