Carleman's condition

In mathematics, Carleman's condition is a sufficient condition for the determinacy of the Hamburger moment problem.

The theorem, proved by Torsten Carleman, states the following:

Let $$\mu$$ be a measure on $$\mathbb{R}$$ such that all the moments


 * $$s_k = \int_{-\infty}^{+\infty} x^k \, d\mu(x)$$

are finite. If
 * $$\sum_{k=1}^\infty s_{2k}^{-\frac{1}{2k}} = + \infty,$$

then the moment problem for $$(s_k)$$ is determinate; that is, $$\mu$$ is the only measure on $$\mathbb{R}$$ with $$(s_k)$$ as its sequence of moments.