  The need to Fourier transform data sets with irregular sampling is shared by
various domains of science. This is the case for example in astronomy or
sismology. Iterative methods have been developed that allow to reach
approximate solutions. Here an exact solution to the problem for band-limited
periodic signals is presented. The exact spectrum can be deduced from the
spectrum of the non-equispaced data through the inversion of a Toeplitz matrix.
The result applies to data of any dimension. This method also provides an
excellent approximation for non-periodic band-limit signals. The method allows
to reach very high dynamic ranges ($10^{13}$ with double-float precision) which
depend on the regularity of the samples.
