Multi-dimensional moment problem

Question

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?

Answer 1

The determinateness problem, i.e., deciding whether a given multi-sequence $(m_k)_{k \in \mathbb Z}$ indeed comes from any measure, is a well known problem for 1-dimensional case, but the multidimensional case still attracts fresh research. A nice recent paper on this problem is by Blekherman and Lasserre . It seems that in your case, however, that is not the issue: you know that you have (truncated) moment data of a measure (that presumably has a density?), and want to reconstruct its density.

From the practical perspective, there has been a considerable "excitement" about entropy optimization methods, which was advocated heavily by Jaynes , in signal processing and applied math communities at the beginning of 80s. In short, these methods treat truncated moment data as constraints on optimization of an entropy functional, yielding ansatz for the reconstructed density to be, for 1D, $\rho(x) = \exp \sum_{n=0}^{N-1} \lambda_n x^n$ for $N$-truncation of the moments.

Good overview is given by Borwein. You might also take a look at papers of T. Georgiou, who works in applied settings and Lasserre.

The maximum entropy approaches have given considerable success in physics literature, but they do have their numerical downsides, which mainly stem from numerical sensitivity when higher number of moments is used. I believe the first approaches by Mead and Papanicolaou handled perhaps 9-13 moments, before the round-off errors kills the convergence of the method. The way around is to take trigonometric moments, instead of monomials, which basically constrains the density by its Fourier coefficients instead of power moments. An excellent survey of numerical issues was given by Abramov who also provides a maxent toolbox on his website (all the way down).

The Stieltjes-Cauchy transform you mentioned brings the moment problem into the complex domain, by producing the function $F(z) = \int (z - x)^{-1}d\mu(x)$. Its power expansion is the generating series for moments of the original measure $d\mu = \rho(x) dx$ on the real line. Conversely, if one is given a Stieltjes transform $F(z)$, taking a certain limit evaluates the density of the measure pointwise $\lim_{\epsilon \downarrow 0} F(x + i\epsilon) - F(x - i\epsilon) = \rho(x)$. Generalizing Stieltjes transform to higher dimensions is non-trivial, as it can be done in more than one way, to yield, e.g., multivariate Stieltjes-Cauchy transform (where the kernel is just a product of Stieltjes-Cauchy kernels along each dimension), or Fantappi`e transform, where kernel contains an inner product of coordinates $(z_0 + z \cdot x)^{-1}$.

However, I have no knowledge about infinite-dimensional domain. All that I've written applies to finite-dimensional (in certain cases, compact) supports of measures. I would be interesting in seeing other answers on infinite-dimensional setup.

Answer 2

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.