Poisson Summation Formulae and the Wave Equation with a Finitely Supported Measure as Initial Velocity

Abstract

New Poisson summation formulae have been recently discovered by Nir Lev and Alexander Olevskii since 2013. But some other examples were concealed in an old paper by Andrew Guinand dating from 1959. This was observed by the second author in 2016. In the present contribution a third approach is proposed. Guinand's work follows from some simple observations on solutions of the wave equation on the three dimensional torus. If the initial velocity is a Dirac mass at the origin, the solution is Guinand's distribution. Using this new approach one can construct a large family of initial velocities which give rise to crystalline measures generalizing Guinand's solution.