Quantitative Approximation by Nonlinear Picard–Choquet, Gauss–Weierstrass–Choquet and Poisson–Cauchy–Choquet Singular Integrals

Abstract

By using the concept of Choquet nonlinear integral with respect to a submodular set function, we introduce the nonlinear Picard–Choquet operators, Gauss–Weierstrass–Choquet operators and Poisson–Cauchy–Choquet operators with respect to a family of submodular set functions. Quantitative approximation results of the order $\omega _{1}(f; t)_{\mathbb {R}}$, $t>0$, are obtained with respect to the Choquet measure $\mu (A)=\sqrt{M(A)}$ where M represents the Lebesgue measure, and with respect to some families of possibility measures. Also, due to the possibilities of choice for the submodular set functions, for some subclasses of functions we prove that these Choquet type operators have essentially better approximation properties than their classical correspondents.