  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.
