Journal article
Solution of the Truncated Hyperbolic Moment Problem
Integral Equations and Operator Theory, Vol.52(2), pp.181-218
06/2005
DOI: 10.1007/s00020-004-1340-6
Abstract
Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β (2n)={ β ij } i,j ≥ 0,i+j ≤ 2n , with β00 > 0, the truncated Q-hyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x, y) = 0, such that $$\beta _{ij} = \int {y^i x^j d\mu } \quad (0 \leq i + j \leq 2n).$$ We prove that β admits a Q-representing measure μ (as above) if and only if the associated moment matrix $$\mathcal{M}(n)(\beta )$$ is positive semidefinite, recursively generated, has a column relation Q(X,Y) = 0, and the algebraic variety $$\mathcal{V}(\beta )$$ associated to β satisfies card $$\mathcal{V}(\beta ) \geq {\text{rank }}\mathcal{M}(n)(\beta ).$$ In this case, $${\text{rank }}\mathcal{M}(n) \leq 2n + 1;$$ if $${\text{rank }}\mathcal{M}(n) \leq 2n,$$ then β admits a rank $$\mathcal{M}(n)$$ -atomic (minimal) Q-representing measure; if $${\text{rank }}\mathcal{M}(n) = 2n + 1,$$ then β admits a Q-representing measure μ satisfying $$2n + 1 \leq {\text{card supp }}\mu \leq 2n + 2.$$
Details
- Title: Subtitle
- Solution of the Truncated Hyperbolic 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 USA
- Resource Type
- Journal article
- Publication Details
- Integral Equations and Operator Theory, Vol.52(2), pp.181-218
- DOI
- 10.1007/s00020-004-1340-6
- ISSN
- 0378-620X
- eISSN
- 1420-8989
- Publisher
- Birkhäuser-Verlag; Basel
- Language
- English
- Date published
- 06/2005
- Academic Unit
- Mathematics
- Record Identifier
- 9983985953202771
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