In this article, we start to recall 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 be both described using some vector spaces
of distributions, on the dual of the group G or on the dual of its Lie algebra.
The morphism from K-theory to cohomology is analyzed and the multiplication by
the Todd class is shown to correspond to the operator (deconvolution) inverting
the semidiscrete 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.Comment: 44 page
