Completeness characterization of Type-I box splines

TL;DR

This paper characterizes the completeness of Type-I box splines on three-directional triangulations, enabling hierarchical space construction and basis development based on their smoothness and polynomial span properties.

Contribution

It provides a novel completeness characterization of Type-I box splines, facilitating hierarchical space construction and basis formulation on multilevel grids.

Findings

Established the local polynomial space spanned by box spline translates.

Proved the completeness of the spline space with respect to local polynomials.

Constructed hierarchical box spline spaces with explicit geometric bases.

Abstract

We present a completeness characterization of box splines on three-directional triangulations, also called Type-I box spline spaces, based on edge-contact smoothness properties. For any given Type-I box spline, of specific maximum degree and order of global smoothness, our results allow to identify the local linear subspace of polynomials spanned by the box spline translates. We use the global super-smoothness properties of box splines as well as the additional super-smoothness conditions at edges to characterize the spline space spanned by the box spline translates. Subsequently, we prove the completeness of this space space with respect to the local polynomial space induced by the box spline translates. The completeness property allows the construction of hierarchical spaces spanned by the translates of box splines for any polynomial degree on multilevel Type-I grids. We provide a basis for these hierarchical box spline spaces under explicit geometric conditions of the domain.