A direct theorem for MKZ-Kantorovich operator (Q2416485)

In the literature there exist many modifications of the classical Bernstein operators and the classical Meyer-König and Zeller operators, aiming to approximate functions in the $L_p$ norm. The most important ones are the Durrmeyer- and Kantorovich-type modifications. Several direct and converse inequalities for the Kantorovich modifications of the Bernstein operators are known. \textit{V. Totik} obtained direct and converse results for a Kantorovich modification of the Meyer-König and Zeller operators in [Math. Z. 182, 425--446 (1983; Zbl 0502.41006)]. Another such modification was introduced by \textit{M. W. Müller} in [Studia Math. 63, 81--88 (1978; Zbl 0389.41009)]. The approximation of functions in $L_p$ norm using these operators is studied. By defining a new $K$-functional, the author proves a direct inequality. The set of functions that can be approximated in this context is essentially wider than $L_p[0,1]$ for which it is possible to use the Kantorovich modification of Bernstein operators.