# Entire analytic functions with entire analytic Fourier transform, and corresponding distributions

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 $$\sup_{|y| is finite for all $$K>0$$. This space is closed under Fourier transform by Theorem IX.13 of Reed and Simon, volume 2.

• Is the theorem you quote supposed to be the Paley-Wiener theorem? The hypotheses look similar but aren't quite the same – Yemon Choi Jan 16 at 20:21
• It's certainly in the realm of Paley-Wiener, though not precisely that theorem: let $f\in L^2(\mathbb{R})$. Then $e^{b|x|}f\in L^2(\mathbb{R})$ for all $b<a$ iff $\hat{f}$ has an analytic continuation to the strip $|\Im z|<a$, such that for $|\eta|<a$, $\hat{f}(\cdot+i\eta)\in L^2(\mathbb{R})$, and for all $b<a$, $$\sup_{|\eta|\leq b} \Vert \hat{f}(\cdot+i\eta) \Vert_2<\infty.$$ – Holographer Jan 17 at 3:23