IJPAM: Volume 65, No. 4 (2010)

SMOOTH CONVEX RESOLUTION OF UNITY AND/OR
SCATTERED-POINT HERMITE INTERPOLATION
BY GENERALIZED EXPO-RATIONAL B-SPLINES, II:
STAR-SHAPED COVERS

Lubomir T. Dechevsky
R&D Group for Mathematical Modelling
Numerical Simulation and Computer Visualization
Faculty of Technology
Narvik University College
2, Lodve Lange's Str., P.O. Box 385, N-8505, Narvik, NORWAY
e-mail: [email protected]
url: http://ansatte.hin.no/ltd/


Abstract.In [#!d-2008!#] a hierarchy of several general constructions of smooth convex resolutions of unity over general partitions of multidimensional domains in $\R^n$, $n\in\N$, was proposed. When the basis functions generating these resolution are not constants but Taylor polynomials centered at the 'peaks' of the basis functions, the linear combination of these basis functions, with this type of functional coefficients, exhibits Hermite interpolation at the 'peaks' and has good approximation properties. These new constructions are all based on generalized expo-rational B-splines (GERBS) [#!d-2008!#,#!d-2009!#]. In a sequence of several papers, of which this is the second one, the essentials of these constructions is provided. Rather than trying to elaborate all details, in these first publications on the new constructions the emphasis will be on lucidity, clarity and conciseness of the exposition of the main ideas. In the first paper [#!d-2010!#] of the sequence, the most general construction is considered: the one corresponding to general simply connected partitions of domains in $\R^n$, $n\in\N$; in the present paper we consider the reductions and upgrades in this construction in the partial case of star-shaped covers.

Received: May 10, 2010

AMS Subject Classification: 41A15, 33B15, 33B20, 33F05, 41A30, 65D05, 65D07, 65D10, 65D20, 65D30

Key Words and Phrases: interpolation, approximation, special function, spline, B-spline, expo-rational, Bernstein polynomial, cumulative distribution function, density, incomplete Euler Beta-function, discontinuous, continuous,absolutely continuous, smooth, infinitely smooth, geometric modelling, computational geometry, computer-aided geometric design

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 65
Issue: 4