Set-Valued Hermite Interpolation

R. Baier, G. Perria: Set-Valued Hermite Interpolation
Journal of Approximation Theory 163 (10), 1349 - 1372, 2011

DOI: 10.1016/j.jat.2010.11.004
MR Nummer: 2832730
Zentralblattnummer: 1252.41002
Keywords: set-valued interpolation; Hermite interpolation; embedding of convex, compact sets; directed sets; derivatives of set-valued maps
Mathematics Subject Classification Code: 65D05 (41A05 54C60 26E25 46G05)
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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.


1. Introduction
2. Directed Sets
2.1 Preliminaries
2.2 Definition of Directed Sets
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

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