Solution of the truncated hyperbolic moment problem

Curto, Raul E.; Fialkow, Lawrence A.

Solution of the truncated hyperbolic moment problem

Authors

Description

Let Q(x,y)=0 be an hyperbola in the plane. Given real numbers $β\equivβ^{2n)}=\{β_{ij}\}_{i,j\geq0,i+j\leq2n}$, with $β_{00}>0$, the truncated Q-hyperbolic moment problem for βentails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x,y)=0, such that $β_{ij}=\int y^{i}x^{j} dμ(0\leq i+j\leq2n)$. We prove that βadmits a Q-representing measure μ(as above) if and only if the associated moment matrix $\mathcal{M}(n)(β)$ is positive semidefinite, recursively generated, has a column relation Q(X,Y)=0, and the algebraic variety $\mathcal{V}(β)$ associated to βsatisfies $card\mathcal{V}(β)\geq\rank\mathcal{M}(n)(β)$. In this case, $rank\mathcal{M}(n)\leq2n+1$; if $rank\mathcal{M}(n)\leq2n$, then βadmits a $rank\mathcal{M}(n)$-atomic (minimal) Q-representing measure; if $rank\mathcal{M}(n)=2n+1$, then βadmits a Q-representing measure μsatisfying $2n+1\leqcard suppμ\leq2n+2$.