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Let consider a sequence of real numbers $(y_k)_{k\in \mathbb{N}^n}$.

Take an order $r\in \mathbb{N}^n$ such that $k\le r$.

Suppose that this numbers come from moment of a measure $\nu$ supported on a compact set $X$ that is, namely for $n=2$, $z=(x_1,x_2)$ $y_{00}=\int_X \nu(dz),\; y_{10}=\int_X x_1\nu(dz),\; y_{01}=\int_X x_2\nu(dz),\; y_{11}=\int_X x_1x_2\nu(dz),\; y_{20}=\int_X x_1^2\nu(dz) \ldots$

Question: How to recover $\nu$ from a given sequence $y$ and order $r$ ?

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    Interesting one. The n=1 case is already quite complex, and most of times it does not have (unique) solution. So this seems a pretty tough question! – 2017-02-08

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