Absolutely and completely monotonic functions and sequences: Difference between revisions
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In mathematics, the notions of an '''absolutely monotonic function''' and a '''completely monotonic function''' are two very closely related concepts. Both imply very strong monotonicity properties. Both types of functions have derivatives of all orders. In the case of an absolutely monotonic function, the function as well as its derivatives of all orders must be non-negative in its ___domain of definition which would imply that the function as well as its derivatives of all orders are monotonically increasing functions in the ___domain of definition. In the case of a completely monotonic function, the function and its derivatives must be alternately non-negative and non-positive in its ___domain of definition which would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions.
Such functions were first studied by S. Bernshtein in 1914 and the terminology is also due to him.<ref name="Enc">{{cite web |title=Absolutely monotonic function |url=https://encyclopediaofmath.org/wiki/Absolutely_monotonic_function |website=encyclopediaofmath.org |publisher=Encyclopedia of Mathematics |access-date=28 December 2023}}</ref><ref>{{cite journal |last1=S. Bernstein |title=Sur la définition et les propriétés des fonctions analytique d'une variable réelle |journal=Mathematische Annalen |date=1914 |volume=75 |issue=4 |pages= The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics. For example, in classical analysis they occur in the proof of the positivity of integrals involving Bessel functions or the positivity of Cesàro means of certain
Jacobi series.<ref>{{cite journal |last1=R. Askey |title=Summability of Jacobi series |journal=Transactions of the American Mathematical Society |date=1973 |volume=179 |pages=
==Definitions==
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====Examples====
The following functions are absolutely monotonic in the specified regions.<ref name=dvw>{{cite book |
# <math>f(x)=c</math>, where <math> c</math> a non-negative constant, in the region <math> -\infty <x < \infty </math>
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where <math>\Delta^k\mu_n = \sum_{m=0}^k (-1)^m {k \choose m}\mu_{n+k-m}</math>.
A sequence <math>\{\mu_n\}_{n=0}^\infty</math> is called a completely monotonic sequence if its elements are non-negative and its successive differences are alternately non-positive and non-negative,<ref name=dvw/>{{
::<math>(-1)^k\Delta^k\mu_n\ge 0, \quad n,k = 0,1,2,\ldots </math>
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Both the extensions and applications of the theory of absolutely monotonic functions derive from theorems.
*
* 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.
*
::<math> f(x) = \int_0^\infty e^{xt}\, d\mu(t)</math>
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* A sequence <math> \{\mu_n\}_0^\infty</math> is completely monotonic if and only if there exists an increasing function <math>\alpha(t)</math> on <math>[0,1]</math> such that
::<math> \mu_n = \int_0^1 t^n \, d\alpha(t), \quad n=0,1,2,\ldots</math>
:The determination of this function from the sequence is referred to as the [[Hausdorff moment problem]].
==Further reading==
The following is a
* {{cite book |last1=
* {{cite book |last1=D. V. Widder |title=The Laplace Transform |date=1946 |publisher=Princeton University Press}} See Chapter III The Moment Problem (pp. 100 - 143) and Chapter IV Absolutely and Completely Monotonic Functions (pp. 144 - 179).
* {{cite book |last1=Milan Merkle |title=Analytic Number Theory, Approximation Theory, and Special Functions |date=2014 |publisher=Springer |pages=
* {{cite journal |last1=Arvind Mahajan and Dieter K Ross |title=A note on completely and absolutely monotonic functions |journal=Canadian Mathematical Bulletin |date=1982 |volume=25 |issue=2 |pages=
* {{cite journal |last1=Senlin Guo, Hari M Srivastava and Necdet Batir |title=A certain class of completely monotonic sequences |journal=Advances in
* {{cite journal |last1=
==See also==
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