Completely monotone function: Difference between revisions
imported>Mateusz (Created page with "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 $x > 0$ a...") |
imported>Mateusz (added "Examples" and "Properties") |
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\[ f(z) = \int_{[0, \infty)} e^{-s z} m(\mathrm d s) . \] | \[ f(z) = \int_{[0, \infty)} e^{-s z} m(\mathrm d s) . \] | ||
Here $m$ is an arbitrary Radon measure such that the above integral is finite for all $z > 0$. | Here $m$ is an arbitrary Radon measure such that the above integral is finite for all $z > 0$. | ||
==Examples== | |||
The following functions of $z$ are completely monotone: | |||
* $z^s$ for $s \le 0$, | |||
* $e^{-t z}$ for $t \ge 0$, | |||
* $\log(1 + \frac{1}{z})$, | |||
* $e^{1 / z}$. | |||
==Properties== | |||
If $f_1, f_2$ are completely monotone and $c > 0$, then also $c f_1$, $f_1 + f_2$, $f_1 f_2$ are completely monotone. | |||
If $f$ is completely monotone and $g$ is a [[Bernstein function]], then $f \circ g$ is completely monotone. | |||
==References== | ==References== | ||
Latest revision as of 06:05, 20 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 $z > 0$.
Examples
The following functions of $z$ are completely monotone:
- $z^s$ for $s \le 0$,
- $e^{-t z}$ for $t \ge 0$,
- $\log(1 + \frac{1}{z})$,
- $e^{1 / z}$.
Properties
If $f_1, f_2$ are completely monotone and $c > 0$, then also $c f_1$, $f_1 + f_2$, $f_1 f_2$ are completely monotone.
If $f$ is completely monotone and $g$ is a Bernstein function, then $f \circ g$ is completely monotone.
References
- ↑ 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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