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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The truncated complex $K$-moment problem
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by Raúl Curto and Lawrence A. Fialkow PDF
Trans. Amer. Math. Soc. 352 (2000), 2825-2855 Request permission


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:
  • Lawrence A. Fialkow
  • Affiliation: Department of Mathematics and Computer Science, State University of New York, New Paltz, New York 12561
  • Email:
  • 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
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 2825-2855
  • MSC (2000): Primary 47A57, 44A60, 30E05; Secondary 15A57, 15-04, 47N40, 47A20
  • DOI:
  • MathSciNet review: 1661305