For any sufficiently large $d\in\mathbb{N}$, does there exist a probability measure $\Psi$ supported on the Euclidean ball in $\mathbb{R}^d$ for which $|\widehat{\Psi}[\omega]|\le C\cdot \exp(-\|\omega\|^{0.99}_2)$ for all $\omega$, or perhaps for all $\omega$ satisfying $\|\omega\|_2 \gtrsim \log(d)$ (here $C$ is any dimension-independent constant)? More modestly, it would even be good to know if there is any such distribution $\Psi$ satisfying super-algebraic decay, perhaps outside of a radius of $d^{o(1)}$.

All this should be possible if one drops the condition that $\Psi$ be nonnegative, as one could take the inverse Radon transform of any smooth, symmetric probability measure over $[-1,1]$ with sufficiently rapid Fourier decay.

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