# Eigenstates of Fourier transformation

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 $$\mathcal F T=T.$$

• – Christian Remling Sep 15 '14 at 18:21
• The question is not Hilbertian, but on $\mathcal S'$: I want also to include the Poisson summation formula. – Bazin Sep 15 '14 at 18:23
• I don't see why this makes any difference. Voting to close as a duplicate. – Michael Renardy Sep 15 '14 at 18:30
• 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). – Joonas Ilmavirta Sep 15 '14 at 18:43
• 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. – Robert Israel Sep 15 '14 at 18:50

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 one-dimensional: 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 translation-invariance, it is (a scalar multiple of) sum-over-evaluation-at-integers. A just-slightly-more-complicated version applies in $n$-dimensions.