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Journal of Operator Theory

Volume 70, Issue 2, Autumn 2013  pp. 401-436.

Recursively determined representing measures for bivariate truncated moment sequences

Authors:  Raul E. Curto (1) and Lawrence A. Fialkow (2)
Author institution: (1) Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, U.S.A.
(2) Departments of Computer Science and Mathematics, State University of New York, New Paltz, New York 12561, U.S.A.

Summary:  A theorem of C. Bayer and J. Teichmann implies that if a finite real multisequence $\beta\equiv \beta^{(2d)}$ has a representing measure, then the associated moment matrix $M_{d}$ admits positive, recursively generated moment matrix extensions $M_{d+1},~M_{d+2},\ldots$. For a bivariate recursively determinate $M_{d}$, we show that the existence of positive, recursively generated extensions $M_{d+1},\ldots,\break M_{2d-1}$ is sufficient for a measure; examples illustrate that all of these extensions may be required. We describe in detail a constructive procedure for determining whether such extensions exist. Under mild additional hypotheses, we show that $M_{d}$ admits an extension $M_{d+1}$ which has many of the properties of a positive, recursively generated extension.

Keywords:  truncated moment sequence, moment matrix, representing measure

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