TITLE: Multi-dimensional moment problem
QUESTION [11 upvotes]: 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?

REPLY [6 votes]: The infinite-dimensional power moment problem was introduced by A.G. Kostyuchenko and B.S. Mityagin in 1960.
A description of this and other interesting generalizations of the power moment problem can be found in the following book: 
[BK] Yu. M. Berezansky and Y. G. Kondratiev, "Spectral Methods in Infinite-Dimensional Analysis", 1988. 
See Chapter 5, Section 2 in this book.
I have only the Russian paper original of this book. Probably it is translated by Kluwer Academic Publishers. 
Some conditions which ensure existence of a representing measure are given in the book. I do not remember any
algorithms in [BK]. However, there are historical notes in [BK] and you can look at them.
On the other hand there are classical books: 
[ST] J. A. Shohat and J. D. Tamarkin "The problem of moments", 
[A] N. I. Akhiezer "The classical moment problem".  
In the case of a finite interval there are several methods to recover the measure, see [ST, p. 90].
For the two-dimensional moment problem I proposed an algorithm in
S.M. Zagorodnyuk, On the two-dimensional moment problem.- Ann. Funct. Anal., 1, no. 1 (2010), 80-104.
Essentially it consists of solving of infinite linear systems of equations with parameters. 
However, I think this algorithm requires powerful computer and much additional technical work to implement it.
Some time ago, one scientist asked me a similar question: how practically reconstruct the measure?
I answered, that one can replace integrals by integral sums and then to solve linear equations. It is not
a rigorous way, but for some cases this will probably have sense.