# Approximation of Univariate Set-Valued Functions - an Overview

• Volume: 33, Issue: 4, page 495-514
• ISSN: 1310-6600

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## Abstract

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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 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.

## How to cite

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Dyn, Nira, Farkhi, Elza, and Mokhov, Alona. "Approximation of Univariate Set-Valued Functions - an Overview." Serdica Mathematical Journal 33.4 (2007): 495-514. <http://eudml.org/doc/281425>.

@article{Dyn2007,
abstract = {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 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.},
author = {Dyn, Nira, Farkhi, Elza, Mokhov, Alona},
journal = {Serdica Mathematical Journal},
keywords = {Compact Sets; Set-Valued Functions; Linear Approximation Operators; Minkowski Sum of Sets; Metric Average; Metric Linear Combinations},
language = {eng},
number = {4},
pages = {495-514},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Approximation of Univariate Set-Valued Functions - an Overview},
url = {http://eudml.org/doc/281425},
volume = {33},
year = {2007},
}

TY - JOUR
AU - Dyn, Nira
AU - Farkhi, Elza
AU - Mokhov, Alona
TI - Approximation of Univariate Set-Valued Functions - an Overview
JO - Serdica Mathematical Journal
PY - 2007
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 33
IS - 4
SP - 495
EP - 514
AB - 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 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.
LA - eng
KW - Compact Sets; Set-Valued Functions; Linear Approximation Operators; Minkowski Sum of Sets; Metric Average; Metric Linear Combinations
UR - http://eudml.org/doc/281425
ER -

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