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Fundamental solution of hyperbolic differential operators and the poisson summation formula

The attempt to subsume the summation of the double series (1) under a more general framework led to a systematic investigation of the existence and the uniqueness of fundamental solutions of linear hyperbolic partial differential operators with constant coefficients on tori and of their Fourier expansions. The restriction to the class of hyperbolic operators implies a representation of the fundamental solution by a locally finite series; further restriction to strictly hyperbolic operators yields convergence results in the scale Hα (Tn ) of Sobolev spaces on the torus Tn . Finally, we apply the theory to some specific many-dimensional series involving Bessel functions.