For $f\in L^2(\mathbb R)$, one may want to expand $f$ in terms of other functions. Fourier transform is a prominent example.
I want to consider another type of `basis functions'. Specifically, for $f\in L^2(\mathbb R)$, I want to find a function $\tilde f_+(a,w)$ and $\tilde f_-(a,w)$ such that $$f(x) = \int_{-\infty}^\infty da \int_0^\infty dw \left( \frac{\tilde f_+(a,w)}{x-a+iw} + \frac{\tilde f_-(a,w)}{x-a-iw} \right). \tag{*}$$ First of all, to have an expression like $(*)$, the set of functions $$\left\{ x\mapsto \frac{1}{x-a+ iw} \right\}_{a\in\mathbb R, w>0} \cup \left\{ x\mapsto \frac{1}{x-a- iw} \right\}_{a\in\mathbb R, w>0} \tag{**}$$ should be dense in $L^2(\mathbb R)$.
Main question: Is the above set dense in $L^2(\mathbb R)$?
As an optional question, if it is dense, then I want to find an inversion formula, i.e., express $\tilde f_\pm(a,w)$ in terms of $f$. (Maybe such $\tilde f_\pm(a,w)$ is not unique. One example is sufficient.)
Note: In $(**)$, if only one collection is considered, say $\left\{ x\mapsto \frac{1}{x-a+ iw} \right\}_{a\in\mathbb R, w>0}$, then the answer is negative. It can be seen by Fourier transform. Note that the Fourier transform is an isometry of $L^2(\mathbb R)$. And, since $\frac{1}{x-a+ iw}$ is analytic on the upper half plane, its Fourier transform is supported on $[0,\infty)$. Hence, a function whose Fourier transform is not supported on $[0,\infty)$ is outside the span of $\left\{ x\mapsto \frac{1}{x-a+ iw} \right\}_{a\in\mathbb R, w>0}$.
Finally, I apologize if my langauge is not precise. I am a physics major, and this question is motivated from physics.