The truncated matrix-valued $K$-moment problem on $\mathbb {R}^d$, $\mathbb {C}^d$, and $\mathbb {T}^d$
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- by David P. Kimsey and Hugo J. Woerdeman PDF
- Trans. Amer. Math. Soc. 365 (2013), 5393-5430 Request permission
Abstract:
The truncated matrix-valued $K$-moment problem on $\mathbb {R}^d$, $\mathbb {C}^d$, and $\mathbb {T}^d$ will be considered. The truncated matrix-valued $K$-moment problem on $\mathbb {R}^d$ requires necessary and sufficient conditions for a multisequence of Hermitian matrices $\{ S_{\gamma } \}_{\gamma \in \Gamma }$ (where $\Gamma$ is a finite subset of $\mathbb {N}_0^d$) to be the corresponding moments of a positive Hermitian matrix-valued Borel measure $\sigma$, and also the support of $\sigma$ must be contained in some given non-empty set $K \subseteq \mathbb {R}^d$, i.e., \begin{equation*} S_{\gamma } = \int _{\mathbb {R}^d} \xi ^{\gamma } d\sigma (\xi ),\;\;\textrm {for}\;\textrm {all}\; \gamma \in \Gamma , \tag {0.1}\end{equation*} and \begin{equation*} \textrm {supp} \; \sigma \subseteq K. \tag {0.2}\end{equation*} Given a non-empty set $K \subseteq \mathbb {R}^d$ and a finite multisequence, indexed by a certain family of finite subsets of $\mathbb {N}_0^d$, of Hermitian matrices we obtain necessary and sufficient conditions for the existence of a minimal finitely atomic measure which satisfies (0.1) and (0.2). In particular, our result can handle the case when $\Gamma = \{ \gamma \in \mathbb {N}_0^d \colon 0 \leq |\gamma | \leq 2n+1 \}$. We will also discuss a similar result in the multivariable complex and polytorus setting.References
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Additional Information
- David P. Kimsey
- Affiliation: Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
- Address at time of publication: Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, POB 26, Rehovot 76100, Israel
- Email: kimsey@drexel.edu, david.kimsey@weizmann.ac.il
- Hugo J. Woerdeman
- Affiliation: Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
- MR Author ID: 183930
- Email: hugo@math.drexel.edu
- Received by editor(s): October 5, 2010
- Received by editor(s) in revised form: February 19, 2012
- Published electronically: May 16, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 5393-5430
- MSC (2010): Primary 47A57; Secondary 30E05, 42A70, 44A60
- DOI: https://doi.org/10.1090/S0002-9947-2013-05812-6
- MathSciNet review: 3074378