The factorization problem for time-limited functions and trigonometric polynomials

The following form of the factorization problem is considered: Given a function
g(t)
vanishing for
|t| > a
, and with a nonnegative Fourier transform
G(j\omega)
, rind a function
f(t)
with energy spectrum
G (j\omega)
and such that
f(t)=O
outside the interval (0, a). A numerical method for determining
f(t)
is developed involving only the discrete Fourier series.