Stieltjes moment problem

In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions that a sequence { μ_"n", : "n" = 0, 1, 2, ... } be of the form

:$mu\_n=int\_0^infty\; x^n,dF(x),$

for some nondecreasing function "F".

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 (−∞, ∞).

Let

:

and

:

Then { μ_"n" : "n" = 1, 2, 3, ... } is a moment sequence of some probability distribution on $[0,infty)$ with infinite support if and only if for all "n", both

:$det(Delta\_n)\; >\; 0\; mathrm\{and\}\; detleft(Delta\_n^\{(1)\}\; ight)\; >\; 0.$

{ μ_"n" : "n" = 1, 2, 3, ... } is a moment sequence of some probability distribution on $[0,infty)$ with finite support of size "m" if and only if for all $n\; leq\; m$, both

:$det(Delta\_n)\; >\; 0\; mathrm\{and\}\; detleft(Delta\_n^\{(1)\}\; ight)\; >\; 0.$

and for all larger $n$

:$det(Delta\_n)\; =\; 0\; mathrm\{and\}\; detleft(Delta\_n^\{(1)\}\; ight)\; =\; 0.$The solution is unique if there are constants "C" and "D" such that for all "n", |μ_"n"|≤ "CD"^"n""(2n)"! harv|Reed|Simon|1975|p=341.

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)