A useful finite-dimensional matrix representation of the derivative of peri
odic functions is obtained by using some elementary facts of trigonometric
interpolation. This N x N matrix becomes a projection of the angular deriva
tive into polynomial subspaces of finite dimension and it can be interprete
d as a generator of discrete rotations associated to the z-component of the
projection of the angular-momentum operator in such subspace, inheriting t
hus some properties of the continuum operator. The group associated to thes
e discrete rotations is the cyclic group of order N. Since the square of th
e quantum angular momentum L-2 is associated to a partial differential boun
dary-value problem in the angular variables: theta and phi whose solution i
s given in terms of the spherical harmonics, we can project such a differen
tial equation to obtain an eigenvalue matrix problem of finite dimension by
extending to several variables a projection technique for solving numerica
lly two point boundary value problems and using the matrix representation o
f the angular derivative found before. The eigenvalues of the matrix repres
enting L-2 are found to have the exact form n(n + 1), counting tile degener
acy, and the eigenvectors are found to coincide exactly with the correspond
ing spherical harmonics evaluated at a certain set of points.