Solution of the Truncated Hyperbolic Moment Problem

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 $2n + 1 \leq {\text{card supp }}\mu \leq 2n + 2.$