Consider some non-negative function $f(x)$ which is integrable over the real line $0 \leq \int_{-\infty}^\infty f(x) dx < \infty$.
Are there simple rules for identifying properties of the functions Hilbert transform $\hat{f}(x): = H[f](x)$. In particular I am interested in the the positions and residues of poles in $\hat{f}$, the positions and values of cusps, discontinuities and extrema in $\hat{f}$, and the asymptotic behaviours of $\hat{f}$.
I can see there are many individually tabulated exact results in Frederick King's "Hilbert Transforms" Vols I and II. From these I can deduce that $\delta$-function components of $f$ lead to simple poles in $\hat{f}$. In addition it seems that discontinuities in $f$ yield logarithmic divergences in $\hat{f}$, and discontinuities in $\frac{d f}{dx}$ yield divergences in $\frac{d\hat{f}}{dx}$ etc, though I am less sure of the generality of these results.
Is such a table of results for more general functions known?