## 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

## 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

- N. I. Akhiezer,
*The classical moment problem and some related questions in analysis*, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR**0184042** - Aharon Atzmon,
*A moment problem for positive measures on the unit disc*, Pacific J. Math.**59**(1975), no. 2, 317–325. MR**402057** - Raúl E. Curto and Lawrence A. Fialkow,
*Recursiveness, positivity, and truncated moment problems*, Houston J. Math.**17**(1991), no. 4, 603–635. MR**1147276** - Raúl E. Curto and Lawrence A. Fialkow,
*Recursively generated weighted shifts and the subnormal completion problem*, Integral Equations Operator Theory**17**(1993), no. 2, 202–246. MR**1233668**, DOI 10.1007/BF01200218 - Raúl E. Curto and Lawrence A. Fialkow,
*Recursively generated weighted shifts and the subnormal completion problem. II*, Integral Equations Operator Theory**18**(1994), no. 4, 369–426. MR**1265443**, DOI 10.1007/BF01200183 - Raúl E. Curto and Lawrence A. Fialkow,
*Solution of the truncated complex moment problem for flat data*, Mem. Amer. Math. Soc.**119**(1996), no. 568, x+52. MR**1303090**, DOI 10.1090/memo/0568 - R.E. Curto and L.A. Fialkow,
*Flat extensions of positive moment matrices: Relations in analytic or conjugate terms*, Oper. Theory Adv. Appl.**104**(1998), 59–82. - Raúl E. Curto and Lawrence A. Fialkow,
*Flat extensions of positive moment matrices: recursively generated relations*, Mem. Amer. Math. Soc.**136**(1998), no. 648, x+56. MR**1445490**, DOI 10.1090/memo/0648 - R.E. Curto and L.A. Fialkow,
*The quadratic moment problem for the unit disc and unit circle*, preprint 1998. - Lawrence Fialkow,
*Positivity, extensions and the truncated complex moment problem*, Multivariable operator theory (Seattle, WA, 1993) Contemp. Math., vol. 185, Amer. Math. Soc., Providence, RI, 1995, pp. 133–150. MR**1332058**, DOI 10.1090/conm/185/02152 - L. Fialkow,
*Minimal representing measures arising from rank-increasing moment matrix extensions*, J. Operator Theory, to appear. - L. Fialkow,
*Multivariable quadrature and extensions of moment matrices*, preprint 1996. - Bent Fuglede,
*The multidimensional moment problem*, Exposition. Math.**1**(1983), no. 1, 47–65. MR**693807** - E.K. Haviland,
*On the momentum problem for distributions in more than one dimension*, Amer. J. Math.**57**(1935), 562–568; II, Amer. J. Math.**58**(1936), 164–168. - M. G. Kreĭn and A. A. Nudel′man,
*The Markov moment problem and extremal problems*, Translations of Mathematical Monographs, Vol. 50, American Mathematical Society, Providence, R.I., 1977. Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development; Translated from the Russian by D. Louvish. MR**0458081** - Mihai Putinar,
*Positive polynomials on compact semi-algebraic sets*, Indiana Univ. Math. J.**42**(1993), no. 3, 969–984. MR**1254128**, DOI 10.1512/iumj.1993.42.42045 - Mihai Putinar and Florian-Horia Vasilescu,
*Problème des moments sur les compacts semi-algébriques*, C. R. Acad. Sci. Paris Sér. I Math.**323**(1996), no. 7, 787–791 (French, with English and French summaries). MR**1416177** - M. Putinar and F.-H. Vasilescu,
*Solving moment problems by dimensional extension*, C. R. Acad. Sci. Paris Sér. Math.**328**(1999), 495–499. - Bruce Reznick,
*Uniform denominators in Hilbert’s seventeenth problem*, Math. Z.**220**(1995), no. 1, 75–97. MR**1347159**, DOI 10.1007/BF02572604 - M. Riesz,
*Sur le problème des moments, Troisième Note*, Arkiv för Matematik, Astronomi och Fysik**17**(1923), no. 16, 1–52. - Konrad Schmüdgen,
*The $K$-moment problem for compact semi-algebraic sets*, Math. Ann.**289**(1991), no. 2, 203–206. MR**1092173**, DOI 10.1007/BF01446568 - Albert Eagle,
*Series for all the roots of a trinomial equation*, Amer. Math. Monthly**46**(1939), 422–425. MR**5**, DOI 10.2307/2303036 - Ju. L. Šmul′jan,
*An operator Hellinger integral*, Mat. Sb. (N.S.)**49 (91)**(1959), 381–430 (Russian). MR**0121662** - J. Stochel, private communication.
- Jan Stochel,
*Moment functions on real algebraic sets*, Ark. Mat.**30**(1992), no. 1, 133–148. MR**1171099**, DOI 10.1007/BF02384866 - J. Stochel and F. Szafraniec,
*The complex moment problem and subnormality: A polar decomposition approach*, J. Funct. Anal.**159**(1998), 432–491. - F.-H. Vasilescu,
*Moment problems for multi-sequences of operators*, J. Math. Anal. Appl.**219**(1998), no. 2, 246–259. MR**1606389**, DOI 10.1006/jmaa.1997.5787 - F.-H. Vasilescu,
*Operator moment problems*, preprint 1998.

## 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
- 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