Compact Sets Set-Valued Functions Linear Approximation Operators Minkowski Sum of Sets Metric Average Metric Linear Combinations
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 33, No 4, (2007), 495p-514p
The paper is an updated survey of our work on the approximation of univariate set-valued functions by samples-based linear approximation operators, beyond the results reported in our previous overview. Our
approach is to adapt operators for real-valued functions to set-valued functions, by replacing operations between numbers by operations between sets.
For set-valued functions with compact convex images we use Minkowski convex combinations of sets, while for those with general compact images metric
averages and metric linear combinations of sets are used. We obtain general
approximation results and apply them to Bernstein polynomial operators,
Schoenberg spline operators and polynomial interpolation operators.