The most accurate determinateness criteria for the multivariate moment problem require the density of polynomials in a weighted Lebesgue space of a generic representing measure. We propose a relaxation of such a criterion to the approximation of a single function, and based on this condition we analyze the impact of the geometry of the support on the uniqueness of the representing measure. In particular we show that a multivariate moment sequence is determinate if its support has dimension one and is virtually compact; a generalization to higher dimensions is also given. Among the one-dimensional sets which are not virtually compact, we show that at least a large subclass supports indeterminate moment sequences. Moreover, we prove that the determinateness of a moment sequence is implied by the same condition (in general easier to verify) of the push-forward sequence via finite morphisms.
@article{ASNSP_2006_5_5_2_137_0, author = {Putinar, Mihai and Scheiderer, Claus}, title = {Multivariate moment problems : geometry and indeterminateness}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {137--157}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 5}, number = {2}, year = {2006}, zbl = {1170.44302}, mrnumber = {2244695}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2006_5_5_2_137_0/} }
TY - JOUR AU - Putinar, Mihai AU - Scheiderer, Claus TI - Multivariate moment problems : geometry and indeterminateness JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2006 DA - 2006/// SP - 137 EP - 157 VL - Ser. 5, 5 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2006_5_5_2_137_0/ UR - https://zbmath.org/?q=an%3A1170.44302 UR - https://www.ams.org/mathscinet-getitem?mr=2244695 LA - en ID - ASNSP_2006_5_5_2_137_0 ER -
Putinar, Mihai; Scheiderer, Claus. Multivariate moment problems : geometry and indeterminateness. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 137-157. http://archive.numdam.org/item/ASNSP_2006_5_5_2_137_0/
[1] “The Classical Moment Problem”, Oliver and Boyd, Edinburgh and London, 1965.
,[2] “Probability Theory”, De Gruyter, Berlin, 1996. | MR 1385460 | Zbl 0868.60001
,[3] “Harmonic Analysis and the Theory of Probability”, Univ. California Press, Berkeley, 1955. | MR 72370 | Zbl 0068.11702
,[4] “Algebra I”, Chapters 1-3. “Elements of Mathematics”, Springer, Berlin, 1989. | MR 979982 | Zbl 0666.13001
,[5] Work in progress on -invariant moment problems.
, and ,[6] Two parameter moment problems, Duke Math. J. 24 (1957), 481-498. | MR 92019 | Zbl 0081.10104
,[7] The multidimensional moment problem, Expo. Math. 1 (1983), 47-65. | MR 693807 | Zbl 0514.44006
,[8] Beiträge zur Konvergenztheorie der Stieltjesschen Kettenbrüche, Math. Z. 4 (1919), 186-222. | JFM 47.0428.01 | MR 1544361
,[9] Über die Konvergenz eines mit einer Potenzreihe assoziierten Kettenbruchs, Math. Ann. 20 (1920), 31-46. | JFM 47.0430.01 | MR 1511955
,[10] On the momentum problem for distribution functions in more than one dimension, II, Amer. J. Math. 58 (1936), 164-168. | JFM 62.0483.01 | MR 1507139
,[11] Bernstein theorems and Radon transform. Application to the theory of production functions, In: “Mathematical Problems of Tomography”, Transl. Math. Monogr. 81, Amer. Math. Soc., Providence, RI, 1990, 189-223. | MR 1104018 | Zbl 0794.44003
and ,[12] Positive-definite functionals on nuclear spaces, Trudy Moskov Mat. Obsc. (in Russian) 9 (1960), 283-316; English translation, I. M. Gelfand and S. G. Gindikin (eds.), in Amer. Math. Soc. Transl. (ser. 2) 93 (1970), 1-43. | MR 124729 | Zbl 0117.09801
and ,[13] Analytic vectors, Ann. of Math. 70 (1959), 572-615. | MR 107176 | Zbl 0091.10704
,[14] A commutativity theorem for unbounded operators in Hilbert space, Trans. Amer. Math. Soc. 140 (1969), 485-491. | MR 242010 | Zbl 0181.40905
,[15] “Die Lehre von den Kettenbrüchen”, Zweite verbesserte Auflage. Chelsea Publ. Comp. (reprint), New York, 1950. | MR 37384 | Zbl 0041.18206
,[16] On the relation between the multidimensional moment problem and the one-dimensional moment problem, Math. Scand. 51 (1982), 361-366. | MR 690537 | Zbl 0514.44007
,[17] Inequalities defining orbit spaces, Invent. math. 81 (1985), 539-554. | MR 807071 | Zbl 0578.14010
and ,[18] Sur le problème des moments. Troisième Note, Ark. Mat. Astr. Fys. 17 (1923), 1-52. | JFM 49.0195.01
,[19] Sums of squares on real algebraic curves, Math. Z. 245 (2003), 725-760. | MR 2020709 | Zbl 1056.14078
,[20] On determinacy notions for the two dimensional moment problem, Ark. Math. 29 (1991), 277-284. | Zbl 0762.44004
,[21] “The Problem of Moments”, Amer. Math. Soc., Providence, R.I., 1943. | MR 8438 | Zbl 0112.06902
and ,