On the classical moment problems

We introduce to the classical moment problem, a short history. We refer to Akhiezer’s book:

N. I, Akhiezer, The Classical Moment Problem and Some Related Questions in
Analysis, Oliver & Boyd, Edinburgh/London, 1965.

and Christiansen’s notes: Moment (on Steven Miller’s page ).

The moment problem is a classical problem in analysis. This problem occurs for the first time in the work of Chebychev in 1873. After that, T.Stieltjes (1894-1895) and A.Markov consider more general case. Chebychev and A.Markov took the moment problem in the relationship with probability theory. The first solution and discussion of extended moment problem is due to Hamburger, he studied Classical moment problem (one-dimensional).
Classical moment problem (one-dimensional) Given an infinite sequence of real numbers {\{s_n\}_n} ({s_0 = 1}). Does there exist a positive Borel measure {\mu} such that:

\displaystyle s_n = \int_{\mathbb{R}}x^nd\mu(x).

In general, we have Classical moment problem (multidimensional)
Given a function {s : \mathbb{N}^k \rightarrow \mathbb{R}}. Does there exist a positive Borel measure {\mu} such that:

\displaystyle s(n) = \int_{\mathbb{R}^k}x_1^{n_1}\dots x_n^{n_k}d\mu(x_1,\dots,x_k)<\infty \quad (*).

In the case one-dimensional moments, the sequence {\{s_n\}} is a function and we have {s_n = s(n), n \in \mathbb{N}}.
Two measure {\mu} and {\nu} are called equivalent if they satisfy:

\displaystyle s_n = \int_{\mathbb{R}^k}x^n d\mu(x) = \int_{\mathbb{R}^k}x^n d\nu(x).

In other words, we say they have same moments.
The measure {\mu} is called determinate if there only exists {\mu} such that {s_n = \int_{\mathbb{R}^k}x^n d\mu(x)} and indeterminate otherwise.
The aims of the multidimensional moment problem are:

  1. To find necessary and sufficient conditions for existence of measure {\mu} satisfying (*).
  2. To be able to decide determinacy.
  3. In the indeterminate case to give a complete description of all measures satisfying (*).
Advertisements