On Szeg\"{o}--Kolmogorov Prediction Theorem

Abstract

The classical Szeg\"{o}--Kolmogorov Prediction Theorem gives necessary and sufficient condition on a weight $w$ on the unite cirlce $T$ so that the exponentials with positive integer frequences span the weighted space $L^2(T,w)$. We consider the problem how many of these exponentials can be removed while still keeping the completeness property.