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Set-valued Hermite interpolation

Title data

Baier, Robert ; Perria, Gilbert:
Set-valued Hermite interpolation.
In: Journal of Approximation Theory. Vol. 163 (2011) Issue 10 . - pp. 1349-1372.
ISSN 1096-0430
DOI: https://doi.org/10.1016/j.jat.2010.11.004

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Project information

Project financing: Bundesministerium für Bildung und Forschung

Abstract in another language

The problem of interpolating a set-valued function with convex images is addressed by means of directed sets. A directed set will be visualised as a usually non-convex set in |R^n consisting of three parts together with its normal directions: the convex, the concave and the mixed-type part. In this Banach space, a mapping resembling the Kergin map is established. The interpolating property and error estimates similar to the point-wise case are then shown; the representation of the interpolant through means of divided differences is given. A comparison to other set-valued approaches is presented. The method developed within the article is extended to the scope of the Hermite interpolation by using the derivative notion in the Banach space of directed sets. Finally, a numerical analysis of the explained technique corroborates the theoretical results.

Further data

Item Type: Article in a journal
Refereed: Yes
Additional notes: Zentralblattnummer: 1252.41002
Contents:
1. Introduction
2. Directed Sets
2.1 Preliminaries
2.2 Definition
2.3 Properties of Directed Sets
3. Set-Valued Derivatives and Divided Differences
4. The (Kergin) Interpolating Map
5. Connections to Other Approaches
6. Numerical Tests
7. Conclusions
Keywords: set-valued interpolation; Hermite interpolation; embedding of convex, compact sets; directed sets; derivatives of set-valued maps
Subject classification: Mathematics Subject Classification Code: 65D05 (41A05 54C60 26E25 46G05)
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 03 Mar 2021 13:26
Last Modified: 26 May 2021 12:47
URI: https://eref.uni-bayreuth.de/id/eprint/63651