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

 

The truncated complex $K$-moment problem


Authors: Raúl Curto and Lawrence A. Fialkow
Journal: Trans. Amer. Math. Soc. 352 (2000), 2825-2855
MSC (2000): Primary 47A57, 44A60, 30E05; Secondary 15A57, 15-04, 47N40, 47A20
Published electronically: February 28, 2000
MathSciNet review: 1661305
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Abstract:

Let $\gamma \equiv \gamma^{\left( 2n\right) }$ denote a sequence of complex numbers $\gamma _{00}, \gamma _{01}, \gamma _{10}, \dots , \gamma _{0,2n}, \dots , \gamma _{2n,0}$ ( $\gamma _{00}>0, \gamma _{ij}=\bar{\gamma}_{ji}$), and let $K$ denote a closed subset of the complex plane $\mathbb{C} $. The Truncated Complex $K$-Moment Problem for $ \gamma $ entails determining whether there exists a positive Borel measure $ \mu $ on $\mathbb{C} $ such that $\gamma _{ij}=\int \bar{z}^{i}z^{j}\,d\mu $ ( $0\leq i+j\leq 2n$) and $\operatorname{supp}\mu \subseteq K$. For $K\equiv K_{\mathcal{P}}$ a semi-algebraic set determined by a collection of complex polynomials $\mathcal{P} =\left\{ p_{i}\left( z,\bar{z}\right) \right\} _{i=1}^{m}$, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix $M\left( n\right) \left( \gamma \right)$and the localizing matrices $M_{p_{i}}$. We prove that there exists a $\operatorname{rank}M\left( n\right)$-atomic representing measure for $\gamma ^{\left( 2n\right) }$supported in $K_{\mathcal{P}}$if and only if $M\left( n\right) \geq 0$and there is some rank-preserving extension $M\left( n+1\right) $for which $M_{p_{i}}\left( n+k_{i}\right) \geq 0$, where $\deg p_{i}=2k_{i} $ or $2k_{i}-1$ $(1\leq i\leq m)$.


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Additional Information

Raúl Curto
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242
Email: curto@math.uiowa.edu

Lawrence A. Fialkow
Affiliation: Department of Mathematics and Computer Science, State University of New York, New Paltz, New York 12561
Email: fialkow@mcs.newpaltz.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02472-7
PII: S 0002-9947(00)02472-7
Keywords: Truncated complex moment problem, moment matrix extension, flat extensions of positive matrices, semi-algebraic sets, localizing matrix
Received by editor(s): May 14, 1998
Published electronically: February 28, 2000
Additional Notes: Research partially supported by NSF grants. The second-named author was also partially supported by the State University of New York at New Paltz Research and Creative Projects Award Program.
Dedicated: Dedicated to Professor Aaron D. Fialkow on the occasion of his eighty-seventh birthday
Article copyright: © Copyright 2000 American Mathematical Society