# Stable deconvolution of a band-limited function from its convolution with a Gaussian

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!

• Could you explain exactly what "stable" means here? Intuitively it is clear that stability (whatever it means) depends also on $a$ not only on $\sigma$. Jul 28, 2021 at 13:25
• 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).
– Dirk
Jul 28, 2021 at 18:11