On Szeg\H{o}'s theorem for a nonclassical case

TL;DR

This paper extends Szeg\

Contribution

It proves Szeg\

Findings

Szeg\

Application of Khrushchev's formula to nonclassical cases

Verification of Verblunsky's theorem in this context

Abstract

In this paper we prove Szeg\H{o}'s Theorem for the case when a finite number of Verblunsky coefficients lie outside the closed unit disk. Although a form of this result was already proved by A.L. Sakhnovich, we use a very different method, which shows that the OPUC machinery can still be applied to deal with such nonclassical cases. The basic tool we use is Khrushchev's formula that in the classical case relates the absolutely continuous part of the measure and the $N$-th iterate of the Schur algorithm. It is noteworthy that Khrushchev's formula makes the proof short and extremely transparent. Also, we discuss Verblunsky's theorem for the case in question.