Absolutely and completely monotonic functions and sequences: Difference between revisions

Revision as of 23:16, 27 August 2024

Latest revision as of 15:32, 16 June 2025

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Both the extensions and applications of the theory of absolutely monotonic functions derive from theorems.

* TheBernstein's 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>