Existence of the Fourier Transform

Conditions for the *existence* of the Fourier transform are
complicated to state in general [36], but it is *sufficient*
for
to be *absolutely integrable*, *i.e.*,

(3.6) |

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

(3.7) |

or, . More generally, it suffices to show for [36, 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 loosely called the
*delta function*), discussed in §B.10.

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