  Zonotopal algebra is the study of a family of pairs of dual vector spaces of
multivariate polynomials that can be associated with a list of vectors X. It
connects objects from combinatorics, geometry, and approximation theory. The
origin of zonotopal algebra is the pair (D(X),P(X)), where D(X) denotes the
Dahmen-Micchelli space that is spanned by the local pieces of the box spline
and P(X) is a space spanned by products of linear forms.
  The first main result of this paper is the construction of a canonical basis
for D(X). We show that it is dual to the canonical basis for P(X) that is
already known.
  The second main result of this paper is the construction of a new family of
zonotopal spaces that is far more general than the ones that were recently
studied by Ardila-Postnikov, Holtz-Ron, Holtz-Ron-Xu, Li-Ron, and others. We
call the underlying combinatorial structure of those spaces forward exchange
matroid. A forward exchange matroid is an ordered matroid together with a
subset of its set of bases that satisfies a weak version of the basis exchange
axiom.
