2009/08/22 by Lawrence A. Fialkow, Fialkow, Lawrence, Jiawang Nie +1 · 1 citation
Mathematics · #44A60 #47A57 #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Mathematical Approximation and Integration #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.0908.3230
openalex publication_date 2009/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence y, we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set K ⊆ \ren if and only if the associated Riesz functional Ly is K-positive. For a determining set K, we prove that if Ly is strictly K-positive, then y admits a representing measure supported in K. As a consequence, we are able to solve the truncated K-moment problem of degree k in the cases: (i) (n,k)=(2,4) and K=\re2; (ii) n≥ 1, k=2, and K is defined by one quadratic equality or inequality. In particular, these results solve the truncated moment problem in the remaining open cases of Hilbert's theorem on sums of squares.