Causality versus stability of a Green function

With inverse Fourier transform $G(t)=\int G(\omega )e^{-i\omega t}d\omega /(2\pi )$, a tempered retarded response vanishes for $t<0$ and is analytic in the upper frequency half-plane with suitable large-frequency growth bounds. Poles strictly below the real axis describe decaying modes; upper-half-plane poles obstruct a bounded retarded real-axis Fourier response. However causality alone does not imply stability: a retarded exponentially growing Green function can exist without an ordinary Fourier transform. Its Laplace transform inversion contour must pass above the growth poles in the frequency plane. Moving the contour to the real axis changes the boundary condition and can introduce a pre-response.