My question is:
Given a function $f$ in the Schwartz class, we are looking for a measure $\mu$ which is a solution of the convolution equation: $f = e^{-|.|^2/2} \ast \mu$, where $e^{-|.|^2/2}$ is the gaussian.
Any ideas?
Thanks in advance.
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1 Answer
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Choosing properly the Gaussian (e.g. $e^{-π\vert x\vert^2}$) and the normalization for the Fourier transform, your equation becomes $$ \hat f(\xi)=e^{-π\vert \xi\vert^2}\hat \mu(\xi). $$ This implies that $\hat f(\xi)e^{π\vert \xi\vert^2}$ must be a tempered distribution. A sufficient condition for this to occur is a strong regularity condition on $f$, requiring that it should belong to a (global) Gevrey class $G^s$ with $s<1/2$: this will ensure a decay for the Fourier transform of type $$ e^{-\delta \vert\xi\vert^{1/s}} $$ and since $1/s>2$, this will absorb the term $e^{π\vert \xi\vert^2}$. These Gevrey $\frac{1}{2}-\epsilon$ functions are far better than analytic with local estimates of type $$ \sup_{x\in K}\vert f^{(\alpha)}(x)\vert\le C_K\rho_K^{-\vert\alpha\vert}(\alpha!)^s $$ for the derivatives ($K$ compact and $\rho_K>0$).
