  In recent years there has been a growing interest in the fractional Fourier
transform driven by its large number of applications. The literature in this
field follows two main routes. On the one hand, the areas where the ordinary
Fourier transform has been applied are being revisited to use this intermediate
time-frequency representation of signals, and on the other hand, fast
algorithms for numerical computation of the fractional Fourier transform are
devised. In this paper we derive a Gaussian-like quadrature of the continuous
fractional Fourier transform. This quadrature is given in terms of the Hermite
polynomials and their zeros. By using some asymptotic formulas, we rewrite the
quadrature as a chirp-fft-chirp transformation, yielding a fast discretization
of the fractional Fourier transform and its inverse in closed form. We extend
the range of the fractional Fourier transform by considering arbitrary complex
values inside the unitary circle and not only at the boundary. We find that,
the chirp-fft-chirp transformation evaluated at z=i, becomes a more accurate
version of the fft which can be used for non-periodic functions.
