On the Szegő-Kolmogorov prediction theorem (Q2130549)

The paper starts from the classical Szegő-Kolmogorov theorem which characterizes the weights \(\omega\) on \(\mathbb T\) such that the family of exponentials \(E(\mathbb N)=\{e^{2\pi i nt}, n\in \mathbb{N}\}\) spans the whole weighted space \(L^2(\mathbb{N},\omega)\), the Hilbert space of all functions \(F\) with the norm \(\|F\|^2_\omega=\int_0^1|F(t)|^2\omega(t)dt\). From the prediction point of view this means that the future of a discrete stationary stochastic process can be predicted from its past if and only if the density \(\omega\) of the spectral measure of the process satisfies the condition \(\int_0^1\log\omega(t)dt=-\infty\). This paper is concerned with the problem if the prediction is possible if some part of the past is not known, and how large this part can be. A condition for a set \(\Gamma\subset\mathbb N\) is found such that the family \(E(\mathbb{N}\setminus\Gamma)\) spans \(L^2(\mathbb{N},\omega)\). Between other results, it is shown that the frequencies \(\{n^3 : n\in\mathbb{N}\}\) can be removed with no effect on the completeness, while \(\{n^2 : n\in\mathbb{N}\}\) cannot. An extension of the main theorem to non-harmonic series is also given.