It is the smoothness. Any fourier series that converges everywhere can be regarded as a maclaurin series of an function that's analytic and bounded within the unit disk (fourier is when restrict to the disk boundary). Well anyhow for the right kind of fourier series. Analytic functions are described by a countable number of coefficients. It is a delusion to think there is an uncountable number of degrees of freedom. So to make that clear, what we want is some theorem saying: Specifying the value of the analytic function at a countable set of points within the disk, will suffice to specify the function. Is such a theorem known? I can't think of one off the top of my head. But here, voila, I proved it: use monte carlo integration to compute the maclaurin series coefficients. With probability=1, any particular series coefficient will get computed in this way with error that in the infinite-points limit goes to zero. That proves that a countable set of points must exist, that works to compute any finite-numbered coefficient exactly; and hence, theorem proven. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)