The -moment problem for continuous linear functionals

Author:
Jean B. Lasserre

Journal:
Trans. Amer. Math. Soc. **365** (2013), 2489-2504

MSC (2010):
Primary 44A60, 13B25, 14P10, 30C10

DOI:
https://doi.org/10.1090/S0002-9947-2012-05701-1

Published electronically:
October 4, 2012

MathSciNet review:
3020106

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Abstract: Given a closed (and not necessarily compact) basic semi-algebraic 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 (degree-truncated) 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:
LAAS-CNRS and Institute of Mathematics, University of Toulouse, LAAS, 7 avenue du Colonel Roche, 31077 Toulouse Cédex 4, France

Email:
lasserre@laas.fr

DOI:
https://doi.org/10.1090/S0002-9947-2012-05701-1

Keywords:
Moment problems,
real algebraic geometry,
positive polynomials,
semi-algebraic 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