Abstract
In the present paper we study the so-called sampling Kantorovich operators in the very general setting of modular spaces. Here, modular convergence theorems are proved under suitable assumptions, together with a modular inequality for the above operators. Further, we study applications of such approximation results in several concrete cases, such as Musielak–Orlicz and Orlicz spaces. As a consequence of these results we obtain convergence theorems in the classical and weighted versions of the L^p and Zygmund (or interpolation) spaces. At the end of the paper examples of kernels for the above operators are presented.
Highlights
The theory of modular spaces, introduced by Nakano[59], was extensively studied by Musielak and Orlicz[56, 58], especially for what concerns the special cases of the so-called Orlicz and Musielak–Orlicz spaces
The main advantages of studying approximation results for the above operators in the setting of Orlicz spaces consists in the possibility of approximating not necessarily continuous functions
In order to reach a wide level of generality and to extend the above approximation theorems to a more general setting, in this paper we study the sampling Kantorovich operators in the general context of modular spaces
Summary
The theory of modular spaces, introduced by Nakano[59], was extensively studied by Musielak and Orlicz[56, 58], especially for what concerns the special cases of the so-called Orlicz and Musielak–Orlicz spaces. The main advantages of studying approximation results for the above operators in the setting of Orlicz spaces consists in the possibility of approximating not necessarily continuous functions This is strongly connected to the expression of Sw in which we have mean values of the function f on the (not necessarily spaced) intervals [tk∕w, tk+1∕w] , w > 0. In order to reach a wide level of generality and to extend the above approximation theorems to a more general setting (including e.g. the above mentioned weighted type spaces), in this paper we study the sampling Kantorovich operators in the general context of modular spaces. Standard assumptions are required on the modular which generates the spaces L , together with suitable compatibility conditions among the kernel and the modulars involved[55] In this setting, we firstly prove a modular convergence theorem in case of operators Swf acting on continuous functions f with compact support. We discuss about a number of kernel functions that satisfy the assumptions here required, such as the Fejér, the Jackson-type, the Bochner–Riesz, the central B-splines kernels, and many others (see e.g.,[5, 11, 24, 40])
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