THE EXTREMAL TRUNCATED MOMENT PROBLEM

Abstract. For a degree 2n real d-dimensional multisequence β ≡ β(2n) = {βi}i∈Zd+,|i|≤2n to have a representing measure µ, it is necessary for the associated moment matrix M(n)(β) to be positive semidefinite and for the algebraic variety associated to β, V ≡ Vβ, to satisfy rank M(n) ≤ card V as well as the following consistency condition: if a polynomial p(x) ≡ ∑|i|≤2n aixi vanishes on V, then∑ |i|≤2n aiβi = 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). 1