  A useful finite-dimensional matrix representation of the derivative of
periodic functions is obtained by using some elementary facts of trigonometric
interpolation. This NxN matrix becomes a projection of the angular derivative
into polynomial subspaces of finite dimension and it can be interpreted as a
generator of discrete rotations associated to the z-component of the projection
of the angular momentum operator in such subspaces, inheriting thus some
properties of the continuum operator. The group associated to these discrete
rotations is the cyclic group of order N. Since the square of the quantum
angular momentum L^2 is associated to a partial differential boundary value
problem in the angular variables $\theta$ and $\phi$ whose solution is given in
terms of the spherical harmonics, we can project such a differential equation
to obtain an eigenvalue matrix problem of finite dimension by extending to
several variables a projection technique for solving numerically two point
boundary value problems and using the matrix representation of the angular
derivative found before. The eigenvalues of the matrix representing L^2 are
found to have the exact form n(n+1), counting the degeneracy, and the
eigenvectors are found to coincide exactly with the corresponding spherical
harmonics evaluated at a certain set of points.
