Absolutely and completely monotonic functions and sequences: Difference between revisions
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→Some important properties: It is called the Hausdorff moment problem. |
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* The little Bernshtein theorem: A function that is absolutely monotonic on a closed interval <math>[a,b]</math> can be extended to an analytic function on the interval defined by <math>|x-a| < b-a</math>.
* A function that is absolutely monotonic on <math>[0,\infty)</math> can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line.
*The [[Bernstein's theorem on monotone functions|big Bernshtein theorem]]: A function <math>f(x)</math> that is absolutely monotonic on <math>(-\infty,0]</math> can be represented there as a Laplace integral in the form
::<math> f(x) = \int_0^\infty e^{xt}\, d\mu(t)</math>
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