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

TL;DR

This paper investigates the Szeg"{o}--Kolmogorov Prediction Theorem, focusing on the minimal set of exponentials needed to maintain completeness in weighted $L^2$ spaces on the unit circle.

Contribution

It extends the classical theorem by analyzing the removal of exponentials while preserving the completeness property in weighted spaces.

Findings

Determines conditions under which exponentials can be removed without losing completeness

Provides new insights into the structure of weighted $L^2$ spaces on the unit circle

Enhances understanding of basis properties in harmonic analysis

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.