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Zonotopal algebra is the study of a family of pairs of dual vector spaces ofmultivariate polynomials that can be associated with a list of vectors X. Itconnects objects from combinatorics, geometry, and approximation theory. Theorigin of zonotopal algebra is the pair (D(X),P(X)), where D(X) denotes theDahmen-Micchelli space that is spanned by the local pieces of the box splineand P(X) is a space spanned by products of linear forms.  The first main result of this paper is the construction of a canonical basisfor D(X). We show that it is dual to the canonical basis for P(X) that isalready known.  The second main result of this paper is the construction of a new family ofzonotopal spaces that is far more general than the ones that were recentlystudied by Ardila-Postnikov, Holtz-Ron, Holtz-Ron-Xu, Li-Ron, and others. Wecall the underlying combinatorial structure of those spaces forward exchangematroid. A forward exchange matroid is an ordered matroid together with asubset of its set of bases that satisfies a weak version of the basis exchangeaxiom.

Zonotopal algebra and forward exchange matroids