Refinability of splines derived from regular tessellations

• Multivariate splines can be derived by convolving indicator functions.
• Refinability (nestedness) is an important property of spline spaces.
• Refinable convolution-derived splines on shift-invariant tessellations are rare.
• Hex-splines and their generalizations are not refinable.
• This is proven via simple, geometric criteria for refinability.

Splines can be constructed by convolving the indicator function of a cell whose shifts tessellate RnRn. This paper presents simple, geometric criteria that imply that, for regular shift-invariant tessellations, only a small subset of such spline families yield nested spaces: primarily the well-known tensor-product and box splines. Among the many non-refinable constructions are hex-splines and their generalization to the Voronoi cells of non-Cartesian root lattices.

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