Suppose that $f : \mathbb R \to \mathbb C$ is a band-limited function, i.e. its Fourier transform $\hat f$ has support in a compact interval $[-a,a]$. Let $\phi(t) = e^{-\frac{t^2}{2\sigma^2}}$ be a Gaussian with variance $\sigma^2$. Further, let $$ F= f* \phi $$ be the convolution of $f$ with $\phi$. Is there a way to stably reconstruct $f$ from $F$? Intuitively if $a$ and $\sigma$ are both small then the deconvolution should be more stable. Are there any results about stability and reconstruction procedures of this inverse problem?

Thank you very much for your help!