Positivity of Riesz functionals and solutions of quadratic and quartic moment problems |
| |
Authors: | Lawrence Fialkow Jiawang Nie |
| |
Institution: | a Department of Computer Science, State University of New York, New Paltz, NY 12561, United States b Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, United States |
| |
Abstract: | 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⊆Rn 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=R2; (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. |
| |
Keywords: | Truncated moment sequence Riesz functional (Strict) K-positivity Determining set Moment matrix Representing measure |
本文献已被 ScienceDirect 等数据库收录! |
|