Carlson's theorem is a uniqueness theorem that states (loosely) that if an entire function $f(z)$ of a complex variable takes on values $f_n$ at the integers $n \in \mathbb{N}$, then any other entire function that takes on the same values $f_n$ at the integers must behave very differently from $f(z)$ as $|z| \to \infty$.
Carlson's theorem only discusses uniqueness, but I'm wondering about the flip side, existence.
- Given a double-sided sequence $f_n \in \mathbb{C}$, is there always an entire interpolating function that takes on those values at the integers? What if we place restrictions on the sequence $f_n$ (e.g. if we assume that it's bounded, or falls off as $n \to \pm\infty$)?
- If we can guarantee the existence of such an entire interpolating function, then (assuming a sufficiently slowly growing sequence $f_n$) can we bound the asymptotic growth of the function? If so, then together with Carlson's theorem, that would guarantee that (for a sufficiently slowly growing sequence $f_n$) there is a unique slowly-growing entire interpolating function.
For question #1, one possible construction would be to take an entire basis function like $g(z) = \frac{\sin(z)}{z}$ and then form the infinite linear combination $\sum \limits_{n \in \mathbb{N}} f_n\, g(z – n)$. I believe that $f_n$ would need to fall off faster than $1/\log(n)$ for large $n$ in order for this series to converge on the real axis, but since this choice of $g(z)$ grows quickly in the imaginary direction, $f_n$ might need to fall off much faster in order to ensure convergence off the real axis. We could modulate the basis function $g(z)$ by something like $\exp \left( -z^2 \right)$ to improve convergence in the real direction, but that would make it worse in the imaginary direction (and vice versa for $\exp \left(z^2 \right)$).
Best Answer
Let $f_n$ be a secuence of values. By Wierstrass Factorization Theorem we can build a function $g(z)$ such that it has zeros of order one at the integers. Let $g_n=(g(z)/(z-n))|_{z=n}$. By Mittag-Leffler's theorem you can build now a function $f(z)$ such that $f$ has at any integer $n$ such that $f_n\neq 0$ a pole of order one and residue $f_n/g_n$ (and such that these are all it's poles). Therefore, $f(z)g(z)$ takes the desired values $f_n$ at the integers. Note that this construction also holds with the requirement that the values $f_n$ must be taken on a discrete subset of $\mathbb{C}$. As you can see, at the start of the construction we could multiply $g$ by any $e^{h(z)}$, with $h$ holomorphic, and then follow the process to get a distinct function which would be also a solution of the problem, so I believe there may be functions taking the arbitrary values $f_n$ which are not of exponential type.