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Transactions of the American Mathematical Society

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Limits of positive flat bivariate moment matrices


Author: Lawrence A. Fialkow
Journal: Trans. Amer. Math. Soc. 367 (2015), 2665-2702
MSC (2010): Primary 47A57, 44A60, 42A70, 30E05; Secondary 15A57, 15-04, 47A20
DOI: https://doi.org/10.1090/S0002-9947-2014-06393-9
Published electronically: December 3, 2014
MathSciNet review: 3301877
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Abstract: The bivariate moment problem for a sequence $ \beta \equiv \beta ^{(6)}$ of degree $ 6$ remains unsolved, but we prove that if the associated $ 10 \times 10$ moment matrix $ M_{3}(\beta )$ satisfies $ M_{3}\succeq 0$ and $ rank~M_{3}\le 6$, then $ \beta $ admits a sequence of approximate representing measures, and $ \beta ^{(5)}$ has a representing measure. More generally, let $ \overline {\mathcal {F}_{d}}$ denote the closure of the positive flat moment matrices of degree $ 2d$ in $ n$ variables. Each matrix in $ \overline {\mathcal {F}_{d}}$ admits computable approximate representing measures, and in 2013, Jiawang Nie and the author began to study concrete conditions for membership in this class. Let $ \beta \equiv \beta ^{(2d)}=\{\beta _{i}\}_{ i\in \mathbb{Z}_{+}^{n},\vert i\vert \leq 2d }$, $ \beta _{0}>0$, denote a real $ n$-dimensional sequence of degree $ 2d$. If the corresponding moment matrix $ M_{d}\equiv M_{d}(\beta )$ is the limit of a sequence of positive flat moment matrices $ \{M_{d}^{(k)}\}$, i.e., $ M_{d}^{(k)}\succeq 0$ and $ rank~M_{d}^{(k)} = rank~M_{d-1}^{(k)}$, then i) $ M_{d}\succeq 0$, ii) $ rank~M_{d} \le \rho _{d-1} \equiv dim~\mathbb{R}[x_{1},\ldots ,x_{n}]_{d-1}$, and iii) $ \beta ^{(2d-1)}$ admits a representing measure. We extend our earlier results by proving, conversely, that for $ n=2$, if $ M_{d}$ satisfies certain positivity and rank conditions related to i)-iii), then $ M_{d}$ is the limit of positive flat moment matrices.


