The dual space of entire functions is known. It is the space of functions analytic around infinity with non-constant term, $\mathcal{O}^\infty_0$. The action of $F\in \mathcal{O}^\infty_0$ on an entire function $\phi$ is defined by $$ \langle f,\phi\rangle = \frac{1}{2\pi i}\oint F(z) \phi(z) dz, $$ where the integral is performed on a circle of big enough radius.

Sometimes these are described as functionals of compact support. This is because for any integrable real function $f$ with compact support and any entire function $\phi$ we can define the pairing $$\langle f,\phi\rangle = \int_{-\infty}^\infty f(x) \phi(x) dx. $$ Then this functional can be included in $\mathcal{O}^\infty_0$ by defining $$ F(z) = \int_{-\infty}^\infty \frac{f(x)}{x-z} d x .$$ The function $F$ is in $\mathcal{O}^\infty_0$ and behaves in the expected way. So in a sense the support of an analytic functional is the set of points on which it is not analytic.

I couldn't find anything on the dual space of entire functions of exponential type. Since this is a smaller space, its dual has to be bigger. Clearly if $\phi$ is of exponential type then the map $$ \phi \mapsto \int_{-\infty}^\infty \phi(x) e^{-x^2} dx $$ defines a continuous functional and clearly it does not have compact support.

So is this space known? Is there a description of it as a functional space?