## The truncated complex $K$-moment problem

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- by Raúl Curto and Lawrence A. Fialkow
- Trans. Amer. Math. Soc.
**352**(2000), 2825-2855 - DOI: https://doi.org/10.1090/S0002-9947-00-02472-7
- Published electronically: February 28, 2000
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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 $2k_{i}-1$ $(1\leq i\leq m)$.## References

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## Bibliographic 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
- 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.
- © 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: https://doi.org/10.1090/S0002-9947-00-02472-7
- MathSciNet review: 1661305

Dedicated: Dedicated to Professor Aaron D. Fialkow on the occasion of his eighty-seventh birthday