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Homogeneous polynomial splines

Published online by Cambridge University Press:  14 November 2011

T. N. T. Goodman  and
S. L. Lee
T. N. T. Goodman
Affiliation:
Department of Mathematics and Computer Science, The University, Dundee DD1 4HN, Scotland, U.K
S. L. Lee
Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511
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Synopsis

We construct functions which are piecewise homogeneous polynomials in the positive octant in three dimensions. These give a rich and elegant theory which combines properties of polynomial box splines see [6] and the references therein) with the explicit representation of simple exponential box splines [11], while enjoying complete symmetry in the three variables. By a linear transformation followed by a projection on suitable planes, one obtains piecewise polynomial functions of two variables on a mesh formed by three pencils of lines. The vertices of these pencils may be finite or one or two may be infinite, i.e. the corresponding pencils may comprise parallel lines. As a limiting case, all three vertices become infinite and one recovers polynomial box splines on a three-direction mesh.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 1-2 , 1991 , pp. 89 - 102
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Boor, C. de and DeVore, R.. Approximation by smooth multivariate splines. Trans. Amer. Math. Soc. 276 (1983), 775–788.CrossRefGoogle Scholar
2Boor, C. de, Hollig, K. and Riemenschneider, S. D.. Bivariate cardinal interpolation by splines on a three-direction mesh. Illinois J. Math. 299 (1985), 533–566.Google Scholar
3Cohen, E., Lyche, T. and Riesenfeld, R.. Subdivision algorithms for the generation of box spline surfaces. Comput. Aided Geom. Design 2 (1984), 131–148.CrossRefGoogle Scholar
4Dahmen, W. and Micchelli, C., Recent progress in multivariate splines. In Approximation Theory IV, eds Chui, C. K., Schumaker, L. L. and Ward, J., 27–121 (New York: Academic Press, 1983).Google Scholar
5Dahmen, W. and Micchelli, C.. Subdivision algorithms for the generation of box spline surfaces. Comput. Aided Geom. Design 1 (1984), 115–129.CrossRefGoogle Scholar
6Hollig, K.. Box splines. In Approximation Theory V, eds Chui, C. K., Schumaker, L. L. and Ward, J. D., 71–95 (New York: Academic Press, 1986).Google Scholar
7Lee, S. L. and Phillips, G. M.. Polynomial interpolation at points of a geometric mesh on a triangle. Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 75–87.CrossRefGoogle Scholar
8Lee, S. L. and Phillips, G. M.. Construction of lattices for Lagrange interpolation in projective spaces. Constr. Approx. (to appear).Google Scholar
9Marsden, M. J.. An identity for spline functions with applications to variation-diminishing spline approximation. J. Approx. Theory 3 (1970), 7–49.CrossRefGoogle Scholar
10Maxwell, E. A.. General homogeneous coordinates in space of three dimensions (Edinburgh: University Press, 1951).Google Scholar
11Ron, A.. Exponential box splines. Constr. Approx. (to appear).Google Scholar
12Schoenberg, I. J.. On polynomial interpolation at the points of a geometric progression. Proc. Roy. Soc. Edinburgh Sect. A 90 (1981), 221–247.CrossRefGoogle Scholar