The XFT is a discrete fractional Fourier transform that was obtained in closed form in  by using finite-dimensional vectors representing Hermite functions and some asymptotic properties of the Hermite polynomials. The XFT is given by the product
is a diagonal matrix with
diagonal element given by
is the standard discrete Fourier transform, and
. The XFT2D can be defined by the Kronecker product
. The XFT is computed with
points in the
directions, respectively. The XFT is as fast as the FFT algorithm used to compute the discrete Fourier transform, but the output of the XFT is more accurate than the output of the FFT because it comes from an algorithm to compute the fast fractional Fourier transform based on a convergent quadrature formula.
 R. G. Campos, J. Rico–Melgoza, and E. Chavez, "RFT: A Fast Discrete Fractional Fourier Transform," submitted.