Approximation of Set-valued Functions with Compact Images—an Overview

Continuous set-valued functions with convex images can be approximated by known positive operators of approximation, such as the Bernstein polynomial operators and the Schoenberg spline operators, with the usual sum between numbers replaced by the Minkowski sum of sets. Yet these operators fail to approximate set-valued functions with general sets as images. The Bernstein operators with growing degree, and the Schoenberg operators, when represented as spline subdivision schemes, converge to set-valued functions with convex images. To obtain approximating operators for set-valued functions with general images, we use a binary average between sets, termed the “metric average”, which was introduced by Artstein for the construction of piecewise linear interpolants to set-valued functions. Representing each of the above mentioned operators in terms of repeated binary averages, and replacing the binary average between numbers by the metric average, we obtain operators for set-valued functions with compact images. In case of the Schoenberg operators, represented either by the de Boor algorithm or by spline subdivision schemes, the operators are approximating. In case of the Bernstein operators, the convergence with the increase of the degree is established only for set-valued functions with 1D images, consisting of the same number of intervals.