On the approximation of several modified Bernstein-type operators

In this paper, we first propose a new class of modified Bernstein Durrmeyer operators, which are independent of one endpoint value of any continuous function. We investigate their approximation rate, and obtain Voronovaskaja?s asymptotic estimation. Then we further introduce two classes of positive linear Bernstein-type operators, to study their approximation performance in both qualitative and quantitative ways. We compare the three mentioned operators, to explore their unique properties such as linearity, positivity, genuineness, and approximation performance both analytically and empirically.