Conditions for the
existence of the Fourier transform are
complicated to state in general [
12], 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

[
12, 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) [
38].
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