Journal article
The truncated complex K-moment problem
Transactions of the American Mathematical Society, Vol.352(6), pp.2825-2855
01/01/2000
DOI: 10.1090/s0002-9947-00-02472-7
Abstract
Let γ ≡ γ(2n) denote a sequence of complex numbers
γ00 , γ01 , γ10, . . . , γ0,2n , . . . , γ2n,0 (γ00 > 0, γij = ¯γji), and let K denote a
closed subset of the complex plane C. The Truncated Complex K-Moment
Problem for γ entails determining whether there exists a positive Borel mea-
sure μ on C such that γij = ∫ ¯zizj dμ (0 ≤ i + j ≤ 2n) and supp μ ⊆ K.
For K ≡ KP a semi-algebraic set determined by a collection of complex poly-
nomials P = {pi (z, ¯z)}m
i=1, we characterize the existence of a finitely atomic
representing measure with the fewest possible atoms in terms of positivity and
extension properties of the moment matrix M (n) (γ) and the localizing matri-
ces Mpi . We prove that there exists a rank M (n)-atomic representing measure
for γ(2n) supported in KP if and only if M (n) ≥ 0 and there is some rank-
preserving extension M (n + 1) for which Mpi (n + ki) ≥ 0, where deg pi = 2ki
or 2ki − 1 (1 ≤ i ≤ m).
Details
- Title: Subtitle
- The truncated complex K-moment problem
- Creators
- Raúl Curto - University of Iowa, MathematicsLawrence A. Fialkow - Department of Mathematics and Computer Science, State University of New York, United States
- Resource Type
- Journal article
- Publication Details
- Transactions of the American Mathematical Society, Vol.352(6), pp.2825-2855
- DOI
- 10.1090/s0002-9947-00-02472-7
- ISSN
- 0002-9947
- eISSN
- 1088-6850
- Publisher
- American Mathematical Society
- Number of pages
- 31
- Language
- English
- Date published
- 01/01/2000
- Academic Unit
- Mathematics
- Record Identifier
- 9983985926602771
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