  Two methods for fast Fourier transforms are used in a quantum context. The
first method is for systems with dimension of the Hilbert space $D=d^n$ with
$d$ an odd integer, and is inspired by the Cooley-Tukey formalism. The `large
Fourier transform' is expressed as a sequence of $n$ `small Fourier transforms'
(together with some other transforms) in quantum systems with $d$-dimensional
Hilbert space. Limitations of the method are discussed. In some special cases,
the $n$ Fourier transforms can be performed in parallel. The second method is
for systems with dimension of the Hilbert space $D=d_0...d_{n-1}$ with
$d_0,...,d_{n-1}$ odd integers coprime to each other. It is inspired by the
Good formalism, which in turn is based on the Chinese reminder theorem. In this
case also the `large Fourier transform' is expressed as a sequence of $n$
`small Fourier transforms' (that involve some constants related to the number
theory that describes the formalism). The `small Fourier transforms' can be
performed in a classical computer or in a quantum computer (in which case we
have the additional well known advantages of quantum Fourier transform
circuits). In the case that the small Fourier transforms are performed with a
classical computer, complexity arguments for both methods show the reduction in
computational time from ${\cal O}(D^2)$ to ${\cal O}(D\log D)$. The second
method is also used for the fast calculation of Wigner and Weyl functions, in
quantum systems with large finite dimension of the Hilbert space.
