I am interested in the space of all holomorphic function of exponential type one, that decay exponentially along the positive real axis. I tried to define it as follows.
Let $$\|f\|_n = \sup_{z\in\mathbb{C}} |f(z)| e^{-(1+\frac1n)|z|} + \sup_{z\in\mathbb{R}^+} |f(z)| e^{(1-\frac1n)z} $$ and let $\mathcal A$ be the space of all functions, $f$, such that $\|f\|_n$ is finite for every $n$. Since $\|\cdot\|_n$ is a family of seminorms, $\mathcal A$ is a Frechet space.
It is clear by the definition that polynomials are not in $\mathcal A$. I would like to find a subset of $\mathcal A$ that is countable and dense.
I think that a good candidate is the span of "monomials" of the type $z^k e^{-z}$, but I couldn't prove that this is a dense set. Is it actually? Is there any other?