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### Existence of the Fourier Transform

Conditions for the *existence* of the Fourier transform are
complicated to state in general [35], but it is *sufficient*
for to be *absolutely integrable*, *i.e.*,

This requirement can be stated as

, meaning that

belongs to the set of all

signals having a finite

norm
(

). It is similarly sufficient for

to be

*square integrable*,

*i.e.*,

or,

. More generally, it suffices to show

for

[

35, p. 47].

There is never a question of existence, of course, for Fourier
transforms of real-world signals encountered in practice. However,
*idealized* signals, such as sinusoids that go on forever in
time, do pose normalization difficulties. In practical engineering
analysis, these difficulties are resolved using Dirac's ``generalized
functions'' such as the *impulse* (also called the
*delta function*), discussed in §B.1.10.

**Previous:** Fourier Transform (FT) and Inverse**Next:** Fourier Theorems for the DTFT

**About the Author: ** Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at

Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See

http://ccrma.stanford.edu/~jos/ for details.