What are completely monotonic functions on an interval $I$? See the picture 1.png
What is the Bernstein-Widder theorem for completely monotonic functions on the infinite interval $I=(0,\infty)$? See the picture 1.png
My question is: is there an anology on the finite interval $I=(a,b)$ of the Bernstein-Widder theorem for completely monotonic functions on the infinite interval $I=(0,\infty)$? In other words, if $f(x)$ is a completely monotonic function on the finite interval $I=(a,b)$, is there an integral representation like (1.2) in the picture 1.png for the completely monotonic function $f(x)$ on the finite interval $(a,b)$?
The answer to this question is very important for me. Anyway, thank everybody who would provide answers and who would pay attention on this question.
Dear Feng,
may be next paper will be useful for You https://www.researchgate.net/publication/245727278_Completely_Monotonic_Functions_Associated_with_the_Gamma_Function_and_Its_q_-_Analogues
Sincerely, Yuri
Article Completely Monotonic Functions Associated with the Gamma Fun...
The formula (1.2) for completely monotonic function on an interval is not valid. For example, f(t) = e-t - e-1 is completely monotonic on [0, 1] with f(1) = 0, and the latter property is impossible for a non-zero function satisfying the integral representation (1.2). In order to have (1.2) you need a function that has completely monotonic extension on the semi-axes (0, \infty). There is really a lot of information about absolutely monotonic / completely monotonic functions in the original Bernstein's paper
S. N. Bernstein (1928). "Sur les fonctions absolument monotones". Acta Mathematica 52: 1–66. doi:10.1007/BF02592679
Maybe some of this information can be of use for you. S. N. Bernstein is one of the most famous mathematicians from my University, and it is always a pleasure for me to mention his works: they contain much more material than the textbooks citing his results.
All the best, Vladimir Kadets
Thank Professor Vladimir Kadets a lot. I will read the paper by S. N. Bernstein (1928).
The function $f(t)=e^{-t}-e$ is completely monotonic on the interval $(-\infty,-1)$ with $f(-1)=0$. The integral representation is valid for all completely monotonic functions on $(0,\infty)$.
Sorry, I meant $f(t)=e^{-t}-e^{-1}$, then it is is completely monotonic on [0, 1] (and on (-\infty, 1] as well) with f(1) = 0. Thanks for pointing this to me, I corrected the misprint.
It is not quite clear what do you mean under ''representation like the formula (1.2) ''. Do you mean exactly formula (1.2)? If so, then for negative values of the argument there can be a problem with the finiteness of the integral. If not, then what kind of representations do you consider to be similar to (1.2)?
It is clear that there is no an integral representation exactly same with (1.2) for a completely monotonic function on a finite integral.
I need an integral representation, from which one can see its complete monotonicity easily and directly, for a completely monotonic function on a finite interval.
For example, can one find an integral represention for the function $(e^x)/(1+x)$ defined on $(-1,0]$ such that its complete monotonicity on $(-1,0]$ can be seen from the integral representation?
Let us speak for simplicity about interval (0, 1) because every finite interval (a, b) reduces to (0,1) by a linear transformation cx + d with c > 0.
So you question can be stated as follows: does there exist a set T and a collection ft(x), t \in T of ''elementary" completely monotonic functions on (0, 1) such that for every completely monotonic function f on (0,1) there exist a positive measure m on T such that
$f(x) = \int_T f_t(x) d m(t)$
and moreover for every natural n the n-th derivative of f has representation
$f^{(n)}(x) = \int_T f_t^{(n)}(x) d m(t)$?
The question looks interesting. It is still an informal statement, because one has to decide what class of completely monotonic functions on (0, 1) should be considered as ''elementary" one. Say, if one takes the class of all completely monotonic functions on (0, 1) as ''elementary", the answer is trivially positive (with m being a $\delta$-measure), but such a representation is completely useless.
Unfortunately, I don't know the answer. I would say that the very first step would be to formulate a hypothesis what is a reasonable ''elementary" class. Maybe, it should contain outside of exponentials $\exp(ax), a < 0$ also functions of the form $\exp(ax) + p(x)$, where p is such a polynomial that $\exp(ax) + p(x)$ is completely monotonic on (0, 1)?
A reasonable approach to the question would be to search for extremals of the cone of all completely monotonic functions on (0, 1) and then to use the Krein-Milman theorem in order to get the requested integral representation through the extremals. Unfortunately, I have no idea what are the extremals of this cone.
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