Box splines and the equivariant index theorem

In this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant K-theory with respect to a compact torus G of various spaces associated to a linear action of G in a vector space M can both be described using some vector spaces of distributions, on the dual of the group G or on the dual of its Lie algebra g. The morphism from K-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a G-transversally elliptic operator on M are determined using the infinitesimal index of the symbol.