\[\mathscr{F}\left(H\right)=\frac{1}{2}\mathscr{F}\left(1+\operatorname{sign}%
\right)=\frac{1}{2}\left[\mathscr{F}\left(1\right)+\mathscr{F}\left(%
\operatorname{sign}\right)\right]=\sqrt{\frac{\pi}{2}}\left(\delta+\frac{%
\mathrm{i}}{\pi x}\right),\]