Logo image
Back
An analogue of the Riesz–Haviland theorem for the truncated moment problem
Journal article   Open access   Peer reviewed

An analogue of the Riesz–Haviland theorem for the truncated moment problem

Raúl E Curto and Lawrence A Fialkow
Journal of functional analysis, Vol.255(10), pp.2709-2731
2008
DOI: 10.1016/j.jfa.2008.09.003
url
https://doi.org/10.1016/j.jfa.2008.09.003View
Published (Version of record) Open Access

Abstract

Let β ≡ β ( 2 n ) = { β i } | i | ⩽ 2 n denote a d-dimensional real multisequence, let K denote a closed subset of R d , and let P 2 n : = { p ∈ R [ x 1 , … , x d ] : deg p ⩽ 2 n } . Corresponding to β, the Riesz functional L ≡ L β : P 2 n → R is defined by L ( ∑ a i x i ) : = ∑ a i β i . We say that L is K-positive if whenever p ∈ P 2 n and p | K ⩾ 0 , then L ( p ) ⩾ 0 . We prove that β admits a K-representing measure if and only if L β admits a K-positive linear extension L ˜ : P 2 n + 2 → R . This provides a generalization (from the full moment problem to the truncated moment problem) of the Riesz–Haviland theorem. We also show that a semialgebraic set solves the truncated moment problem in terms of natural “degree-bounded” positivity conditions if and only if each polynomial strictly positive on that set admits a degree-bounded weighted sum-of-squares representation.
 Moment matrix extension Riesz–Haviland theorem Semialgebraic sets Positive functional K-moment problems Flat extensions of positive matrices Riesz functional Localizing matrices Truncated moment problem

Details

Metrics

46 Record Views
41 Times Cited - Web of Science
Logo image