Stieltjes moment problem explained

In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (m₀, m₁, m₂, ...) to be of the form

m _{n} = \int ^{inf ty} _{0} x ^{nd\mu(x)}

for some measure μ. If such a function μ exists, one asks whether it is unique.

The essential difference between this and other well-known moment problems is that this is on a half-line [0, ∞), whereas in the Hausdorff moment problem one considers a bounded interval [0, 1], and in the Hamburger moment problem one considers the whole line (−∞, ∞).

Existence

Let

\Delta _{n=\left[\begin{matrix} m} _{0} & m _{1} & m _{2} & … & m _{n} \\ m _{1} & m _{2} & m _{3} & … & m _{n+1} \\ m _{2&} m _{3} & m _{4} & … & m _{n+2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ m _{n} & m _{n+1} & m _{n+2} & … & m _{2n} \end{matrix}\right]

be a Hankel matrix, and

\Delta ^{( 1 )} _{n} = \left[\begin{matrix} m _{1} & m _{2} & m _{3} & … & m _{n+1} \\ m _{2} & m _{3} & m _{4} & … & m _{n+2} \\ m _{3} & m _{4} & m _{5} & … & m _{n+3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ m _{n+1} & m _{n+2} & m _{n+3} & … & m _{2n+1} \end{matrix}\right].

Then is a moment sequence of some measure on

[ 0 , inf ty) with infinite support if and only if for all n, both\det(\Delta _{n)} > 0 and \det\left(\Delta ^{( 1 )} _{n} \right) > 0 .

is a moment sequence of some measure on

[ 0 , inf ty) with finite support of size m if and only if for all n \leq m , both\det(\Delta _{n)} > 0 and \det\left(\Delta ^{( 1 )} _{n} \right) > 0

and for all larger

n \det(\Delta _{n)} = 0 and \det\left(\Delta ^{( 1 )} _{n} \right) = 0 .

Uniqueness

There are several sufficient conditions for uniqueness, for example, Carleman's condition, which states that the solution is unique if

\sum _{n} m ^{-1 /( 2 n)} _{n} = inf ty~.