Geometric approach to prediction and periodic systems

The classical prediction problem is analyzed via a geometric approach rather than measure theoretic approach. The main tool used is the Naimark dilation. The problem is tackled from operator theory and analytic function theory instead of stochastic and probabilistic point of view. The classical convergence results of Helson-Lowdenslager are proved using geometric techniques. The formulas for computing the maximal outer spectral factors and matrix of amplitudes of sinusoid are derived. These turn out to be extensions of the classical result from Levinson algorithm and Geronimus-Capon formula on the unit disc respectively. The connection between this problem and the positive Toeplitz matrix extension problem is drawn. Lower triangular Cholesky factorization method for computing the maximal outer spectral factor and matrix of amplitudes of sinusoid is discussed. Moreover, the analysis of cyclic time invariant representation of linear periodic systems is given. A state space transformation of cyclic time invariant state space matrices is shown. The structures of the corresponding Toeplitz matrices are given. Furthermore, necessary and sufficient conditions for the cyclic time invariant transfer function to be inner or outer are given. Finally, some examples of cyclic time invariant transfer function in classical control applications are demonstrated.