Dedicated to the memory of our colleague Vasil Popov January 14, 1942 – May 31, 1990
* Partially supported by ISF-Center of Excellence, and by The Hermann Minkowski Center for Geometry at Tel Aviv University, Israel
Attempts at extending spline subdivision schemes to operate
on compact sets are reviewed. The aim is to develop a procedure for
approximating a set-valued function with compact images from a finite set of
its samples. This is motivated by the problem of reconstructing a...
We introduce Lipschitz continuity of set-valued maps with respect to a given set difference. The existence of Lipschitz selections that pass through any point of the graph of the map and inherit its Lipschitz constant is studied. We show that the Lipschitz property of the set-valued map with
respect to the Demyanov difference with a given constant is characterized
by the same property of its generalized Steiner selections. For a univariate
multifunction with only compact values in R^n, we characterize...
We introduce one-sided Lipschitz (OSL) conditions of setvalued maps with respect to given set differences. The existence of selections of such maps that pass through any point of their graphs and inherit uniformly their OSL constants is studied. We show that the OSL property of a convex-valued set-valued map with respect to the Demyanov difference with a given constant is characterized by the same property of the generalized Steiner selections. We prove that an univariate OSL map with compact images...
2000 Mathematics Subject Classification: 26E25, 41A35, 41A36, 47H04, 54C65.
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...
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...
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