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

:$$Delta_n=left [egin{matrix}1 & mu_1 & mu_2 & cdots & mu_{n} \mu_1 & mu_2 & mu_3 & cdots & mu_{n+1} \mu_2& mu_3 & mu_4 & cdots & mu_{n+2} \vdots & vdots & vdots & ddots & vdots \mu_{n} & mu_{n+1} & mu_{n+2} & cdots & mu_{2n}end{matrix} ight] .

and

:$$Delta_n^{(1)}=left [egin{matrix}mu_1 & mu_2 & mu_3 & cdots & mu_{n+1} \mu_2 & mu_3 & mu_4 & cdots & mu_{n+2} \mu_3 & mu_4 & mu_5 & cdots & mu_{n+3} \vdots & vdots & vdots & ddots & vdots \mu_{n+1} & mu_{n+2} & mu_{n+3} & cdots & mu_{2n+1}end{matrix} ight] .

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)