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Mathematics of Computation

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Multivariate refinable Hermite interpolant

Authors: Bin Han, Thomas P.-Y. Yu and Bruce Piper
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Journal: Math. Comp. 73 (2004), 1913-1935
MSC (2000): Primary 41A05, 41A15, 41A63, 42C40, 65T60, 65F15
Published electronically: December 22, 2003
MathSciNet review: 2059743
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Abstract: We introduce a general definition of refinable Hermite interpolants and investigate their general properties. We also study a notion of symmetry of these refinable interpolants. Results and ideas from the extensive theory of general refinement equations are applied to obtain results on refinable Hermite interpolants. The theory developed here is constructive and yields an easy-to-use construction method for multivariate refinable Hermite interpolants. Using this method, several new refinable Hermite interpolants with respect to different dilation matrices and symmetry groups are constructed and analyzed.

Some of the Hermite interpolants constructed here are related to well-known spline interpolation schemes developed in the computer-aided geometric design community (e.g., the Powell-Sabin scheme). We make some of these connections precise. A spline connection allows us to determine critical Hölder regularity in a trivial way (as opposed to the case of general refinable functions, whose critical Hölder regularity exponents are often difficult to compute).

While it is often mentioned in published articles that ``refinable functions are important for subdivision surfaces in CAGD applications", it is rather unclear whether an arbitrary refinable function vector can be meaningfully applied to build free-form subdivision surfaces. The bivariate symmetric refinable Hermite interpolants constructed in this article, along with algorithmic developments elsewhere, give an application of vector refinability to subdivision surfaces. We briefly discuss several potential advantages offered by such Hermite subdivision surfaces.

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Additional Information

Bin Han
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Thomas P.-Y. Yu
Affiliation: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180-3590

Bruce Piper
Affiliation: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180-3590

Keywords: Hermite interpolation, refinable function, vector refinability, subdivision scheme, shift invariant subspace, multivariate spline, Bernstein-B\'ezier form, wavelet, subdivision surface
Received by editor(s): January 22, 2002
Received by editor(s) in revised form: March 27, 2003
Published electronically: December 22, 2003
Article copyright: © Copyright 2003 American Mathematical Society