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The Extremal Truncated Moment Problem

Authors:	Raúl E Curto Lawrence A Fialkow H Michael Möller

Institution:	1. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242-1419, USA 2. Department of Computer Science, State University of New York, New Paltz, NY, 12561, USA 3. FB Mathematik der Universit?t Dortmund, Dortmund, 44221, Germany

Abstract:	For a degree 2n real d-dimensional multisequence $\beta \equiv \beta^{(2n)} = \{\beta_i\}_{i\in{Z}^{d}_{+},\|i\|\leq 2n}$ to have a representing measure μ, it is necessary for the associated moment matrix ${\mathcal{M}}(n)(\beta)$ to be positive semidefinite and for the algebraic variety associated to β, ${\mathcal{V}} \equiv {\mathcal{V}}_{\beta}$ , to satisfy rank ${\mathcal{M}}(n) \leq$ card ${\mathcal{V}}$ as well as the following consistency condition: if a polynomial $p(x) \equiv \sum_{\|i\|\leq 2n} a_{i}x^{i}$ vanishes on ${\mathcal{V}}$ , then $\sum_{\|i\|\leq 2n} a_{i}{\beta_i} = 0$ . We prove that for the extremal case $(\rm{rank}\,{\mathcal{M}}(n) = \rm{card}\,{\mathcal{V}})$ , positivity of ${\mathcal{M}}(n)$ and consistency are sufficient for the existence of a (unique, rank ${\mathcal{M}}(n)$ -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of ${\mathcal{M}}(n)$ . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.

Keywords:	Primary 47A57 44A60 42A70 30E05 Secondary 15A57 15-04 47N40 47A20
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