Possible gaps for a function and its Fourier transform

Ok, I think I managed to prove that the answer is no for the discrete version, I put it here for anyone interested (likely generalises to any dimension, didn't look into the continuous version).

Assume $ \psi \subset L^{2}( \mathbb T ^{2})$ has a gap in each coordinate, i.e. vanishes on $T_\varepsilon^c$ where $ T_{\varepsilon }:=( \mathbb T \setminus [-\varepsilon ,\varepsilon ])^{2}$ for some $ \varepsilon >0.$ Then if $ \hat \psi $ vanishes on a quarter space, then $ \psi \equiv 0$.

Let us write the proof by contradiction. Assume wlog that $ \hat \psi $ vanishes on $ (-\mathbb{N}^{*})^{2}$ and that $ \hat \psi (0,0)\neq 0.$

Let $ C_{\varepsilon } \subset \mathbb C $ the unit circle except points of angle in $ [-\varepsilon ,\varepsilon ]$. According to Mergelyan's theorem, there are polynomials $P_{n}( z)\to z^{-1}$ uniformly on $ C_{\varepsilon }$. Putting $ Q_{n}( u)=e^{iu}P_{n}( e^{iu}),$ $ Q_{n}( u)\to 1$ on $ \mathbb T \setminus [-\varepsilon ,\varepsilon ]$, hence $$ Q_{n}(u)Q_{n}(v)\psi (u,v)\to \psi (u,v)$$ in $ T_{\varepsilon } $, hence on $ L^{2} ( \mathbb T ^{2})$ because $ \psi \equiv 0$ on $ T_{\varepsilon }^c$, and $ \hat \psi \star( \hat Q_{n} \otimes \hat Q_{n})\to \hat \psi $ in $ L^{2}( \mathbb{Z} ^{2})$.

Therefore, since $ Q_{n}( u)=\sum_{m>0}a_{m} e^{imu}, \hat Q_{m}=\sum_{m>0}a_{m}\delta _{m}$, \begin{align*} 0\neq \hat \psi ( 0,0)=\lim_n\sum_{k,l\in \mathbb{Z} ^{2}}\underbrace{ \hat \psi ( k,l) }_{=0\text{ if }k<0,l<0}\underbrace{ \hat Q_{n}(- k) \hat Q_{n}( -l)}_{=0\text{ if }k\geqslant0\text{ or }l\geqslant0}=0. \end{align*}