I am trying to solve the following deconvolution problem where $g(s)$ is a known real valued function and has finite energy:
$$g(s) = \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}f(t)e^{-(t-s)^2/2}dt$$
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
$$\hat{f}=\text{argmin}_f \left(\int_{-\infty}^{\infty}[g(s) - \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}f(t)e^{-(t-s)^2/2}dt]^2 ds+ \alpha\int_{-\infty}^{\infty} |f(t)|^2dt\right)$$
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 \, d\omega $ (up to a factor of $2\pi$). So, Fourier transform the problem and you can write the objective in Fourier space. The result then follows straightaway.