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It is well-known that, besides the standard Gaussian $e^{-|x|^2/2}$, there are many interesting functions which are eigenfunctions of the Fourier transform, for example the Hermite functions.

Mainly out of curiosity, I am wondering if there is any literature on a systematic study of $L^2$-functions whose Fourier transform is $L^2$-orthogonal to themselves, i.e., functions $f\in L^2(\mathbb R^n)$ such that $$\left< f,\hat f\right>_{L^2(\mathbb R^n)}=0.$$ Note that there do exist non-zero functions with this property: For example, take $f=\mathrm{rect}(x-a)+\mathrm{rect}(x+a)$, where $a\geq 0$ and $\mathrm{rect}:\mathbb R\to \mathbb R$ is the standard rectangle function taking the value $1$ on the interval $[-1/2,1/2]$ and $0$ elsewhere. Then $\hat f(x)=\cos(ax)\mathrm{sinc}(x/2)$, and for an appropriate choice of $a$ (approximately $a=2.8157073021$) one finds that $\left< f,\hat f\right>_{L^2(\mathbb R)}=0$.

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Let $\{\psi_n\}$ the orthonormal basis of $L^2(\mathbb{R}^d)$ consisting of the (normalized) Hermite functions. It is well known that $ \widehat{\psi_n}=(-i)^n\psi_n.$ Consider the closed subspaces $V_k = \overline{span}\{\psi_n: n\equiv k\mod4 \},k=0,1,2,3.$ Then functions satisfying the orthogonality relation in the question are exactly the functions $f\in L^(\mathbb{R}^d)$ that can be written as $f = f_0+f_1+f_2+f_3$, where $f_k\in V_k$ and $\Vert f_0 \Vert = \Vert f_2 \Vert$ and $ \Vert f_1 \Vert = \Vert f_3 \Vert $.

In fact \begin{equation*} \langle f , \hat{f} \rangle = \langle f , f_0-if_1-f_2+if_3 \rangle = \Vert f_0 \Vert^2 -\Vert f_2 \Vert ^2 + i(\Vert f_1 \Vert^2-\Vert f_3 \Vert ^2), \end{equation*}

which is zero if and only if the condition in the norms hold. Certainly there is an abundance of such functions.

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