I'm interested in the Fourier transform on a space of distributions that includes more than the usual tempered distributions, and in particular allows for $\delta$-distributions supported at complex values, and with exponential growth. This means I'd like to work with a space of entire analytic test functions, which is closed under Fourier transform.

Before I reinvent the wheel, is there a good reference where this, or a similar space closed under Fourier transform, has been studied?

The best I can come up with is the space of entire analytic functions such that $x\mapsto e^{\lambda x}\psi(x)$ is in $L^2(\mathbb{R})$ for every $\lambda\in\mathbb{R}$, and $\sup_{|y|<K}\Vert\psi(\cdot+iy)\Vert_2$ is finite for all $K>0$. This space is closed under Fourier transform by Theorem IX.13 of Reed and Simon, volume 2.