Why does Fourier transformation of Doppler shift give thermal radiation?

[exponent137]

I read article
http://arxiv.org/abs/quant-ph/0401170
It obtains how Dopler frequency change exponentialy with time for accelerated observer.
Then it does Fourier's transformation of this waving and it gets thermical radiation.
But, where in quantum mechanics it is supported that we make Fourier's transformation that change one monotonic waving in accidenal radiation?

 

[exponent137]

Or asking different: Fourier transformation is basic in quantum mechanics (QM). It describe also principle of uncertainty.
But, how this implies on that classical waving at different times gives accidental thermical photons at the same time.