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.

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

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

References

• Akhiezer (1965)