A survey on recent results in Korovkin’s approximation theory in modular spaces

Abstract

In this paper, we give a survey about recent versions of Korovkin-type theorems for modular function spaces, a class which includes L p , Orlicz, Musielak-Orlicz spaces and many others. We consider various kinds of modular convergence, using certain summability processes, like triangular matrix statistical convergence, and filter convergence (which are generalizations of the statistical convergence). Finally, we consider an abstract axiomatic convergence which includes the previous ones and even almost convergence, which is not generated by any filter, as we show by an example.

A note on convergence results for varying interval valued multisubmeasures

“…Clearly, the above numerical discretization can be viewed as a kind of approximation of a suitable sequence of interval valued multifunctions G n (as happens in case of real valued functions, see e.g., [29,40,42]) which converges (in some sense) to a (multi-)signal G corresponding to the original analogue signal. Obviously, the interval valued multifunctions G n are discontinuous (see, e.g., [1,2,4,5]) hence a notion of convergence under a suitable definition of integrals of interval valued multifunctions is needed. Finally the convergence of measures is also used for example in [3] in order to obtain an approximation scheme for the Kantorovich-Rubinstein transportation problem.…”