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Can we write every (tempered) distribution $\psi$, say on $\mathbb{R}$, as the sum of two distributions
$\psi = \psi_1 + \psi_2$
such that $\psi_1$ and the Fourier transform of $\psi_2$ are actually measurable functions of moderate growth. If so, under which additional conditions are the choices $\psi_1$ and $\psi_2$ unique?