Proof: The steps are the same as in the Convolution Theorem.
This theorem also bears on the use of FFT windows. It says that windowing in the time domain corresponds to smoothing in the frequency domain. That is, the spectrum of is simply filtered by , or, . This smoothing reduces sidelobesassociated with the rectangular window (which is the window one gets implicitly when no window is explicitly used). FFT windows are covered inMusic 420.