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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!

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  • $\begingroup$ Could you explain exactly what "stable" means here? Intuitively it is clear that stability (whatever it means) depends also on $a$ not only on $\sigma$. $\endgroup$ Jul 28, 2021 at 13:25
  • $\begingroup$ The inverse of F has an explicit formula via Fourier transform. It is straightforward to calculate the operator norm of the inverse (e.g. in the setting where F maps from L² to L²). $\endgroup$
    – Dirk
    Jul 28, 2021 at 18:11

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