My question is, given a real random variable $u$, then can we find a discrete random variable $\xi$ with finite possible values, such that $u$ and $\xi$ have the same first $2k$ moments?
According to my intuition, it seems like that there exists an even distribution $$P(\xi=x_i)=\frac{1}{k+1},\ k=1,\dots,n$$ satisfies the requirement. But I have no idea about how to prove it.
The key point is that how to decide whether the following equations has at least one solution over $\mathbb{C}$, $$\begin{align} x_1+\dots+x_n&=n\cdot\mathrm{E}u,\\ &\ \ \vdots\\ x_1^{2k}+\dots+x_n^{2k}&=n\cdot\mathrm{E}u^{2k}. \end{align}$$
If it actually exists a solution, we can prove the solution is real by using Hankel matrix.
Is there any conclusion in algebraic-geometry can be used to state the existence of the solution?