Solution of the truncated hyperbolic moment problem

2005/07/04 by Raul E. Curto, Lawrence A. Fialkow · 2 citations
Mathematics · #math.FA #msc:47A57 #msc:44A60 #msc:42A70 #msc:30A05

paper · pdf

published as Integral Equations Operator Theory 52(2005), 181-218

arxiv created 2005/07/04 · arxiv updated 2009/12/01

Abstract

Let Q(x,y)=0 be an hyperbola in the plane. Given real numbers β≡β²ⁿ⁾=\β_ij\_{i,j≥0,i+j≤2n}, with β₀₀>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=∫ yⁱx^j dμ(0≤ i+j≤2n). We prove that βadmits a Q-representing measure μ(as above) if and only if the associated moment matrix M(n)(β) is positive semidefinite, recursively generated, has a column relation Q(X,Y)=0, and the algebraic variety V(β) associated to βsatisfies cardV(β)≥\rankM(n)(β). In this case, rankM(n)≤2n+1; if rankM(n)≤2n, then βadmits a rankM(n)-atomic (minimal) Q-representing measure; if rankM(n)=2n+1, then βadmits a Q-representing measure μsatisfying 2n+1\leqcard suppμ≤2n+2.

Cited by

• Constructive approach to the truncated moment problem on reducible cubic curves: Hyperbolic type relations
• The pure Y=X^d truncated moment problem

Related

• The Tracial Moment Problem on Quadratic Varieties
• The extremal truncated moment problem
• The pure Y=X^d truncated moment problem
• On the truncated two-dimensional moment problem
• Positive polynomials and the truncated moment problem on plane cubics