Suppose we have a positive integer $n$ and a finite list of real numbers $\{a_1,\,a_2,\,\dots,\,a_n\}$. We want to find a real-analytic function $f:[1,n]\to\mathbb R$ such that $f(m)=a_m$ for all $m\in\mathbb N\cap[1,n]$, i.e. $f(1)=a_1$, $f(2)=a_2$, $\dots$, $f(n)=a_n$. Of course, many such functions exist — for example, we can take the Lagrange interpolating polynomial.
Let's add another requirement on $f$ — it must be completely monotone, i.e. the function itself and all its derivative must be monotone. Obviously, it is not always possible to find such a function. For example, there is no such function if the list $\{a_m\}$ itself is not monotone. I would like to know under what exact conditions on $\{a_m\}$ such a function exists.
Bcause my question below concerns an effective decidability, the list $\{a_m\}$ has to be provided in some explicit finitary form. Let's, for the sake of simplicity, assume it is a list of explicit rational numbers. You are welcome to relax this condition, and assume, for example, that it is a list of real algebraic numbers defined by their minimal polynomials and rational isolating intervals, or some broader class of real numbers that can be defined in an explicit finitary form (e.g. periods, or numbers that can be represented using periods and elementary functions, etc).
Is there an algorithm, which, given a list of numbers $\{a_m\}$ as an input, decides whether there is a completely monotone real-analytic function on $[1,n]$ that reproduces these numbers in integer points $\mathbb N\cap[1,n]$? The answer has to be in a "yes" or "no" form, and the function need not be produced in any explicit form.
If yes, could you please give an outline of such an algorithm?