On Krein's theorem for indeterminacy of the classical moment problem (Q1268715)

Krein's sufficient condition for indeterminacy of the classical Hamburger moment problem states that a positive measure on the real line, having moments of all orders, is indeterminate provided it has density with respect to Lebesgue measure and that this density has a finite logarithmic integral. The author generalizes this result and also gives a discrete analog. He shows that it is sufficient to assume convergence of the logarithmic integral over some subset of so-called positive lower uniform density. The proof is based on an estimate of harmonic measure due to \textit{L. Carleson} [Ann. Acad. Sci. Fenn., Ser. A I 7, 25-32 (1982; Zbl 0521.30026)] and on results of \textit{P. Koosis} [The logarithmic integral. Vol. I (1988; Zbl 0665.30038); Chaps. VI and VIII, Sect. A] on weighted approximation over subsets of this form. The author also gives an analogue of Krein's theorem for discrete measures concentrated on the integers; he shows that such a measure is indeterminate provided that the logarithmic sum of masses at the integers converges.