In this monograph we combine operator techniques with state space methods to solve
factorization spectral estimation and interpolation problems arising in control and signal
processing. We present both the theory and algorithms with some Matlab code to solve these
problems. A classical approach to spectral factorization problems in control theory is based on
Riccati equations arising in linear quadratic control theory and Kalman ?ltering. One advantage
of this approach is that it readily leads to algorithms in the non-degenerate case. On the
other hand this approach does not easily generalize to the nonrational case and it is not
always transparent where the Riccati equations are coming from. Operator theory has developed
some elegant methods to prove the existence of a solution to some of these factorization and
spectral estimation problems in a very general setting. However these techniques are in
general not used to develop computational algorithms. In this monograph we will use operator
theory with state space methods to derive computational methods to solve factorization sp-
tral estimation and interpolation problems. It is emphasized that our approach is geometric
and the algorithms are obtained as a special application of the theory. We will present two
methods for spectral factorization. One method derives al- rithms based on ?nite sections of a
certain Toeplitz matrix. The other approach uses operator theory to develop the Riccati
factorization method. Finally we use isometric extension techniques to solve some
interpolation problems.
