Any ``reasonable'' *probability density function* (PDF) (§C.1.3)
has a Fourier transform that looks like
near its tip. Iterating
convolutions then corresponds to
, which becomes
[#!Abel06!#]

(D.27) |

for large , by the definition of [#!MDFT!#]. This proves that the th power of approaches the Gaussian function defined in §D.1 for large .

Since the inverse Fourier transform of a Gaussian is another Gaussian (§D.8), we can define a time-domain function as being ``sufficiently regular'' when its Fourier transform approaches in a sufficiently small neighborhood of . That is, the Fourier transform simply needs a ``sufficiently smooth peak'' at that can be expanded into a convergent Taylor series. This obviously holds for the DTFT of any discrete-time window function (the subject of Chapter 3), because the window transform is a finite sum of continuous cosines of the form in the zero-phase case, and complex exponentials in the causal case, each of which is differentiable any number of times in .

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University