Journal article
The Extremal Truncated Moment Problem
Integral Equations and Operator Theory, Vol.60(2), pp.177-200
02/2008
DOI: 10.1007/s00020-008-1557-x
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 $${\mathcal{M}}(n)$$ .
Details
- Title: Subtitle
- The Extremal Truncated Moment Problem
- Creators
- Raúl Curto - Department of Mathematics The University of Iowa Iowa City IA 52242-1419 USALawrence Fialkow - Department of Computer Science State University of New York New Paltz NY 12561 USAH Möller - FB Mathematik der Universität Dortmund Dortmund 44221 Germany
- Resource Type
- Journal article
- Publication Details
- Integral Equations and Operator Theory, Vol.60(2), pp.177-200
- DOI
- 10.1007/s00020-008-1557-x
- ISSN
- 0378-620X
- eISSN
- 1420-8989
- Publisher
- SP Birkhäuser Verlag Basel; Basel
- Language
- English
- Date published
- 02/2008
- Academic Unit
- Mathematics
- Record Identifier
- 9983985828102771
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