The truncated complex $K$-moment problem
HTML articles powered by AMS MathViewer
- 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
- PDF | 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.
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