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.

  • $\begingroup$ Is the theorem you quote supposed to be the Paley-Wiener theorem? The hypotheses look similar but aren't quite the same $\endgroup$ – Yemon Choi Jan 16 at 20:21
  • $\begingroup$ 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 $b<a$, $$\sup_{|\eta|\leq b} \Vert \hat{f}(\cdot+i\eta) \Vert_2<\infty.$$ $\endgroup$ – Holographer Jan 17 at 3:23

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