Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}\exp(-x^2/2)$ be the Gaussian density and $f:\mathbb{R}\to\mathbb{R}$ another measurable function.

Under what conditions can $f$ be recovered from its convolution with $\varphi$? In other words, under what conditions does $f\ast\varphi=0$ imply that $f$ is zero a.e?

If $f\in L^1(\mathbb{R})$, then it has a Fourier transform and the statement follows since $\varphi$ has a Fourier inverse. What about other conditions on $f$? For example, what if it is bounded by a polynomial? Or a subexponential function?