The moment problem for continuous linear functionals
Author:
Jean B. Lasserre
Journal:
Trans. Amer. Math. Soc. 365 (2013), 24892504
MSC (2010):
Primary 44A60, 13B25, 14P10, 30C10
Published electronically:
October 4, 2012
MathSciNet review:
3020106
Fulltext PDF
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Additional Information
Abstract: Given a closed (and not necessarily compact) basic semialgebraic set , we solve the moment problem for continuous linear functionals. Namely, we introduce a weighted norm on , and show that the closures of the preordering and quadratic module (associated with the generators of ) is the cone of polynomials nonnegative on . We also prove that and solve the moment problem for continuous linear functionals and completely characterize those continuous linear functionals nonnegative on and (hence on ). When has a nonempty interior, we also provide in explicit form a canonical projection for any polynomial , on the (degreetruncated) preordering or quadratic module. Remarkably, the support of is very sparse and does not depend on ! This enables us to provide an explicit Positivstellensatz on . And last but not least, we provide a simple characterization of polynomials nonnegative on , which is crucial in proving the above results.
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Additional Information
Jean B. Lasserre
Affiliation:
LAASCNRS and Institute of Mathematics, University of Toulouse, LAAS, 7 avenue du Colonel Roche, 31077 Toulouse Cédex 4, France
Email:
lasserre@laas.fr
DOI:
http://dx.doi.org/10.1090/S000299472012057011
PII:
S 00029947(2012)057011
Keywords:
Moment problems,
real algebraic geometry,
positive polynomials,
semialgebraic sets
Received by editor(s):
July 19, 2011
Received by editor(s) in revised form:
September 5, 2011, and September 7, 2011
Published electronically:
October 4, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
