Completely monotone function: Difference between revisions

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Revision as of 05:42, 19 July 2012

A function $f : (0, \infty) \to [0, \infty)$ is said to be completely monotone (totally monotone, completely monotonic, totally monotonic) if $(-1)^k f^{(k)} \ge 0$ for $x > 0$ and $k = 0, 1, 2, ...$[1]

Representation

By Bernstein's theorem, a function $f$ is completely monotone if and only if it is the Laplace transform of a nonnegative measure, \[ f(z) = \int_{[0, \infty)} e^{-s z} m(\mathrm d s) . \] Here $m$ is an arbitrary Radon measure such that the above integral is finite for all $z > 0$.

References

  1. Schilling, R.; Song, R.; Vondraček, Z. (2010), Bernstein functions. Theory and Applications, Studies in Mathematics, 37, de Gruyter, Berlin, doi:10.1515/9783110215311, http://dx.doi.org/10.1515/9783110215311 

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