ABSTRACT KOROVKIN THEOREMS VIA RELATIVE MODULAR CONVERGENCE FOR DOUBLE SEQUENCES OF LINEAR OPERATORS

We will obtain an abstract version of the Korovkin type approximation theorems with respect to the concept of statistical relative convergence in modular spaces for double sequences of positive linear operators. We will give an application showing that our results are stronger than classical ones. We will also study an extension to non-positive operators.

Convergence of sampling Kantorovich operators in modular spaces with applications

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 $$L^p$$ L p and Zygmund (or interpolation) spaces. At the end of the paper examples of kernels for the above operators are presented.

A General Approximation Approach for the Simultaneous Treatment of Integral and Discrete Operators

In this paper, we give a unitary approach for the simultaneous study of the convergence of discrete and integral operators described by means of a family of linear continuous functionals acting on functions defined on locally compact Hausdorff topological groups. The general family of operators introduced and studied includes very well-known operators in the literature. We give results of uniform convergence and modular convergence in the general setting of Orlicz spaces. The latter result allows us to cover many other settings as the {L^{p}}-spaces, the interpolation spaces, the exponential spaces and many others.

Approximation Results for a General Class of Kantorovich Type Operators

We introduce and study a family of integral operators in the Kantorovich sense acting on functions defined on locally compact topological groups. We obtain convergence results for the above operators with respect to the pointwise and uniform convergence and in the setting of Orlicz spaces with respect to the modular convergence. Moreover, we show how our theory applies to several classes of integral and discrete operators, as sampling, convolution and Mellin type operators in the Kantorovich sense, thus obtaining a simultaneous approach for discrete and integral operators. Further, we obtain general convergence results in particular cases of Orlicz spaces, as L