An algebraic characterization of B-splines

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) $N(a_0,\dots ,a_k)$ has local support; (2) $N(a_0,\dots ,a_k)$ allows refinement, i.e. for every $a\in {\cup }_{j=0}^{k-1}(a_j,a_{j+1})$ we have that if $({\alpha }_0,\dots ,{\alpha }_{k+1})$ is the increasing rearrangement of the points $\{a_0,\dots ,a_k,a\}$, the 'old' function $N(a_0,\dots ,a_k)$ is a linear combination of the 'new' functions $N({\alpha }_0,\dots ,{\alpha }_k)$ and $N({\alpha }_1,\dots ,{\alpha }_{k+1})$; (3) $N$ is translation and dilation invariant. It is easy to see that derivatives of $N(a_0,\dots ,a_k)$ 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.