Carleman's condition

In mathematics, particularly, in analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem. That is, if a measure μ satisfies Carleman's condition, there is no other measure ν having the same moments as μ. The condition was discovered by Torsten Carleman in 1922.^[1]

Hamburger moment problem

For the Hamburger moment problem (the moment problem on the whole real line), the theorem states the following:

Let μ be a measure on R such that all the moments

$m_{n}=\int _{{-\infty }}^{{+\infty }}x^{n}\,d\mu (x)~,\quad n=0,1,2,\cdots$

are finite. If

$\sum _{{n=1}}^{\infty }m_{{2n}}^{{-{\frac {1}{2n}}}}=+\infty ,$

then the moment problem for m_n is determinate; that is, μ is the only measure on R with (m_n) as its sequence of moments.

Stieltjes moment problem

For the Stieltjes moment problem, the sufficient condition for determinacy is

$\sum _{{n=1}}^{\infty }m_{{n}}^{{-{\frac {1}{2n}}}}=+\infty .\,$

Notes

1. Akhiezer (1965)

References

• Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd.