Carleman's condition

From Wikipedia, the free encyclopedia

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 $μ$ 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, $μ$ is the only measure on $\mathbb{R}$ with $(s k)$ as its sequence of moments.

[edit] See also

The problem of moments

[edit] References

N.I.Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver & Boyd, 1965

Categories: Mathematical analysis | Probability theory | Mathematical theorems

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