This function is between the function $f_1 = 1$ and $f_2 = 1/(1+ x^2)$. For the former, the Fourier transform is a delta function (a distribution, or a generalized function). For the latter, the Fourier transform is something like $a e^{-b|k|}$. I thus expect something between a delta function and a regular function.
What is it exactly? Is it in the form of $a \delta(k) + b(k)$, where $a$ is some constant and $b$ some regular function?