The polynomial space ℋ
spanned by the integer translates of a box spline M admits a well-known
characterization as the joint kernel of a set of homogeneous differential operators
with constant coefficients. The dual space ℋ∗ has a convenient representation by a
polynomial space 𝒫, explicitly known, which plays an important role in box spline
theory as well as in multivariate polynomial interpolation.
In this paper we characterize the dual space 𝒫 as the joint kernel of
simple differential operators, each one a power of a directional derivative.
Various applications of this result to multivariate polynomial interpolation,
multivariate splines and duality between polynomial and exponential spaces are
discussed.
