Multivariate moment problems : geometry and indeterminateness

Putinar, Mihai; Scheiderer, Claus

Putinar, Mihai ; Scheiderer, Claus

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 137-157.

Résumé

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.

MR 2244695 | Zbl 1170.44302

@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 = {archive.numdam.org/item/ASNSP_2006_5_5_2_137_0/}
}

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/

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