Refinability of splines derived from regular tessellations

TL;DR

This paper investigates the refinability of splines derived from regular tessellations, identifying which families produce nested spaces and highlighting non-refinable examples like hex-splines.

Contribution

It provides simple, non-algebraic criteria to determine refinability of tessellation-based splines, emphasizing the special status of tensor-product and box splines.

Findings

Tensor-product and box splines are the primary refinable families.

Many other tessellation-based splines, such as hex-splines, are non-refinable.

Criteria for refinability help distinguish between different spline constructions.

Abstract

Splines can be constructed by convolving the indicator function of a cell whose shifts tessellate $\R^k$. This paper presents simple, non-algebraic 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.