Sum Rules and the Szego Condition for Orthogonal Polynomials on the Real Line

Abstract

We study the Case sum rules, especially $C_0$, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat's theorem to cases with an infinite point spectrum and a proof that if $\lim n(a_n-1)=\alpha$ and $\lim n{b}_n=\beta$ exist and $2α<\absβ$, then the Szegő condition fails.

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