Stieltjes moment problem

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In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence { mn, : n = 0, 1, 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 [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}&\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]

and

\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].

Then { mn : n = 1, 2, 3, ... } 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  {and}}\ \det \left(\Delta _{n}^{{(1)}}\right)>0.

{ mn : n = 1, 2, 3, ... } 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  {and}}\ \det \left(\Delta _{n}^{{(1)}}\right)>0

and for all larger n

\det(\Delta _{n})=0\ {\mathrm  {and}}\ \det \left(\Delta _{n}^{{(1)}}\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\geq 1}}m_{n}^{{-1/(2n)}}=\infty ~.

References

  • Reed, Michael; Simon, Barry (1975), Fourier Analysis, Self-Adjointness, Methods of modern mathematical physics 2, Academic Press, p. 341 (exercise 25), ISBN 0-12-585002-6 
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