Uniform and Pointwise Quantitative Approximation by Kantorovich–Choquet Type Integral Operators with Respect to Monotone and Submodular Set Functions

Abstract

In this paper, for the univariate Bernstein–Kantorovich, Szász–Mirakjan–Kantorovich and Baskakov–Kantorovich operators written in terms of the Choquet integral with respect to a monotone and submodular set function, we obtain quantitative approximation estimates, uniform and pointwise in terms of the modulus of continuity. In addition, we show that for large classes of functions, the Kantorovich–Choquet type operators approximate better than their classical correspondents. Also, we construct new Szász–Mirakjan–Kantorovich–Choquet and Baskakov–Kantorovich–Choquet operators, which approximate uniformly f in each compact subinterval of $[0, +\infty )$ with the order $\omega _{1}(f; \sqrt{\lambda _{n}})$, where $\lambda _{n}\searrow 0$ arbitrary fast.