Are time- and frequency-domain nonlinear spectroscopies related by a Fourier transform?

It is commonly believed that nonlinear-spectroscopic techniques involving nearly monochromatic beams, in which one or more beam frequencies are varied, are in some sense the "Fourier transforms" of techniques involving ultrashort pulses, in which one or more pulse delays are varied. We find that, while the above Fourier transform does hold for classical problems, it fails badly in semi-classical nonlinear spectroscopy