In mathematics, particularly, in analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem.
μ
satisfies Carleman's condition, there is no other measure
having the same moments as
μ .
The condition was discovered by Torsten Carleman in 1922.
[1] For the Hamburger moment problem (the moment problem on the whole real line), the theorem states the following: Let
such that all the moments
then the moment problem for
as its sequence of moments.
For the Stieltjes moment problem, the sufficient condition for determinacy is
In,[2] Nasiraee et al. showed that, despite previous assumptions,[3] when the integrand is an arbitrary function, Carleman's condition is not sufficient, as demonstrated by a counter-example.
In fact, the example violates the bijection, i.e. determinacy, property in the probability sum theorem.
When the integrand is an arbitrary function, they further establish a sufficient condition for the determinacy of the moment problem, referred to as the generalized Carleman's condition.