The flatness of power spectral zeros and their significance in quadratic estimation

In optimal prediction as well as in optimal smoothing the variance of the optimal estimator is impacted predominantly by the frequency segments where the power spectrum is small or negligible. Indeed, the Szegö´s celebrated theorem characterizes deterministic process as those whose power spectral density fails to be log-integrable by virtue of sufficiently flat spectral zeros. Likewise Kolmogorov´s formula gives an analogous condition for optimal smoothing. We discuss how the flatness of spectral zeros suggests a nested stratification of families of spectral where estimation of a stochastic process over a window of a given size is possible with negligible variance based on observations outside the interval. We then focus on the more general problem of estimating missing data in observation records which are not necessarily contiguous. A key result in the paper (Theorem 3) provides a sufficient condition for being able to estimate missing data with arbitrarily small error variance, in terms of the flatness of the spectral zeros.