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An algorithm for constructing a basis for \(C^{r}\)-spline modules over polynomial rings. (English) Zbl 0914.65004

\(C^r(\square)\) denotes the class of multivariate splines of smoothness \(r\) over \(\square\) where \(\square\) is a polyhedral \(d\)-complex embedded in the Euclidean space \(E^d\) [see D. Satya, J. Approximation Theory 84, No. 1, 12-30 (1996; Zbl 0842.41010)]. \(C^r(\square)\) is an \(R\)-module over the polynomial ring \(R\) and conditions under which it is free have been studied by L. J. Billera and L. L. Rose [Math. Z. 209, No. 4, 485-497 (1992; Zbl 0891.13004)]. The authors provide an algorithm for writing down a basis for the free \(R\)-module \(C^r(\square)\) where \(\square\) consists of a grid in the plane obtained by crossing a set of parallel lines by another set of parallel lines. In particular, this includes the general polyhedral case of rectangular grids not covered in the case of a simplicial complex.

MSC:

65D07 Numerical computation using splines
13C10 Projective and free modules and ideals in commutative rings
41A15 Spline approximation
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References:

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