Let $\gamma$ be defined on $\mathbb R^n$ by $\gamma (x)=e^{π x^2}$. With $\mathcal F$ standing for the Fourier transformation defined on the Schwartz space by $$ (\mathcal F u)(\xi)=\int e^{2iπ x\cdot \xi} u(x) dx, $$ we have $ \mathcal F\gamma =\gamma. $ We can also define $\mathcal F$ for the tempered distribution ($\mathscr S'$) with the duality formula $$ \langle \widehat T,\phi\rangle_{\mathscr S',\mathscr S}=\langle T,\widehat{\phi}\rangle_{\mathscr S',\mathscr S}. $$ For instance, the Poisson summation formula is $\widehat S=S$ with $S=\sum_{k\in \mathbb Z^n}\delta_k$. Finally the question: determine all the tempered distributions $T$ such that $$ \mathcal F T=T. $$

2$\begingroup$ Related: mathoverflow.net/questions/12045/… $\endgroup$ – Christian Remling Sep 15 '14 at 18:21

2$\begingroup$ The question is not Hilbertian, but on $\mathcal S'$: I want also to include the Poisson summation formula. $\endgroup$ – Bazin Sep 15 '14 at 18:23

1$\begingroup$ I don't see why this makes any difference. Voting to close as a duplicate. $\endgroup$ – Michael Renardy Sep 15 '14 at 18:30

1$\begingroup$ A tempered distribution satisfies $T=\hat T$ iff $\langle T,\hat\phi\phi\rangle=0$ for all Schwartz functions $\phi$. If the Schwartz space is decomposed into eigenspaces of the Fourier transform as $E_1\oplus E_{1}\oplus E_i\oplus E_{i}$, then tempered distributions with eigenvalue one correspond to the dual space of $E_1$ (by extension by zero to the Schwartz space). $\endgroup$ – Joonas Ilmavirta Sep 15 '14 at 18:43

1$\begingroup$ That is, you want all the tempered distributions that annihilate $E_{1}$, $E_i$, and $E_{i}$, and equivalently the test functions $H_j(x) \exp(x^2/2)$ for $j \not\equiv 0 \mod 4$ where $H_j$ are the Hermite polynomials. $\endgroup$ – Robert Israel Sep 15 '14 at 18:50
In case the specific distributiontheoretic argument is not clear... as it hadn't really been overtly mentioned in comments or answers, and is not really suggested by the classical argument as in Titchmarsh and such:
E.g., for Poisson summation on $\mathbb R$: observe that the distribution "sum over integers" is annihilated by multiplication by $e^{2\pi ix}1$, and is translation invariant. Observe that these two conditions are interchanged by Fourier transform. Show that the space of such distributions is onedimensional: the multiplication annihilation shows that any such distribution is of order $0$ and supported at integers. By classification of distributions supported at a point, it is a sum of Dirac deltas at integers. By translationinvariance, it is (a scalar multiple of) sumoverevaluationatintegers. A justslightlymorecomplicated version applies in $n$dimensions.
This question is addressed in great detail in the book of Titchmarsh, Introduction to the theory of Fourier integrals. Such functions are called selfreciprocal, and there is a separate chapter about them. Of course Titchmarsh did not use the language of distributions, but it is easy to translate. A recent paper on the related subject is http://arxiv.org/abs/1203.2427.