Abstract
We study the truncated multidimensional moment problem with a general type of truncations. The operator approach to the moment problem is presented. The case where the associated operators form a commuting self-adjoint tuple is characterized in terms of the given moments. The case of the dimensional stability is characterized in terms of the prescribed moments as well. Some sufficient conditions for the solvability of the moment problem are presented. A construction of the corresponding solution is described by algorithms. Numerical examples of the construction are provided.
References
[1] Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators, (Amer. Math. Soc., Providence, RI, 1968)10.1090/mmono/017Search in Google Scholar
[2] C. Berg, J. P. R. Christensen, P. Ressel, Harmonic Analysis on Semigroups, (Springer-Verlag, New York, 1984)10.1007/978-1-4612-1128-0Search in Google Scholar
[3] D. Cichon, J. Stochel, F. H. Szafraniec, Extending positive definiteness, Trans. Amer. Math. Soc., 363, No. 1 (2011), 545-577.Search in Google Scholar
[4] R. Curto, L. Fialkow, Solution of the Truncated Complex Moment Problem for Flat Data, (Memoirs Amer. Math. Soc. 119, no. 568 (1996), x+52 pp.)10.1090/memo/0568Search in Google Scholar
[5] R. Curto, L. Fialkow, Flat Extensions of Positive Moment Matrices: Recursively Generated Relations, (Memoirs Amer. Math. Soc. 136, no. 648 (1998), x+56 pp.)10.1090/memo/0648Search in Google Scholar
[6] L. A. Fialkow, Solution of the truncated moment problem with variety y = x3, Trans. Amer. Math. Soc., 363, No. 6 (2011), 3133-3165.Search in Google Scholar
[7] B. Fuglede, The multidimensional moment problem, Expo. Math., 1 no. 4 (1983), 47-65.Search in Google Scholar
[8] D. P. Kimsey, M. Putinar, Complex orthogonal polynomials and numerical quadrature via hyponormality, Comput. Methods Funct. Theory, (2018), 1-16.10.1007/s40315-018-0237-3Search in Google Scholar
[9] D. P. Kimsey, H. J. Woerdeman, The truncated matrix-valued K-moment problem on Rd, Cd, and Td, Trans. Am. Math. Soc. 365, 10 (2013), 5393-5430.10.1090/S0002-9947-2013-05812-6Search in Google Scholar
[10] D. P. Kimsey, The subnormal completion problem in several variables, J. Math. Anal. Appl. 434, 2 (2016), 1504-1532.10.1016/j.jmaa.2015.09.063Search in Google Scholar
[11] J.-B. Lasserre, Moments, Positive Polynomials and Their Applications, (World Scientific, 2010)10.1142/p665Search in Google Scholar
[12] M. Laurent, B. Mourrain, A generalized flat extension theorem for moment matrices, Arch. Math. 93, (2009), 87-98.10.1007/s00013-009-0007-6Search in Google Scholar
[13] M.Marshall, Positive Polynomials and Sums of Squares, (Amer.Math. Soc.,Math. Surveys and Monographs, Vol. 146, 2008)10.1090/surv/146Search in Google Scholar
[14] I. E. Ovcharenko, Two-dimensional power moment sequences, (in Russian) Ukrain. Mat. Zh. 36, no. 1 (1984), 51-56.Search in Google Scholar
[15] I. E. Ovcharenko, Two-dimensional Hausdorff moment sequences, (in Russian) Ukrain. Mat. Zh. 36, no. 6 (1984), 729-733.Search in Google Scholar
[16] M. Putinar, F.-H. Vasilescu, Solving moment problems by dimensional extension, Annals of Math., 149 (1999), 1087-1107.10.2307/121083Search in Google Scholar
[17] K. Schmüdgen, The K-moment problem for compact semi-algebraic sets, Math. Ann., 289 (1991), 203-206.10.1007/BF01446568Search in Google Scholar
[18] K. Schmüdgen, On the moment problem of closed semi-algebraic sets, J. reine angew. Math., 558 (2003), 225-234.Search in Google Scholar
[19] Ph. J. di Dio, Konrad Schmüdgen, The multidimensional truncated moment problem: atoms, determinacy, and core variety, arXiv:1703.01497Search in Google Scholar
[20] J. A. Shohat, J. D. Tamarkin, The Problem of Moments, (Amer. Math. Soc., 531 West 116th Street, New York City, 1943)10.1090/surv/001Search in Google Scholar
[21] F.-H. Vasilescu, Hamburger and Stieltjes moment problems in several variables, Trans. Amer.Math. Soc., 354 3 (2001), 1265-1278.Search in Google Scholar
[22] F.-H. Vasilescu, Dimensional stability in truncated moment problems, J. Math. Anal. Appl., 388 (2012), 219-230.10.1016/j.jmaa.2011.11.063Search in Google Scholar
[23] F.-H. Vasilescu, An idempotent approach to truncated moment problems, Integr. Equ. Oper. Theory, 79 Issue 3 (2014), 301-335.Search in Google Scholar
[24] S. Yoo, Sextic moment problems on 3 parallel lines, Bull. Korean. Math. Soc., 54, No. 1 (2017), 299-318.Search in Google Scholar
[25] S. M. Zagorodnyuk, On the two-dimensional moment problem, Ann. Funct. Anal. , 1, no. 1 (2010), 80-104.Search in Google Scholar
[26] S.M. Zagorodnyuk, Generalized resolvents of symmetric and isometric operators: the Shtraus approach, Annals of Functional Analysis, 4, No. 1 (2013), 175-285.Search in Google Scholar
[27] S. M. Zagorodnyuk, The two-dimensional moment problem in a strip,Methods Funct. Anal. Topology, 19, no. 1 (2013), 40-54.Search in Google Scholar
[28] S. M. Zagorodnyuk, The Nevanlinna-type parametrization for the operator Hamburger moment problem, J. Adv. Math. Stud., 10, No. 2 (2017), 183-199.Search in Google Scholar
[29] S. Zagorodnyuk, On the truncated two-dimensional moment problem, Adv. Oper. Theory, 3, no. 2 (2018), 63-74.Search in Google Scholar
© by Sergey M. Zagorodnyuk, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 Public License.
