  A versatile method is described for the practical computation of the discrete
Fourier transforms (DFT) of a continuous function $g(t)$ given by its values
$g_{j}$ at the points of a uniform grid $F_{N}$ generated by conjugacy classes
of elements of finite adjoint order $N$ in the fundamental region $F$ of
compact semisimple Lie groups. The present implementation of the method is for
the groups SU(2), when $F$ is reduced to a one-dimensional segment, and for
$SU(2)\times ... \times SU(2)$ in multidimensional cases. This simplest case
turns out to result in a transform known as discrete cosine transform (DCT),
which is often considered to be simply a specific type of the standard DFT.
Here we show that the DCT is very different from the standard DFT when the
properties of the continuous extensions of these two discrete transforms from
the discrete grid points $t_j; j=0,1, ... N$ to all points $t \in F$ are
considered. (A) Unlike the continuous extension of the DFT, the continuous
extension of (the inverse) DCT, called CEDCT, closely approximates $g(t)$
between the grid points $t_j$. (B) For increasing $N$, the derivative of CEDCT
converges to the derivative of $g(t)$. And (C), for CEDCT the principle of
locality is valid. Finally, we use the continuous extension of 2-dimensional
DCT to illustrate its potential for interpolation, as well as for the data
compression of 2D images.
