Multivariate moment problems : geometry and indeterminateness

Mihai Putinar; Claus Scheiderer

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 2, page 137-157
  • ISSN: 0391-173X

Abstract

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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.

How to cite

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Putinar, Mihai, and Scheiderer, Claus. "Multivariate moment problems : geometry and indeterminateness." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.2 (2006): 137-157. <http://eudml.org/doc/240335>.

@article{Putinar2006,
abstract = {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.},
author = {Putinar, Mihai, Scheiderer, Claus},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {137-157},
publisher = {Scuola Normale Superiore, Pisa},
title = {Multivariate moment problems : geometry and indeterminateness},
url = {http://eudml.org/doc/240335},
volume = {5},
year = {2006},
}

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
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 2
SP - 137
EP - 157
AB - 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.
LA - eng
UR - http://eudml.org/doc/240335
ER -

References

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