Moment problem for discrete distributions

Question

Let $x_1, \dots, x_N \in \mathbb R$ and consider the discrete distribution $\mu := \frac{1}{N} \sum_{i=1}^N \delta_{x_i}$, where $\delta_x$ denotes the Dirac measure, i.e. for any measurable set $B \subset \mathbb R$, we have $\delta_x(B) = 1$ if $x \in B$ and zero otherwise.

The definition of moments for discrete distribution is included in the general definition: In the case of discrete distributions, we have $\mathbb E(x^p) = \sum_{i=1}^N x_i^p$.

I am interested in the (simple-looking) moment problem of determining $\mu$ from the moments $\mathbb E(x^p)$. In particular, is it true that $\mathbb E(x), \mathbb E(x^2), \dots, \mathbb E(x^N)$ determine $\mu$ uniquely?

Secondly, if the considered points are vectors, i.e. $x_i \in \mathbb R^n$, can I consider only $\mathbb E(x^\alpha)$, where $\alpha$ is a multi-index of the form $(0, \dots, p, \dots, 0)$ or must I consider all moments related to all multi-indices of a given degree $p$?

Answer 1

The answer for vectors uses the theory of multi-symmetric polynomials and in particular power sums. You can learn about that in Emmanuel Briand's thesis mentioned in myprevious MO answer Generalizing the Fundamental Theorem of Symmetric Polynomials Often people attibute the theory to Weyl (in his book on classical groups), however it is much older, with major contributions by Poisson, Schlaffli and Junker.

Answer 2

The single-variable case is indeed easy, as pointed out in the comment.

The multivariate case has been studied by Curto and Fialkow and then M.Laurent gave a more algebraic approach to this.