Open Access

The multi-dimensional truncated moment problem: maximal masses

     Article (.pdf)


Given a subset $\mathcal K$ of $\mathbb R^d$ and a linear functional $L$ on the polynomials $\mathbb R^d_{2n}[\underline{x}]$ in $d$ variables and of degree at most $2n$ the truncated $\mathcal K$-moment problem asks when there is a positive Borel measure $\mu$ supported by $\mathcal K$ such that $L(p)=\int p\, d\mu$ for $p\in \mathbb R^d_{2n}[\underline{x}]$. For compact sets $\mathcal K$ we investigate the maximal mass of all representing measures at a given point of $\mathcal K$. Various characterizations of this quantity and related properties are developed and a close link to zeros of positive polynomials is established.

Key words: Moment problem, maximal mass, positive polynomial.

Full Text

Article Information

TitleThe multi-dimensional truncated moment problem: maximal masses
SourceMethods Funct. Anal. Topology, Vol. 21 (2015), no. 3, 266-281
MathSciNet MR3521697
zbMATH 06630273
MilestonesReceived 11/02/2015; Revised 28/02/2015
CopyrightThe Author(s) 2015 (CC BY-SA)

Authors Information

Konrad Schmudgen
Universitat Leipzig, Mathematisches Institut, Augustusplatz 10/11, D-04109 Leipzig, Germany

Citation Example

Konrad Schmüdgen, The multi-dimensional truncated moment problem: maximal masses, Methods Funct. Anal. Topology 21 (2015), no. 3, 266-281.


@article {MFAT783,
    AUTHOR = {Schmüdgen, Konrad},
     TITLE = {The multi-dimensional truncated moment problem: maximal masses},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {3},
     PAGES = {266-281},
      ISSN = {1029-3531},
  MRNUMBER = {MR3521697},
 ZBLNUMBER = {06630273},
       URL = {},

Google Scholar Metrics

Citing articles in Google Scholar
Similar articles in Google Scholar

Export article

Save to Mendeley


  1. N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965.  MathSciNet
  2. Aharon Ben-Tal, Arkadi Nemirovski, Lectures on modern convex optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2001. Analysis, algorithms, and engineering applications  MathSciNet CrossRef
  3. Ju. M. Berezans′kii, Expansions in eigenfunctions of selfadjoint operators, American Mathematical Society, Providence, R.I., 1968.  MathSciNet
  4. Christian Berg, Jens Peter Reus Christensen, Paul Ressel, Harmonic analysis on semigroups, Springer-Verlag, New York, 1984. Theory of positive definite and related functions  MathSciNet CrossRef
  5. G. Blekherman, J. B. Lasserre, The truncated K-moment problem for closure of open sets, J. Funct. Anal. 263 (2012), no. 11, 3604-3616.  MathSciNet CrossRef
  6. G. Choquet, Lectures on Analysis, Vol. III, Benjamin, New York, 1969.
  7. Man Duen Choi, Tsit Yuen Lam, Bruce Reznick, Real zeros of positive semidefinite forms. I, Math. Z. 171 (1980), no. 1, 1-26.  MathSciNet CrossRef
  8. Ral E. Curto, Lawrence A. Fialkow, Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc. 119 (1996), no. 568, x+52.  MathSciNet CrossRef
  9. Ral E. Curto, Lawrence A. Fialkow, Flat extensions of positive moment matrices: recursively generated relations, Mem. Amer. Math. Soc. 136 (1998), no. 648, x+56.  MathSciNet CrossRef
  10. Ral E. Curto, Lawrence A. Fialkow, The truncated complex $K$-moment problem, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2825-2855.  MathSciNet CrossRef
  11. R. G. Douglas, On extremal measures and subspace density, Michigan Math. J. 11 (1964), 243-246.  MathSciNet
  12. Lawrence A. Fialkow, Truncated multivariable moment problems with finite variety, J. Operator Theory 60 (2008), no. 2, 343-377.  MathSciNet
  13. William J. Helton, Jiawang Nie, A semidefinite approach for truncated $K$-moment problems, Found. Comput. Math. 12 (2012), no. 6, 851-881.  MathSciNet CrossRef
  14. M. G. Krein, A. A. Nudel′man, The Markov moment problem and extremal problems, American Mathematical Society, Providence, R.I., 1977. Ideas and problems of P. L. \vCeby\vsev and A. A. Markov and their further development, Translated from the Russian by D. Louvish, Translations of Mathematical Monographs, Vol. 50  MathSciNet
  15. Jean Bernard Lasserre, Moments, positive polynomials and their applications, Imperial College Press, London, 2010.  MathSciNet
  16. Monique Laurent, Sums of squares, moment matrices and optimization over polynomials, in: Emerging applications of algebraic geometry, Springer, New York, 2009.  MathSciNet CrossRef
  17. J. Matzke, Mehrdimensionale Momentenprobleme und Positivitatskegel, Dissertation, Universitat Leipzig, 1992.
  18. Murray Marshall, Positive polynomials and sums of squares, American Mathematical Society, Providence, RI, 2008.  MathSciNet CrossRef
  19. Hans Richter, Parameterfreie Abschatzung und Realisierung von Erwartungswerten, Bl. Deutsch. Ges. Versicherungsmath. 3 (1957), 147-162.  MathSciNet
  20. Raphael M. Robinson, Some definite polynomials which are not sums of squares of real polynomials, in: Selected questions of algebra and logic (collection dedicated to the memory of A. I. Mal′ cev) (Russian), Izdat. ``Nauka'' Sibirsk. Otdel., Novosibirsk, 1973.  MathSciNet
  21. W. W. Rogosinski, Moments of non-negative mass, Proc. Roy. Soc. London Ser. A 245 (1958), 1-27.  MathSciNet
  22. Konrad Schmudgen, Unbounded self-adjoint operators on Hilbert space, Springer, Dordrecht, 2012.  MathSciNet CrossRef
  23. Vladimir Tchakaloff, Formules de cubatures mecaniques \`a coefficients non negatifs, Bull. Sci. Math. (2) 81 (1957), 123-134.  MathSciNet

All Issues