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 $$\mu$$ satisfies Carleman's condition, there is no other measure $$\nu$$ having the same moments as $$\mu.$$ 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 $$\mu$$ 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, $$\mu$$ 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 $$\sum_{n=1}^\infty m_{n}^{-\frac{1}{2n}} = + \infty.$$