Fast and exact evaluation of box splines via the PP-form

TL;DR

This paper provides a constructive proof that non-degenerate box splines are piecewise polynomial and introduces an explicit scheme to construct the polynomial pieces within their interior regions.

Contribution

It offers a new constructive proof and an explicit construction scheme for polynomial pieces of non-degenerate box splines.

Findings

Box splines are confirmed to be piecewise polynomial.

An explicit construction scheme for polynomial pieces is proposed.

The proof is constructive, enabling practical computation.

Abstract

For the class of non-degenerate box splines, we prove that these box splines are piecewise polynomial. This is not a new result, it is in fact a well known and useful property of box splines. However, our proof is constructive, and the main result of this work is a corollary that follows from this proof, namely one that gives an explicit construction scheme for the polynomial pieces in the interior regions of any non-degenerate box spline.