  We develop a "motivic integration" version of the Poisson summation formula
for function fields, with values in the Grothendieck ring of definable
exponential sums. We also study division algebras over the function field, and
obtain relations among the motivic Fourier transforms of a test function at
different completions. We use these to prove, in a special case, a motivic
version of a theorem of Deligne-Kazhdan-Vigneras.
