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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Existence of smooth solutions to the classical moment problems


Author: Palle E. T. Jorgensen
Journal: Trans. Amer. Math. Soc. 332 (1992), 839-848
MSC: Primary 44A60; Secondary 42A70, 43A35, 46N99, 47A57
MathSciNet review: 1059709
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Abstract: Let $ s(0),s(1), \ldots $ be a given sequence, and define $ s(n) = \overline {s( - n)} $ for $ n < 0$. If $ \Sigma \Sigma {\overline \xi _n}{\xi _m}s(m - n) \geq 0$ holds for all finite sequences $ {({\xi _n})_{n \in \mathbb{Z}}}$, then it is known that there is a positive Borel measure $ \mu $ on the circle $ \mathbb{T}$ such that $ s(n) = \smallint_{ - \pi }^\pi {{e^{int}}d\mu (t)} $, and conversely. Our main theorem provides a necessary and sufficient condition on the sequence $ (s(n))$ that the measure $ \mu $ may be chosen to be smooth. A measure $ \mu $ is said to be smooth if it has the same spectral type as the operator $ id/dt$ acting on $ {L^2}(\mathbb{T})$ with respect to Haar measure $ dt$ on $ \mathbb{T}$: Equivalently, $ \mu $ is a superposition (possibly infinite) of measures of the form $ \vert w(t){\vert^2}dt$ with $ w \in {L^2}(\mathbb{T})$ such that $ dw/dt \in {L^2}(\mathbb{T})$. The condition is stated purely in terms of the initially given sequence $ (s(n))$: We show that a smooth representation exists if and only if, for some $ \varepsilon \in {\mathbb{R}_ + }$, the a priori estimate

$\displaystyle \sum {\sum {s(m - n){{\overline \xi }_n}{\xi _m} \geq \varepsilon {{\left\vert {\sum {ns(n){\xi _n}} } \right\vert}^2}} } $

is valid for all finite double sequences $ ({\xi _n})$. An analogous result is proved for the determinate (Hamburger) moment problem on the line. But the corresponding result does not hold for the indeterminate moment problem.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1059709-1
PII: S 0002-9947(1992)1059709-1
Article copyright: © Copyright 1992 American Mathematical Society