Previous |  Up |  Next

Article

Title: On a variational approach to truncated problems of moments (English)
Author: Ambrozie, C.-G.
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 138
Issue: 1
Year: 2013
Pages: 105-112
Summary lang: English
.
Category: math
.
Summary: We characterize the existence of the $L^1$ solutions of the truncated moments problem in several real variables on unbounded supports by the existence of the maximum of certain concave Lagrangian functions. A natural regularity assumption on the support is required. (English)
Keyword: problem of moments
Keyword: representing measure
MSC: 30E05
MSC: 44A60
MSC: 49J99
idZBL: Zbl 1274.44010
idMR: MR3076224
DOI: 10.21136/MB.2013.143233
.
Date available: 2013-03-02T18:58:29Z
Last updated: 2020-07-29
Stable URL: http://hdl.handle.net/10338.dmlcz/143233
.
Reference: [1] Akhiezer, N. I.: The Classical Moment Problem and Some Related Questions in Analysis.Univ. Math. Monographs. Oliver & Boyd, Edinburgh (1965) (English translation). Zbl 0135.33803, MR 0184042
Reference: [2] Ambrozie, C.-G.: Maximum entropy and moment problems.Real Anal. Exchange 29 (2003/04), 607-627. MR 2083800, 10.14321/realanalexch.29.2.0607
Reference: [3] Ambrozie, C.-G.: Multivariate truncated moments problems and maximum entropy, arXiv:1111.7123..
Reference: [4] Ambrozie, C.-G.: A Riesz-Haviland type result for truncated moment problems with solutions in $L^1$, arXiv:1111.6555.(to appear) in J. Operator Theory.
Reference: [5] Blekherman, G., Lasserre, J. B.: The truncated $K$-moment problem for closure of open sets.J. Funct. Anal. 263 3604-3616. MR 2984077, 10.1016/j.jfa.2012.09.001
Reference: [6] Borwein, J. M.: Maximum entropy and feasibility methods for convex and nonconvex inverse problems.Optimization 61 (2012), 1-33. Zbl 1241.45009, MR 2875572, 10.1080/02331934.2011.632502
Reference: [7] Borwein, J. M., Lewis, A. S.: Convergence of best entropy estimates.SIAM J. Optim. 1 (1991), 191-205. Zbl 0756.41037, MR 1098426, 10.1137/0801014
Reference: [8] Borwein, J. M., Lewis, A. S.: Duality relationships for entropy-like minimization problems.SIAM J. Control Optimization 29 (1991), 325-338. Zbl 0797.49030, MR 1092730, 10.1137/0329017
Reference: [9] Campenhout, J. M. Van, Cover, T. M.: Maximum entropy and conditional probability.IEEE Trans. Inf. Theory IT--27 (1981), 483-489. MR 0635527
Reference: [10] Cassier, G.: Problème des moments sur un compact de $\mathbb{R}^{n}$ et décomposition de polynômes à plusieurs variables.J. Funct. Analysis 58 (1984), 254-266 French. MR 0759099, 10.1016/0022-1236(84)90042-9
Reference: [11] Cichoń, D., Stochel, J., Szafraniec, F. H.: Riesz-Haviland criterion for incomplete data.J. Math. Anal. Appl. 380 (2011), 94-104. Zbl 1214.30020, MR 2786187, 10.1016/j.jmaa.2011.02.035
Reference: [12] Curto, R. E., Fialkow, L. A.: An analogue of the Riesz-Haviland theorem for the truncated moment problem.J. Funct. Anal. 255 (2008), 2709-2731. Zbl 1158.44003, MR 2464189, 10.1016/j.jfa.2008.09.003
Reference: [13] Curto, R. E., Fialkow, L. A., Moeller, H. M.: The extremal truncated moment problem.Integral Equations Oper. Theory 60 (2008), 177-200. Zbl 1145.47012, MR 2375563, 10.1007/s00020-008-1557-x
Reference: [14] Fuglede, B.: The multidimensional moment problem.Exp. Math. 1 (1983), 47-65. Zbl 0514.44006, MR 0693807
Reference: [15] Haviland, E. K.: On the momentum problem for distributions in more than one dimension.Amer. J. Math. 57 (1935), 562-568. MR 1507095, 10.2307/2371187
Reference: [16] Hauck, C. D., Levermore, C. D., Tits, A. L.: Convex duality and entropy-based moment closures; characterizing degenerate densities.SIAM J. Control Optim. 47 (2008), 1977-2015. Zbl 1167.49033, MR 2421338, 10.1137/070691139
Reference: [17] Junk, M.: Maximum entropy for reduced moment problems.Math. Models Methods Appl. Sci. 10 (2000), 1001-1025. Zbl 1012.44005, MR 1780147, 10.1142/S0218202500000513
Reference: [18] Kagan, A. M., Linnik, Y. V., Rao, C. R.: Characterization Problems in Mathematical Statistics.Wiley, New York (1983). MR 0346969
Reference: [19] Kuhlmann, S., Marshall, M., Schwartz, N.: Positivity, sums of squares and the multi-dimensional moment problem. II.Adv. Geom. 5 (2005), 583-606. Zbl 1095.14055, MR 2174483
Reference: [20] Kullback, S.: Information Theory and Statistics.John Wiley, New York (1959). Zbl 0088.10406, MR 0103557
Reference: [21] Lewis, A. S.: Consistency of moment systems.Can. J. Math. 47 (1995), 995-1006. Zbl 0840.90106, MR 1350646, 10.4153/CJM-1995-052-2
Reference: [22] Léonard, C.: Minimization of entropy functionals.J. Math. Anal. Appl. 346 (2008), 183-204. Zbl 1152.49039, MR 2428283, 10.1016/j.jmaa.2008.04.048
Reference: [23] Mead, L. R., Papanicolaou, N.: Maximum entropy and the problem of moments.J. Math. Phys. 25 (1984), 2404-2417. MR 0751523, 10.1063/1.526446
Reference: [24] Moreau, J. J.: Sur la fonction polaire d'une fonction semicontinue supérieurement.C. R. Acad. Sci., Paris 258 (1964), 1128-1130 French. Zbl 0144.30501, MR 0160093
Reference: [25] Putinar, M., Vasilescu, F.-H.: Solving moment problems by dimensional extension.Ann. Math., II. Ser. 149 (1999), 1087-1107. Zbl 0939.44003, MR 1709313, 10.2307/121083
Reference: [26] Rockafellar, R. T.: Extension of Fenchel's duality for convex functions.Duke Math. J. 33 (1966), 81-89. MR 0187062, 10.1215/S0012-7094-66-03312-6
Reference: [27] Rockafellar, R. T.: Convex Analysis.Princeton University Press, Princeton, New Jersey (1970). Zbl 0193.18401, MR 0274683
.

Files

Files Size Format View
MathBohem_138-2013-1_9.pdf 253.9Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo