On Szegő's theorem for a nonclassical case

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• 作者：Maxim Derevyagin^a ; ^{derevyagin.m@gmail.com} ; Brian Simanek^b ; ^{Brian_Simanek@baylor.edu}
• 关键词：primary ; 42C05 ; secondary ; 30D30 ; 46C20
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：272
• 期：6
• 页码：2487-2503
• 全文大小：393 K
• 卷排序：272

文摘

In this paper we prove Szegő'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.