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Dimension of \(C^1\)-splines on type-6 tetrahedral partitions. (English) Zbl 1062.65014
Multivariate polynomial spline spaces are most important in the theory of approximation and its applications. Especially when the breakpoints/lines between the individual polynomial parts are not gridded, they are difficult to analyse, for instance with respect to their dimension. In many cases, even in just two dimensions, only upper and lower bounds are known so far (while in one dimension, the exact dimensions are known).
Here, three dimensional spline spaces are studied, for arbitrary polynomial degree. Formulae for the spaces’ dimension are found, as well as minimal determining sets (for particular cases). The partitions for the piecewise polynomials which are used are still regular (“type-6 tetrahedral partitions”) but they are not just an equispaced lattice.

MSC:
65D07 Numerical computation using splines
65D17 Computer-aided design (modeling of curves and surfaces)
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
41A15 Spline approximation
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