The pointwise limit of metric integral operators approximating set-valued functions

Abstract: For set-valued functions (SVFs, multifunctions), mapping a compact interval $[a,b]$ into the space of compact non-empty subsets of ${\mathbb R}^d$, we study approximation based on the metric approach that includes metric linear combinations, metric selections and weighted metric integrals. In our earlier papers we considered convergence of metric Fourier approximations and metric adaptations of some classical integral approximating operators for SVFs of bounded variation with compact graphs. While the pointwise limit of a sequence of these approximants at a point of continuity $x$ of the set-valued function $F$ is $F(x)$, the limit set at a jump point was earlier described in terms of the metric selections of the multifunction. Here we show that, under certain assumptions on $F$, the limit set at $x$ equals the metric average of the left and the right limits of $F$ at $x$, thus extending the case of real-valued functions.