# IJPAM: Volume 65, No. 4 (2010)

SMOOTH CONVEX RESOLUTION OF UNITY AND/OR
SCATTERED-POINT HERMITE INTERPOLATION
BY GENERALIZED EXPO-RATIONAL B-SPLINES, I:
GENERAL SIMPLY CONNECTED
COVERS AND PARTITIONS

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: ltd@hin.no
url: https://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 , , 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 first 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 this first paper, only the most general construction is considered: the one corresponding to general simply connected partitions of domains in , ; the more highly specialized constructions will be considered in the subsequent papers of this sequence.