Dual of the Convolution TheoremThe dual7.18 of the convolution theorem says that multiplication in the time domain is convolution in the frequency domain:
Proof: The steps are the same as in the convolution theorem. This theorem also bears on the use of FFT windows. It implies 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 sidelobes associated with the rectangular window, which is the window one is using implicitly when a data frame is considered time limited and therefore eligible for ``windowing'' (and zero-padding). See Chapter 8 and Book IV  for further discussion.