This question is motivated by one physical problem, which deals with a complex function $F(\omega)$ that fulfills the Kramers–Kronig relations.
Let $F(\omega)=f_1(\omega)+i f_2(\omega)$ be some complex-valued function, where its real part is $f_1(\omega)$. The imaginary part of $F(\omega)$ is given by the Hilbert transform of $f_1(\omega)$, and is denoted as $f_2(\omega)$.
There are a number of physical constraints:
$f_2(\omega)>0$
$f_1(\omega)$ is an odd function.
$f_1(\omega)\stackrel{\omega\rightarrow\infty}{\longrightarrow} -1/\omega^n$, where $n\ge 3$.
A prototype example would be $f_1=-\pi \dot{\delta}(\omega)$ and $f_2=1/\omega^2$.
This is depicted here
In reality, there are always some de-phasing processes present that smoothen the singularities. I am curious about some analytical family of $f_1^{(\alpha)}$ and $f_2^{(\alpha)}$ that would converge in some sense to the given functions preserving the constraints above. That is
$$ f_{1,2}(\omega)=\lim_{\alpha\rightarrow 0} f_{1,2}^{(\alpha)}(\omega). $$
As the simplest possibility, I was thinking about
$$f_1^{(\alpha)}(\omega)=-\frac{2\omega}{(\alpha^2+\omega^2)^2}.$$
However, its Hilbert transform,
$$f_2^{(\alpha)}(\omega)=\frac{\alpha^2-\omega^2}{(\alpha^2+\omega^2)^2},$$
is no longer positive. This situation is illustrated here for $\alpha=1$.
Can someone suggest a family of such function pairs?