Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment $$ m_{\bar i} = \int x_i^{i_i} x_2^{i_2} \cdots x_n^{i_n} d\mu .$$ I'm interested in the inverse problem of reconstructing $\mu$ from the collection of moments $\{m_{\bar i}\}$. I'm mainly interested in the case $n=\infty$, but I would be happy to learn of good references for any $n$, even $n=1$. I'm vaguely aware that one way to tackle the $n=1$ moment problem is via the Stieljes transform. Is there something similar for $n>1$?

I'm mainly interested in this problem from a practical, computational point of view. I can generate various instances of $\{m_{\bar i}\}$, and I would like to know in each case what the corresponding measure $\mu$ is.

Summary: How, as a practical matter, does one go about solving moment problems, especially multidimensional moment problems?

Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment $$ m_{\bar i} = \int x_i^{i_1} x_2^{i_2} \cdots x_n^{i_n} d\mu .$$ I'm interested in the inverse problem of reconstructing $\mu$ from the collection of moments $\{m_{\bar i}\}$. I'm mainly interested in the case $n=\infty$, but I would be happy to learn of good references for any $n$, even $n=1$. I'm vaguely aware that one way to tackle the $n=1$ moment problem is via the Stieljes transform. Is there something similar for $n>1$?

I'm mainly interested in this problem from a practical, computational point of view. I can generate various instances of $\{m_{\bar i}\}$, and I would like to know in each case what the corresponding measure $\mu$ is.

Summary: How, as a practical matter, does one go about solving moment problems, especially multidimensional moment problems?