Interpolation by holomorphic functions of small exponential type on a half-plane

Question

Let $\{a_n\}_{n=1}^\infty$ be a sequence of complex numbers satisfying $|a_n|\le n^2$ and $|a_n|\to \infty$. I'm looking for a function $h(z)$ such that:

(a) $h$ is holomorphic on a half-plane $\{\Re(z) > - \epsilon\}$ for some $\epsilon >0$;

(b) $h(n)=a_n$ for all $n\ge 1$;

(c) $h$ is of exponential type $c<\pi$ on $\{\Re(z) > - \epsilon\}$.

I have tried to construct such an $h$ by using a function $g(z)$ with zeroes of order 1 at each $n$ and letting $$h(z) = \sum_{n=1}^\infty\, a_n \, \frac{g(z)}{g'(n)\, (z-n)}\, e^{\delta(z-n)}$$ for any $\delta>0$. The problem is that I've found only entire functions $g$ with the requested properties and this implies the exponential type of $h$ to be $\ge \pi$. I hope the request that $h$ is holomorphic only on a domain and is not necessarily entire helps.

Answer 1

This is not always possible under your conditions. For example, if $a_n=0$ for $n\geq 2$, then any function of exponential type $<\pi$ interpolating this sequence must be zero by Carlson's theorem, so if we choose $a_1=1$, interpolation by a function of this class is impossible.

For Carlson's theorem, see for example, Levin, Distribution of zeros of entire functions, AMS 1970, introduction to Chap IV.

The question has been modified, but the answer is still negative. Suppose that $a_n=P(n), \; n\geq 2$ where $P$ is a polynomial, and $a_1\neq P(1)$. Then one can apply Carlson's theorem to $f-P$.

Remarks. a) Carlson's theorem applies to functions holomorphic in a half-plane. b) One can generalize the last example, by taking $P$ to be any function of type less than $\pi$.