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