Archiv der Mathematik

, Volume 93, Issue 1, pp 87–98 | Cite as

A generalized flat extension theorem for moment matrices

  • Monique Laurent
  • Bernard Mourrain


In this note we prove a generalization of the flat extension theorem of Curto and Fialkow (Memoirs of the American Mathematical Society, vol. 119. American Mathematical Society, Providence, 1996) for truncated moment matrices. It applies to moment matrices indexed by an arbitrary set of monomials and its border, assuming that this set is connected to 1. When formulated in a basis-free setting, this gives an equivalent result for truncated Hankel operators.

Mathematics Subject Classification (2000)

Primary 30E05 Secondary 12D10 


Truncated moment problem Moment matrix Hankel operator Polynomial optimization 


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  1. 1.
    Akhiezer N.I.: The Classical Moment Problem. Hafner, New York (1965)Google Scholar
  2. 2.
    Berg C., Christensen J.P.R., Ressel P.: Positive definite functions on Abelian semigroups. Math. Ann. 223, 253–272 (1976)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berg C., Maserick P.H.: Exponentially bounded positive definite functions. Illinois J. Math. 28, 162–179 (1984)MATHMathSciNetGoogle Scholar
  4. 4.
    R. E. Curto and L. A. Fialkow, Solution of the truncated complex moment problem for flat data, Memoirs of the American Mathematical Society, 119, Amer. Math. Soc., Providence, RI, 1996.Google Scholar
  5. 5.
    R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: recursively generated relations, Memoirs of the American Mathematical Society, 648, Amer. Math. Soc., Providence, RI, 1998.Google Scholar
  6. 6.
    Curto R.E., Fialkow L.A.: The truncated complex K-moment problem. Trans. Amer. Math. Soc. 352, 2825–2855 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fuglede B.: The multidimensional moment problem. Expositiones Mathematicae 1, 47–65 (1983)MATHMathSciNetGoogle Scholar
  8. 8.
    Hamburger H.: Über eine Erweiterung des Stieltjesschen Momentproblems. Parts I, II, III. Math. Ann. 81, 235–319 (1920) 82 (1921), 20–164, 168–187CrossRefMathSciNetGoogle Scholar
  9. 9.
    D. Henrion and J. B. Lasserre, Detecting global optimality and extracting solutions in GloptiPoly, In: Positive Polynomials in Control, D. Henrion and A. Garulli (eds.), Lecture Notes on Control and Information Sciences 312 (2005), 293–310, Springer, Berlin.Google Scholar
  10. 10.
    A. Kehrein, M. Kreuzer, and L. Robbiano, An algebraist’s view on border bases. Solving Polynomial Equations—Foundations, Algorithms and Applications, A. Dickenstein and I. Z. Emiris (eds.), pp. 169–202. Springer, Berlin, 2005.Google Scholar
  11. 11.
    J. B. Lasserre et al., Moment matrices, border bases and real radical ideals, In preparation.Google Scholar
  12. 12.
    Lasserre J.B.: Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization 11, 796–817 (2001)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lasserre J.B., Laurent M., Rostalski P.: Semidefinite characterization and computation of zero-dimensional real radical ideals. Found. Comput. Math. 8, 607–647 (2008)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lasserre J.B., Laurent M., Rostalski P.: A prolongation-projection algorithm for computing the finite real variety of an ideal. Theoret. Comput. Sci. 410, 2685–2700 (2009)MATHCrossRefGoogle Scholar
  15. 15.
    Laurent M.: Revisiting two theorems of Curto and Fialkow on moment matrices. Proceedings of the American Mathematical Society 133, 2965–2976 (2005)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    M. Laurent, Sums of squares, moment matrices and optimization over polynomials, in: Emerging Applications of Algebraic Geometry, IMA Volumes in Mathematics and its Applications, 149, M. Putinar and S. Sullivant (eds.), Springer-Verlag, Berlin, pp. 157–270, 2009.Google Scholar
  17. 17.
    M. Marshall, Positive Polynomials and Sums of Squares, Mathematical Surveys and Monographs, 146 AMS, Providence, 2008.Google Scholar
  18. 18.
    B. Mourrain, A new criterion for normal form algorithms, In: Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, H. Imai, S. Lin, and A. Poli (eds.) Lecture Notes In Computer Science, 1719 pp. 430–443. Springer-Verlag, Berlin, 1999.Google Scholar
  19. 19.
    Mourrain B., Pan V.Y.: Multivariate Polynomials, Duality, and Structured Matrices. J. Complexity 16, 110–180 (2000)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Stochel J.: Solving the truncated moment problem solves the moment problem. Glasgow Journal of Mathematics 43, 335–341 (2001)MATHMathSciNetGoogle Scholar
  21. 21.
    Vandenberghe L., Boyd S.: Semidefinite Programming. SIAM Review 38, 49–95 (1996)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Wolkowicz, H., Saigal, R., Vandeberghe, L. (eds): Handbook of Semidefinite Programming. Kluwer, Dordrecht (2000)Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Centrum Wiskunde& Informatica (CWI)AmsterdamThe Netherlands
  2. 2.GALAAD, INRIA MéditerranéeSophia AntipolisFrance

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