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 [math]\displaystyle{ \mu }[/math] satisfies Carleman's condition, there is no other measure [math]\displaystyle{ \nu }[/math] having the same moments as [math]\displaystyle{ \mu. }[/math] 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 [math]\displaystyle{ \mu }[/math] be a measure on [math]\displaystyle{ \R }[/math] such that all the moments [math]\displaystyle{ m_n = \int_{-\infty}^{+\infty} x^n \, d\mu(x)~, \quad n = 0,1,2,\cdots }[/math] are finite. If [math]\displaystyle{ \sum_{n=1}^\infty m_{2n}^{-\frac{1}{2n}} = + \infty, }[/math] then the moment problem for [math]\displaystyle{ (m_n) }[/math] is determinate; that is, [math]\displaystyle{ \mu }[/math] is the only measure on [math]\displaystyle{ \R }[/math] with [math]\displaystyle{ (m_n) }[/math] as its sequence of moments.

Stieltjes moment problem

For the Stieltjes moment problem, the sufficient condition for determinacy is [math]\displaystyle{ \sum_{n=1}^\infty m_{n}^{-\frac{1}{2n}} = + \infty. }[/math]

References

• Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd. 
• Chapter 3.3, Durrett, Richard. Probability: Theory and Examples. 5th ed. Cambridge Series in Statistical and Probabilistic Mathematics 49. Cambridge ; New York, NY: Cambridge University Press, 2019.