On a variational approach to truncated problems of moments

C.-G. Ambrozie

Mathematica Bohemica (2013)

  • Volume: 138, Issue: 1, page 105-112
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Ambrozie, C.-G.. "On a variational approach to truncated problems of moments." Mathematica Bohemica 138.1 (2013): 105-112. <http://eudml.org/doc/252472>.

@article{Ambrozie2013,
abstract = {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.},
author = {Ambrozie, C.-G.},
journal = {Mathematica Bohemica},
keywords = {problem of moments; representing measure; moment problem; representing measure; Lagrangian},
language = {eng},
number = {1},
pages = {105-112},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a variational approach to truncated problems of moments},
url = {http://eudml.org/doc/252472},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Ambrozie, C.-G.
TI - On a variational approach to truncated problems of moments
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 1
SP - 105
EP - 112
AB - 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.
LA - eng
KW - problem of moments; representing measure; moment problem; representing measure; Lagrangian
UR - http://eudml.org/doc/252472
ER -

References

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  1. Akhiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis, Univ. Math. Monographs. Oliver & Boyd, Edinburgh (1965) (English translation). Zbl0135.33803MR0184042
  2. Ambrozie, C.-G., 10.14321/realanalexch.29.2.0607, Real Anal. Exchange 29 (2003/04), 607-627. (2003) MR2083800DOI10.14321/realanalexch.29.2.0607
  3. Ambrozie, C.-G., Multivariate truncated moments problems and maximum entropy, arXiv:1111.7123, . 
  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. 
  5. Blekherman, G., Lasserre, J. B., 10.1016/j.jfa.2012.09.001, J. Funct. Anal. 263 3604-3616. MR2984077DOI10.1016/j.jfa.2012.09.001
  6. Borwein, J. M., 10.1080/02331934.2011.632502, Optimization 61 (2012), 1-33. (2012) Zbl1241.45009MR2875572DOI10.1080/02331934.2011.632502
  7. Borwein, J. M., Lewis, A. S., 10.1137/0801014, SIAM J. Optim. 1 (1991), 191-205. (1991) Zbl0756.41037MR1098426DOI10.1137/0801014
  8. Borwein, J. M., Lewis, A. S., 10.1137/0329017, SIAM J. Control Optimization 29 (1991), 325-338. (1991) Zbl0797.49030MR1092730DOI10.1137/0329017
  9. Campenhout, J. M. Van, Cover, T. M., Maximum entropy and conditional probability, IEEE Trans. Inf. Theory IT--27 (1981), 483-489. (1981) MR0635527
  10. Cassier, G., 10.1016/0022-1236(84)90042-9, J. Funct. Analysis 58 (1984), 254-266 French. (1984) MR0759099DOI10.1016/0022-1236(84)90042-9
  11. Cichoń, D., Stochel, J., Szafraniec, F. H., 10.1016/j.jmaa.2011.02.035, J. Math. Anal. Appl. 380 (2011), 94-104. (2011) Zbl1214.30020MR2786187DOI10.1016/j.jmaa.2011.02.035
  12. Curto, R. E., Fialkow, L. A., 10.1016/j.jfa.2008.09.003, J. Funct. Anal. 255 (2008), 2709-2731. (2008) Zbl1158.44003MR2464189DOI10.1016/j.jfa.2008.09.003
  13. Curto, R. E., Fialkow, L. A., Moeller, H. M., 10.1007/s00020-008-1557-x, Integral Equations Oper. Theory 60 (2008), 177-200. (2008) Zbl1145.47012MR2375563DOI10.1007/s00020-008-1557-x
  14. Fuglede, B., The multidimensional moment problem, Exp. Math. 1 (1983), 47-65. (1983) Zbl0514.44006MR0693807
  15. Haviland, E. K., 10.2307/2371187, Amer. J. Math. 57 (1935), 562-568. (1935) MR1507095DOI10.2307/2371187
  16. Hauck, C. D., Levermore, C. D., Tits, A. L., 10.1137/070691139, SIAM J. Control Optim. 47 (2008), 1977-2015. (2008) Zbl1167.49033MR2421338DOI10.1137/070691139
  17. Junk, M., 10.1142/S0218202500000513, Math. Models Methods Appl. Sci. 10 (2000), 1001-1025. (2000) Zbl1012.44005MR1780147DOI10.1142/S0218202500000513
  18. Kagan, A. M., Linnik, Y. V., Rao, C. R., Characterization Problems in Mathematical Statistics, Wiley, New York (1983). (1983) MR0346969
  19. Kuhlmann, S., Marshall, M., Schwartz, N., Positivity, sums of squares and the multi-dimensional moment problem. II, Adv. Geom. 5 (2005), 583-606. (2005) Zbl1095.14055MR2174483
  20. Kullback, S., Information Theory and Statistics, John Wiley, New York (1959). (1959) Zbl0088.10406MR0103557
  21. Lewis, A. S., 10.4153/CJM-1995-052-2, Can. J. Math. 47 (1995), 995-1006. (1995) Zbl0840.90106MR1350646DOI10.4153/CJM-1995-052-2
  22. Léonard, C., 10.1016/j.jmaa.2008.04.048, J. Math. Anal. Appl. 346 (2008), 183-204. (2008) Zbl1152.49039MR2428283DOI10.1016/j.jmaa.2008.04.048
  23. Mead, L. R., Papanicolaou, N., 10.1063/1.526446, J. Math. Phys. 25 (1984), 2404-2417. (1984) MR0751523DOI10.1063/1.526446
  24. Moreau, J. J., Sur la fonction polaire d'une fonction semicontinue supérieurement, C. R. Acad. Sci., Paris 258 (1964), 1128-1130 French. (1964) Zbl0144.30501MR0160093
  25. Putinar, M., Vasilescu, F.-H., 10.2307/121083, Ann. Math., II. Ser. 149 (1999), 1087-1107. (1999) Zbl0939.44003MR1709313DOI10.2307/121083
  26. Rockafellar, R. T., 10.1215/S0012-7094-66-03312-6, Duke Math. J. 33 (1966), 81-89. (1966) MR0187062DOI10.1215/S0012-7094-66-03312-6
  27. Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey (1970). (1970) Zbl0193.18401MR0274683

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