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The extremal truncated moment problem

Raul E. Curto·Lawrence A. Fialkow·H. Michael Moeller
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Abstract

For a degree 2n real d-dimensional multisequence β^(2n) to have a representing measure, it is necessary for the associated moment matrix M(n) to be positive semidefinite and for the algebraic variety V = V(β) associated to βto satisfy rank M(n) <= card V as well as the following consistency condition: if a polynomial p vanishes on V, then p(β) = 0. We prove that for the extremal case (rank M(n) = card V), positivity of M(n) and consistency are sufficient for the existence of a (unique, rank M(n)-atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of M(n).

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Raul E. Curto, Lawrence A. Fialkow, H. Michael Moeller. 2006-10-28. The extremal truncated moment problem. https://arxiv.org/abs/math/0610882

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