Nonlinear multivariate sampling Kantorovich operators: a study of their approximation properties in modular spaces

The goal of the present PhD thesis is to examine and analyze the convergence properties and the order of approximation of nonlinear multivariate sampling Kantorovich operators in Orlicz spaces, and to extend the convergence properties within the more general framework of modular spaces. The latter spaces have been firstly introduced in by Nakano as a generalization of Orlicz spaces, which, in turn, have been introduced as a natural extension of the classical Lebesgue spaces. Firstly, we deal with the problem of the order of approximation. In particular, we estimate the rate of convergence through both quantitative and qualitative analysis in the space of bounded and uniformly continuous functions and in the setting of Orlicz spaces. In this respect, a crucial role is played by the basical properties of the modulus of continuity and the modulus of smoothness, respectively. Further, we provide convergence result in the more general setting of modular spaces via a density approach. First we prove a modular convergence theorem, as well as a Luxemburg norm convergence result, for the nonlinear multivariate sampling Kantorovich operators acting on the space of continuous functions with compact support, then we obtain a modular-type inequality, and finally we exploit a well-known density result for the continuous function with compact support in the modular spaces. The choice to work within modular spaces is driven by the fact that they enable us to provide a unifying approach to several settings of approximation problems. In fact, modular spaces include Musielak-Orlicz spaces, which contain, for instance, weighted-Orlicz spaces and Orlicz spaces, as well as spaces of functions equipped by modulars that are not of integral type.