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I am looking for a sequence of functions $f_n,n\geq 1$ in $L^2(\mathbb R)$ such that $f_n$ is equal to $1$ on $[-n,n]$ and $\hat{f_n}$ vanishes on $[-1,1]$.

Actually, I would also like $f_n$ to be $C_c^2$ but I can relax the spectral gap assumption to $\int_{-1}^1 \hat{f_n}^2$ can be arbitrarily small (i.e. smaller than $a_n$ for any sequence of positive numbers)

EDIT: I am also interested in a result saying that this is impossible

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    $\begingroup$ Are you imposing any conditions on how much $f_n$ may grow at infinity as $n\to\infty$? If you impose such conditions, then $f_n$ converges to 1 in ${\cal S}'$. This would imply that $\hat f_n$ converges to a delta function, which is not consistent with vanishing on $[-1,1]$. $\endgroup$
    – Michael Renardy
    Commented Jul 1, 2020 at 16:46
  • $\begingroup$ Maybe this is helpful: math.stackexchange.com/q/2332292/99220 mathoverflow.net/q/275072/107094 $\endgroup$
    – Hyperplane
    Commented Jul 1, 2020 at 21:30
  • $\begingroup$ I am not imposing any condition, but as I indicated in my edit I am also interested in a negative result $\endgroup$
    – kaleidoscop
    Commented Jul 2, 2020 at 5:59

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