An algebraic characterization of B-splines

TL;DR

This paper explores whether the fundamental algebraic properties of B-splines, such as local support, refinement, and invariance, are sufficient to uniquely characterize B-splines and their derivatives.

Contribution

The paper investigates if the core algebraic properties of B-splines are enough to uniquely define them and their derivatives, providing a potential characterization framework.

Findings

Properties (1)-(3) are necessary for B-splines.

Derivatives of B-splines also satisfy these properties.

The sufficiency of these properties for characterization is analyzed.

Abstract

B-splines of order $k$ can be viewed as a mapping $N$ taking a $(k+1)$-tuple of increasing real numbers $a_0 < \cdots < a_k$ and giving as a result a certain piecewise polynomial function. Looking at this mapping $N$ as a whole, basic roperties of B-spline functions imply that it has the following algebraic properties: (1) $N(a_0,\ldots,a_k)$ has local support; (2) $N(a_0,\ldots,a_k)$ allows refinement, i.e. for every $a\in \cup_{j=0}^{k-1} (a_j,a_{j+1})$ we have that if $(\alpha_0,\ldots, \alpha_{k+1})$ is the increasing rearrangement of the points $\{a_0,\ldots,a_k,a\}$, the 'old' function $N(a_0,\ldots,a_k)$ is a linear combination of the 'new' functions $N(\alpha_0,\ldots,\alpha_k)$ and $N(\alpha_1,\ldots,\alpha_{k+1})$; (3) $N$ is translation and dilation invariant. It is easy to see that derivatives of $N(a_0,\ldots,a_k)$ satisfy properties (1)-(3) as well. In this paper we investigate if properties (1)-(3) are already sufficient to characterize B-splines and their derivatives.