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Solution of the truncated moment problem with variety y = x³

By Lawrence A. Fialkow


We show that positivity, consistency, and the variety condition, which are sufficient to solve the truncated moment problem on planar curves of degree 2, are not sufficient for curves of higher degree. Using new, partly algorithmic, conditions based on positive moment matrix extensions, we present a concrete solution to the truncated moment problem on the curve y = x³. We also use moment matrix extensions to solve (in a less concrete sense) truncated moment problems on curves of the form y = g(x) and yg(x) =1(g ∈ R[x]), leading to degree-bounded weighted sum-of-squares representations for polynomials which are strictly positive on such curves

Year: 2011
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