 Solution

Apply Fourier inversion to the global relation on the real axis, extending $u$ by zero to negative $x$. At an interior point $x>0$ this givesThe finite-time spectral boundary transforms are entire functions of $k$. Define the upper decay domainOrient $\partial D^{+}$ from its left infinite end through $i\alpha$ to its right infinite end, so that $D^{+}$ lies to the left. Its finite point is $i\alpha$, not zero. On the contour, $\mathrm{Re}\omega =0$; the factor $e^{ikx}$ decays because $x>0$.

For the boundary term, write $e^{-\omega t}{G}_j={\int }_0^t{e}^{-\omega (t-s)}{g}_j(s)ds$. Between the real axis and $\partial D^{+}$, $\mathrm{Re}\omega \ge 0$, so this factor has no exponential growth. Contour deformation and Jordan lemma, with the usual cutoffs for oscillatory integrals, therefore giveThis is the requested complex-plane representation with the unknown finite-time spectral boundary transform still present. The decay domain must agree with the sign of the drift in the dispersion relation.