Quantitative approximation by nonlinear Angheluta-Choquet singular integrals

Abstract

By using the concept of nonlinear Choquet integral with respect to a capacity and as a generalization of the Poisson-Cauchy-Choquet operators, we introduce the nonlinear Angheluta-Choquet singular integrals with respect to a family of submodular set functions. Quantitative approximation results in terms of the modulus of continuity are obtained with respect to some particular possibility measures and with respect to the Choquet measure \(\mu(A)=\sqrt{M(A)}\), where \(M\) represents the Lebesgue measure. For some subclasses of functions we prove that these Choquet type operators can have essentially better approximation properties than their classical correspondents. The paper ends with the important, independent remark that for Choquet-type operators which are comonotone additive too, like Kantorovich-Choquet operators, Szasz-Mirakjan-Kantorovich-Choquet operators and Baskakov-Kantorovich-Choquet operators studied in previous papers, the approximation results remain identically valid not only for non-negative functions, but also for all functions which take negative values too, if they are lower bounded.