recently I've been working on some estimation problems in $L_2$ space and came across the problem of computing the following Fourier transform for constants $a,b>0$: $$ F(w) := \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \exp\left(-\sqrt{-1}w x\right) \exp(-ax^2 -b|x|)\, dx. $$ What I tried:
We can use the fact that a Fourier transform of the product is a convolution of Fourier transforms. Fourier transforms of the Gaussian and Laplacian densities are known. We get the following: $$ F(w) = \sqrt{\frac{\pi}{a}} \int_{-\infty}^{\infty} \exp\left(-\frac{x^2}{4a}\right) \frac{2b}{b^2 + (x-w)^2} dx. $$ I spent some time searching through the literature and various tables of integrals, but could not find a closed-form expression for this one.
What I need:
I'll be glad to have any sort of closed-form expression. Perhaps, it will depend on erf $\Phi(x)$ (error function), which is also fine for me!