Stieltjes Moment Problem

In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (''m''₀, ''m''₁, ''m''₂, ...) to be of the form

:$m_n = \int_0^\infty x^n\,d\mu(x)$

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 /nowiki>0, ∞), whereas in the bounded interval [0, 1">Hausdorff moment problem one considers a Interval_(mathematics)#Terminology">bounded interval [0, 1 and in the Hamburger moment problem one considers the whole line (−∞, ∞).

Existence

Let

:$\Delta_n=\left[\begin m_0 & m_1 & m_2 & \cdots & m_ \\ m_1 & m_2 & m_3 & \cdots & m_ \\ m_2& m_3 & m_4 & \cdots & m_ \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ m_ & m_ & m_ & \cdots & m_ \end\right]$

and

:\Delta_n^=\left begin
m_1 & m_2 & m_3 & \cdots & m_ \\
m_2 & m_3 & m_4 & \cdots & m_ \\
m_3 & m_4 & m_5 & \cdots & m_ \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
m_ & m_ & m_ & \cdots & m_
\end\right

Then is a moment sequence of some measure on $[0,\infty)$ with infinite support if and only if for all ''n'', both

:$\det(\Delta_n) > 0\ \mathrm\ \det\left(\Delta_n^\right) > 0.$

is a moment sequence of some measure on $[0,\infty)$ with finite support of size ''m'' if and only if for all $n \leq m$, both

:$\det(\Delta_n) > 0\ \mathrm\ \det\left(\Delta_n^\right) > 0$

and for all larger $n$

:$\det(\Delta_n) = 0\ \mathrm\ \det\left(\Delta_n^\right) = 0.$

Uniqueness

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

:$\sum_ m_n^ = \infty~.$

References

*{{citation, first=Michael, last=Reed, first2=Barry, last2=Simon, title=Fourier Analysis, Self-Adjointness, year=1975, ISBN=0-12-585002-6, series=Methods of modern mathematical physics, volume=2, publisher=Academic Press, page= 341 (exercise 25)

Probability problems
Mathematical analysis
Moment (mathematics)
Mathematical problems