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# The multi-dimensional truncated moment problem: maximal masses

### Abstract

Given a subset $\mathcal K$ of $\mathbb R^d$ and a linear functional $L$ on the polynomials $\mathbb R^d_{2n}[\underline{x}]$ in $d$ variables and of degree at most $2n$ the truncated $\mathcal K$-moment problem asks when there is a positive Borel measure $\mu$ supported by $\mathcal K$ such that $L(p)=\int p\, d\mu$ for $p\in \mathbb R^d_{2n}[\underline{x}]$. For compact sets $\mathcal K$ we investigate the maximal mass of all representing measures at a given point of $\mathcal K$. Various characterizations of this quantity and related properties are developed and a close link to zeros of positive polynomials is established.

Key words: Moment problem, maximal mass, positive polynomial.

### Article Information

 Title The multi-dimensional truncated moment problem: maximal masses Source Methods Funct. Anal. Topology, Vol. 21 (2015), no. 3, 266-281 MathSciNet MR3521697 zbMATH 06630273 Milestones Received 11/02/2015; Revised 28/02/2015 Copyright The Author(s) 2015 (CC BY-SA)

### Authors Information

Universitat Leipzig, Mathematisches Institut, Augustusplatz 10/11, D-04109 Leipzig, Germany

### Citation Example

Konrad Schmüdgen, The multi-dimensional truncated moment problem: maximal masses, Methods Funct. Anal. Topology 21 (2015), no. 3, 266-281.

### BibTex

@article {MFAT783,
TITLE = {The multi-dimensional truncated moment problem: maximal masses},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {21},
YEAR = {2015},
NUMBER = {3},
PAGES = {266-281},
ISSN = {1029-3531},
MRNUMBER = {MR3521697},
ZBLNUMBER = {06630273},
URL = {http://mfat.imath.kiev.ua/article/?id=783},
}

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