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

:$s\_k\; =\; int\_\{-infty\}^\{+infty\}\; x^k\; ,\; dmu(x)$

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.

ee also

* The problem of moments

References

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