I'm searching for a (could be complex) function $f_\nu(k), k,\nu\in \mathbb{R}$ that has the following properties:
(1) Always continuous in $k$ (for fixed $\nu$). Continuous in $\nu$ except possibly at a finite number of points $(k_i,\nu_i)$.
(2) $$f_\nu(k)\to (\nu>0)k+(\nu\leq 0)~\text{when}~ k\to 0,$$
where the symbol $(P)$ means: $(P)=1$ if $P$ is true and $(P)=0$ if $P$ is false.
(3) The 2D Fourier transform $$\tilde{f}_\nu(\vec{r})=\int d^2 \vec{k} e^{i\vec{k}\cdot\vec{r}} f_\nu(|\vec{k}|)$$ exists, and should be continuous and differentiable (w.r.t $\vec{r}$, except possibly at $\vec{r}=0$). Should decay at least as fast as $1/r$ as $r\to \infty$.
Met this problem in my theoretical physics research. $\tilde{f}_\nu(\vec{r})$ is going to represent a wave function or interaction, potential, so should hopefully be as good-looking, well-behaved (a rational function is perfect!) as possible.
Any kind of construction or idea is appreciated! Know you mathematicians are genius :)
Note: A function $f_\nu(\vec{k})$ would be equally good, with $f_\nu(\vec{k})\to (\nu>0)|\vec{k}|+(\nu\leq 0)$ when $ |\vec{k}|\to 0$, and continuous in $k_x,k_y$.