Existence of the Fourier Transform
Conditions for the existence of the Fourier transform are complicated to state in general , 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 [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.
Linearity of the DTFT