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
![{\displaystyle m_{n}=\int _{0}^{\infty }x^{n}\,d\mu (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97cc668c5caef5d0958785c04b992499ddf40e68)
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
![{\displaystyle \Delta _{n}=\left[{\begin{matrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\m_{2}&m_{3}&m_{4}&\cdots &m_{n+2}\\\vdots &\vdots &\vdots &\ddots &\vdots \\m_{n}&m_{n+1}&m_{n+2}&\cdots &m_{2n}\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8206d416e2354bde00ecd3217b257609539b19c0)
be a
Hankel matrix, and
![{\displaystyle \Delta _{n}^{(1)}=\left[{\begin{matrix}m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\m_{2}&m_{3}&m_{4}&\cdots &m_{n+2}\\m_{3}&m_{4}&m_{5}&\cdots &m_{n+3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\m_{n+1}&m_{n+2}&m_{n+3}&\cdots &m_{2n+1}\end{matrix}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/067bad63dc066fda432dc92c73a1724826746860)
Then { mn : n = 1, 2, 3, ... } is a moment sequence of some measure on
with infinite support if and only if for all n, both
![{\displaystyle \det(\Delta _{n})>0\ \mathrm {and} \ \det \left(\Delta _{n}^{(1)}\right)>0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b67058826b1e65f0363689d04b7e5e42631d9e26)
{ mn : n = 1, 2, 3, ... } is a moment sequence of some measure on
with finite support of size m if and only if for all
, both
![{\displaystyle \det(\Delta _{n})>0\ \mathrm {and} \ \det \left(\Delta _{n}^{(1)}\right)>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abf4db3cba427f80d6c7c9f6ce29394c836eb0c7)
and for all larger
![{\displaystyle \det(\Delta _{n})=0\ \mathrm {and} \ \det \left(\Delta _{n}^{(1)}\right)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c85ce6c176d3641899f4798f09ff1f4b7c147cde)
Uniqueness
There are several sufficient conditions for uniqueness, for example,
Carleman's condition, which states that the solution is unique if
![{\displaystyle \sum _{n\geq 1}m_{n}^{-1/(2n)}=\infty ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad79ae978337a55dc8f2756963f3961e90d2fa88)
References
- Reed, Michael; Simon, Barry (1975), Fourier Analysis, Self-Adjointness, Methods of modern mathematical physics, vol. 2, Academic Press, p. 341 (exercise 25),
ISBN
0-12-585002-6