Cfrgtkky

I am trying to solve the following deconvolution problem where $g(s)$ is a known real valued function and has finite energy:

$$g(s) = int_{-infty}^{infty} frac{1}{sqrt{2pi}}f(t)e^{-(t-s)^2/2}dt$$

Since the integral kernel is a difference kernel the solution is available via convolution theorem.

$$f(t) = F^{-1}left[frac{F[g](w)}{e^{-w^2/2}}right]$$

where $F$ denotes the fourier transform.

Now on top of this I would like to impose finite energy condition on $f$, so I wrote the following regularized error minimization

$$hat{f}=text{argmin}_f left(int_{-infty}^{infty}[g(s) - int_{-infty}^{infty} frac{1}{sqrt{2pi}}f(t)e^{-(t-s)^2/2}dt]^2 ds+ alphaint_{-infty}^{infty} |f(t)|^2dtright)$$

I would like to show that:

$$hat{f}(t) = F^{-1}left[frac{F[g](w)}{e^{-w^2/2}+alpha}right]$$

or something in the line of this.

Yes, this is broadly correct and the main step that you need is that the Fourier transformation is an isometry so $int_{-infty}^infty |f(t)|^2 , dt = int_{-infty}^infty |F(omega)|^2 , domega $ (up to a factor of $2pi$). So, Fourier transform the problem and you can write the objective in Fourier space. The result then follows straightaway.