Refinability of splines from lattice Voronoi cells

TL;DR

This paper investigates the conditions under which lattice Voronoi cell-based splines can be refined, revealing that only certain well-known spline families are refinable, while others like hex-splines are not, impacting their approximation properties.

Contribution

It provides simple criteria for spline refinability from lattice Voronoi cells, distinguishing between refinable and non-refinable spline families, including new insights on hex-splines.

Findings

Only box splines and tensor-product splines are refinable.

Hex-splines are non-refinable and can increase approximation error upon refinement.

Non-refinable splines may exhibit worse approximation with lattice refinement.

Abstract

Splines can be constructed by convolving the indicator function of the Voronoi cell of a lattice. This paper presents simple criteria that imply that only a small subset of such spline families can be refined: essentially the well-known box splines and tensor-product splines. Among the many non-refinable constructions are hex-splines and their generalization to non-Cartesian lattices. An example shows how non-refinable splines can exhibit increased approximation error upon refinement of the lattice.