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Given a real closed polytope $P$, we first describe the Fourier transform ofits indicator function by using iterations of Stokes' theorem. We then use theensuing Fourier transform formulations, together with the Poisson summationformula, to give a new algorithm to count fractionally-weighted lattice pointsinside the one-parameter family of all real dilates of $P$. The combinatoricsof the face poset of $P$ plays a central role in the description of the Fouriertransform of $P$. We also obtain a closed form for the codimension-1coefficient that appears in an expansion of this sum in powers of the realdilation parameter $t$. This closed form generalizes some known results aboutthe Macdonald solid-angle polynomial, which is the analogous expressiontraditionally obtained by requiring that $t$ assumes only integer values.Although most of the present methodology applies to all real polytopes, aparticularly nice application is to the study of all real dilates of integer(and rational) polytopes.

Fourier transforms of polytopes, solid angle sums, and discrete volume