Exact factorization of the spectral density ann its application to wrf,castiilg and time series analysis

Let f be the spectral density function of a purely nondeterministic stationary stochastic process and be the optimal (canonical) fator of f. The role of the coefficients c_n and d_n (n ≥ 0) of φ and φ^?1 respectivey, in prediction, filtering and control theory is well-knwn. We show that the c_n's and d_n's can be obtained recursively in terms of the Fourier coefficients of log f. Also, recursive and updating formulae fr the kolmogorovwiener predictor similar to those Box-Jenkins are provided..