Hermite Interpolation with Directed Sets

R. Baier, G. Perria: Hermite Interpolation with Directed Sets
University of Bonn, December 2008

Smart-Link: http://www.hausdorff-research-institute.uni-bonn.de/files/preprints/Baier_Perria_Hermite_Interpolation_HIM.pdf
Keywords: Hermite interpolation; derivatives of set-valued maps; divided differences; embedding of convex compact sets into a vector space
Mathematics Subject Classification Code: 65D05 (28B20 52A20 41A05 54C60 49J53)
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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 nonconvex set in |R^n consisting of three parts, the convex, the concave and the mixed-type part together with its normal directions. In this Banach space, a mapping resembling the Kergin map is established. The interpolating property and error estimates similar to the pointwise case are then shown based on the representation of the interpolant through means of divided differences. A comparison to other set-valued approaches is included. 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. Derivatives
4. Set-Valued Divided Differences
4.1 Hermite-Genocchi Formula
4.2 Estimates for the Divided Differences
4.3 Coinciding Points
5. The (Kergin) Interpolating Map
6. Connections to Other Approaches
7. Numerical Tests

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