A Poisson summation formula for integrals over quadratic surfaces

Abstract:

Let $S(t)$ denote Lebesgue measure on the sphere of radius $t > 0$ in ${{\mathbf {R}}^n}$, and \[ {S_k}(t) = {\left ( {\frac {\partial } {{\partial t}}\quad \frac {1} {t}} \right )^k}S(t).\] Let $P{\sum _k} = {S_k}(0) + 2\sum _{m = 1}^\infty {S_k}(m)$. Theorem. If $n$ is odd and $j$ and $k$ are nonnegative integers with $j + k = (n - 1) / 2$, then the Fourier transform of $P{\sum _j}$ is ${(2\pi )^{j - k}}P{\sum _k}$. There is an analogous, although slightly different, identity involving integrals over hyperboloids in odd dimensions. These results were inspired by recent work of M. Vergne.