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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The approximation order of box spline spaces


Authors: A. Ron and N. Sivakumar
Journal: Proc. Amer. Math. Soc. 117 (1993), 473-482
MSC: Primary 41A15; Secondary 41A25, 41A63
MathSciNet review: 1110553
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Abstract: Let $ M$ be a box spline associated with an arbitrary set of directions and suppose that $ S(M)$ is the space spanned by the integer translates of $ M$. In this note, the subspace of all polynomials in $ S(M)$ is shown to be the joint kernal of a certain collection of homogeneous differential operators with constant coefficients. The approximation order from the dilates of $ S(M)$ to smooth functions is thereby characterized. This extends a well-known result of de Boor and Höllig ($ B$-splines from parallelepipeds, J. Analyse Math. 42 (1982/83), 99-115), on box splines with integral direction sets.

The argument used is based on a new relation, valid for any compactly supported distribution $ \phi $, between the semidiscrete convolution $ \phi \ast'$ and the distributional convolution $ \phi\ast$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1110553-2
PII: S 0002-9939(1993)1110553-2
Keywords: Box splines, polynomials, multivariate splines, approximation order
Article copyright: © Copyright 1993 American Mathematical Society