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 satisfies Carleman's condition, there is no other measure having the same moments as The condition was discovered by Torsten Carleman in 1922.

Hamburger moment problem

For the Hamburger moment problem, the theorem states the following:
Let be a measure on such that all the moments
are finite. If
then the moment problem for is determinate; that is, is the only measure on with as its sequence of moments.

Stieltjes moment problem

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

Generalized Carleman's condition

In Nasiraee et al. showed that, despite previous assumptions, 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.