Solution of the truncated moment problem with variety 𝑦=𝑥³

Fialkow, Lawrence

doi:10.1090/S0002-9947-2011-05262-1

Solution of the truncated moment problem with variety $y = x^{3}$
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Trans. Amer. Math. Soc. 363 (2011), 3133-3165 Request permission

Abstract:

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 $y=x^{3}$. 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 $yg(x)=1$ ($g\in \mathbb {R}[x]$), leading to degree-bounded weighted sum-of-squares representations for polynomials which are strictly positive on such curves.

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