Digital filtering and prolate functions

A class of trigonometric polynomials
p(x) = \sum_{n=-N}^{N} a_{n} e^{j n \pi x}
of unit energy is introduced such that their energy concentration
\alpha = \int_{-e}^{e} p^{2}(x) dx
in a specified interval
(- \epsilon, \epsilon)
is maximum. It is shown that the coefficients
a_{n}
must be the eigenvectors of the system
\sum_{m=-N}^{N} \frac{\sin (n - m)\pi \epsilon}{(n - m)\epsilon} a_{m} = \lambda a_{n}
. corresponding to the maximum eigenvalue X. These polynomials are determined for
N = 1, \cdots , 10
and
\epsilon = 0.025, \cdots , 0.5
. The resulting family of periodic functions forms the discrete version of the familiar prolate spheroidal wave functions.