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  • [A] N. I. Akhiezer, The classical moment problem and some related questions in analysis, translated by N. Kemmer, Hafner Publishing Co., New York, 1965. MR 0184042 (32 #1518)
  • [BT] Christian Bayer and Josef Teichmann, The proof of Tchakaloff's theorem, Proc. Amer. Math. Soc. 134 (2006), no. 10, 3035-3040 (electronic). MR 2231629 (2007d:44004), https://doi.org/10.1090/S0002-9939-06-08249-9
  • [B] Sterling K. Berberian, Lectures in functional analysis and operator theory, Graduate Texts in Mathematics, No. 15, Springer-Verlag, New York, 1974. MR 0417727 (54 #5775)
  • [Bl] Grigoriy Blekherman, Positive Gorenstein ideals, Proc. Amer. Math. Soc. 143 (2015), no. 1, 69-86. MR 3272733, https://doi.org/10.1090/S0002-9939-2014-12253-2
  • [Co] John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713 (91e:46001)
  • [CF1] Raúl E. Curto and Lawrence A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math. 17 (1991), no. 4, 603-635. MR 1147276 (93a:47016)
  • [CF2] 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 (96g:47009), https://doi.org/10.1090/memo/0568
  • [CF3] Raúl E. Curto and Lawrence A. Fialkow, Flat extensions of positive moment matrices: relations in analytic or conjugate terms, Nonselfadjoint operator algebras, operator theory, and related topics, Oper. Theory Adv. Appl., vol. 104, Birkhäuser, Basel, 1998, pp. 59-82. MR 1639649 (99i:47026)
  • [CF4] 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 (99d:47015), https://doi.org/10.1090/memo/0648
  • [CF5] Raúl E. Curto and Lawrence A. Fialkow, Solution of the singular quartic moment problem, J. Operator Theory 48 (2002), no. 2, 315-354. MR 1938799 (2003j:47017)
  • [CF6] Raúl E. Curto and Lawrence A. Fialkow, Solution of the truncated parabolic moment problem, Integral Equations Operator Theory 50 (2004), no. 2, 169-196. MR 2099788 (2005f:47041), https://doi.org/10.1007/s00020-003-1275-3
  • [CF7] Raúl E. Curto and Lawrence A. Fialkow, Truncated $ K$-moment problems in several variables, J. Operator Theory 54 (2005), no. 1, 189-226. MR 2168867 (2006e:47032)
  • [CF8] Raúl E. Curto and Lawrence A. Fialkow, Solution of the truncated hyperbolic moment problem, Integral Equations Operator Theory 52 (2005), no. 2, 181-218. MR 2216081 (2007a:47017), https://doi.org/10.1007/s00020-004-1340-6
  • [CF9] Raúl E. Curto and Lawrence A. Fialkow, An analogue of the Riesz-Haviland theorem for the truncated moment problem, J. Funct. Anal. 255 (2008), no. 10, 2709-2731. MR 2464189 (2009i:47039), https://doi.org/10.1016/j.jfa.2008.09.003
  • [CF10] Raúl E. Curto and Lawrence A. Fialkow, Recursively determined representing measures for bivariate truncated moment sequences, J. Operator Theory 70 (2013), no. 2, 401-436. MR 3138363, https://doi.org/10.7900/jot.2011sep06.1943
  • [EF] Chirakkal Easwaran and Lawrence Fialkow, Positive linear functionals without representing measures, Oper. Matrices 5 (2011), no. 3, 425-434. MR 2858497 (2012m:47033), https://doi.org/10.7153/oam-05-30
  • [F] Lawrence A. Fialkow, Solution of the truncated moment problem with variety $ y=x^3$, Trans. Amer. Math. Soc. 363 (2011), no. 6, 3133-3165. MR 2775801 (2012c:47044), https://doi.org/10.1090/S0002-9947-2011-05262-1
  • [FN1] Lawrence Fialkow and Jiawang Nie, Positivity of Riesz functionals and solutions of quadratic and quartic moment problems, J. Funct. Anal. 258 (2010), no. 1, 328-356. MR 2557966 (2010j:47017), https://doi.org/10.1016/j.jfa.2009.09.015
  • [FN2] Lawrence Fialkow and Jiawang Nie, On the closure of positive flat moment matrices, J. Operator Theory 69 (2013), no. 1, 257-277. MR 3029497, https://doi.org/10.7900/jot.2010may11.1890
  • [FN3] Lawrence Fialkow and Jiawang Nie, The truncated moment problem via homogenization and flat extensions, J. Funct. Anal. 263 (2012), no. 6, 1682-1700. MR 2948227, https://doi.org/10.1016/j.jfa.2012.06.004
  • [H] E. K. Haviland, On the momentum problem for distribution functions in more than one dimension. II, Amer. J. Math. 58 (1936), no. 1, 164-168. MR 1507139, https://doi.org/10.2307/2371063
  • [KN] M. G. Kreĭn and A. A. Nudelman, The Markov moment problem and extremal problems. Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development; Translated from the Russian by D. Louvish; Translations of Mathematical Monographs, Vol. 50, American Mathematical Society, Providence, R.I., 1977. MR 0458081 (56 #16284)
  • [HN] J. William Helton and Jiawang Nie, A semidefinite approach for truncated $ K$-moment problems, Found. Comput. Math. 12 (2012), no. 6, 851-881. MR 2989475, https://doi.org/10.1007/s10208-012-9132-x
  • [HeLa] Didier Henrion and Jean-Bernard Lasserre, GloptiPoly: global optimization over polynomials with Matlab and SeDuMi, ACM Trans. Math. Software 29 (2003), no. 2, 165-194. MR 2000881 (2004g:90084), https://doi.org/10.1145/779359.779363
  • [Her] Domingo A. Herrero, Approximation of Hilbert space operators. Vol. 1, 2nd ed., Pitman Research Notes in Mathematics Series, vol. 224, Longman Scientific & Technical, Harlow, 1989. MR 1088255 (91k:47002)
  • [Las] Jean B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11 (2000/01), no. 3, 796-817. MR 1814045 (2002b:90054), https://doi.org/10.1137/S1052623400366802
  • [Lau] Monique Laurent, Sums of squares, moment matrices and optimization over polynomials, Emerging applications of algebraic geometry, IMA Vol. Math. Appl., vol. 149, Springer, New York, 2009, pp. 157-270. MR 2500468 (2010j:13054), https://doi.org/10.1007/978-0-387-09686-5_7
  • [Rez] Bruce Reznick, Some concrete aspects of Hilbert's 17th Problem, Real algebraic geometry and ordered structures (Baton Rouge, LA, 1996), Contemp. Math., vol. 253, Amer. Math. Soc., Providence, RI, 2000, pp. 251-272. MR 1747589 (2001i:11042), https://doi.org/10.1090/conm/253/03936
  • [R] M. Riesz, Sur le problème des moments, Troisième Note, Arkiv für Matematik, Astronomi och Fysik 17 (1923), 1-52.
  • [SH] S. L. Salas and E. Hille, Calculus: One and several variables, Fourth Edition, John Wiley and Sons, 1982.
  • [S1] Konrad Schmüdgen, An example of a positive polynomial which is not a sum of squares of polynomials. A positive, but not strongly positive functional, Math. Nachr. 88 (1979), 385-390. MR 543417 (81b:12024), https://doi.org/10.1002/mana.19790880130
  • [S2] Konrad Schmüdgen, The $ K$-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), no. 2, 203-206. MR 1092173 (92b:44011), https://doi.org/10.1007/BF01446568
  • [T] Vladimir Tchakaloff, Formules de cubatures mécaniques à coefficients non négatifs, Bull. Sci. Math. (2) 81 (1957), 123-134 (French). MR 0094632 (20 #1145)

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Additional Information

Lawrence A. Fialkow
Affiliation: Department of Computer Science, State University of New York, New Paltz, New York 12561
Email: fialkowl@newpaltz.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06393-9
Keywords: Truncated moment problem, Riesz functional, moment matrix extension, flat extensions of positive matrices, positive functional
Received by editor(s): February 10, 2013
Published electronically: December 3, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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