Matrix two-dimensional spectral factorization

The existence of an asymmetric half-plane spectral factorization for a nonnegative two-dimensional matrix-valued spectral density is established under the conditions that the density and the logarithm of the determinant of the density are absolutely integrable on the torus. The proof gives a construction of the factors in terms of a family of one-dimensional spectral factorizations. An efficient algorithm for calculating the two-dimensional spectral factorization is thereby obtained; G.T. Wilson´s (1972) one-dimensional factorization algorithm appears to be the most suitable for this purpose. In the case where the initial array is of finite extent, it is shown that the factors have constant (minimal) order in the `causal´ direction, and that, if a finite-support array has a quarter-plane spectral factorization, the spectral factors must also be of finite support