# Approximation of Univariate Set-Valued Functions - an Overview

Dyn, Nira; Farkhi, Elza; Mokhov, Alona

Serdica Mathematical Journal (2007)

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

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topDyn, 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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