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

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

are finite. If

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

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

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

Notes[]

References[]

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