Box splines and the equivariant index theorem

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.

💡 Research Summary

The paper “Box Splines and the Equivariant Index Theorem” develops a novel bridge between the theory of box splines—piecewise polynomial functions arising in approximation theory—and the equivariant index theory of transversally elliptic operators on a real vector space equipped with a linear action of a compact torus G. The authors begin by recalling the inversion formula for the semi‑discrete convolution with a box spline. Given a finite list X =