Existence of smooth solutions to the classical moment problems
Author:
Palle E. T. Jorgensen
Journal:
Trans. Amer. Math. Soc. 332 (1992), 839848
MSC:
Primary 44A60; Secondary 42A70, 43A35, 46N99, 47A57
MathSciNet review:
1059709
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Abstract: Let be a given sequence, and define for . If holds for all finite sequences , then it is known that there is a positive Borel measure on the circle such that , and conversely. Our main theorem provides a necessary and sufficient condition on the sequence that the measure may be chosen to be smooth. A measure is said to be smooth if it has the same spectral type as the operator acting on with respect to Haar measure on : Equivalently, is a superposition (possibly infinite) of measures of the form with such that . The condition is stated purely in terms of the initially given sequence : We show that a smooth representation exists if and only if, for some , the a priori estimate is valid for all finite double sequences . 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/S00029947199210597091
PII:
S 00029947(1992)10597091
Article copyright:
© Copyright 1992
American Mathematical Society
