Carleman's condition

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. ==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''. ==Notes==