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?



Decomposition of distributions
The [Dirac Comb](http://en.wikipedia.org/wiki/Dirac_comb), an infinite sum of delta functions, is an example of a tempered distribution that cannot be thusly decomposed (its Fourier transform is another Dirac Comb). 


[Added:] There is a positive result in this direction that I (among others) only partly-remembered: Any distribution can be written as a locally finite sum of derivatives of continuous functions. If the distribution has finite order, then the sum is finite. See Rudin's Functional Analysis, Theorem 6.28.


