= Solution
Apply the <Fourier transform of a derivative> twice to the <damped harmonic oscillator> equation. Under the corresponding convergence assumptions,
$$
(-\omega^2+2i\kappa\omega+\Omega^2)\widetilde y(\omega)=\widetilde f(\omega).
$$
Thus
$$
\boxed{\widetilde y(\omega)=\widetilde h(\omega)\widetilde f(\omega),\qquad
\widetilde h(\omega)=\frac1{\Omega^2-\omega^2+2i\kappa\omega}}.
$$
Since $\kappa,\Omega>0$, this denominator has no real zero: its imaginary part could vanish only at $\omega=0$, where its real part is positive. The transform therefore defines a regular response on the real frequency axis.
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