Solution of the Truncated Hyperbolic Moment Problem |
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Authors: | Raúl E Curto Lawrence A Fialkow |
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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 |
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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
We prove that β admits a Q-representing measure μ (as above) if and only if the associated moment matrix
is positive semidefinite, recursively generated, has a column relation Q(X,Y) = 0, and the algebraic variety
associated to β satisfies card
In this case,
if
then β admits a rank
-atomic (minimal) Q-representing measure; if
then β admits a Q-representing measure μ satisfying
![$$2n + 1 \leq {\text{card supp }}\mu \leq 2n + 2.$$](/content/n543mt526206w141/20_2004_Article_1340_TeX2GIFIEq9.gif) |
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Keywords: | Primary 47A57 44A60 42A70 30A05 Secondary 15A57 15-04 47N40 47A20 |
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