Abstract
The Fourier transform of a function f(x) exists if f(x) is absolutely integrable, i. e., if
is convergent and if f(x) satisfies the Dirichlet conditions1:
-
(a)
f(x) is single valued
-
(b)
f(x) is piecewise continuous
-
(c)
f(x) has a finite number of maxima and minima.
To be absolutely integrable, |f(x)| must decrease more rapidly than 1/x as x approaches ± ∞, and f(x) must not have poles within the interval of integration. If these conditions are not satisfied, it is senseless to talk about a Fourier spectrum; and all the results that are obtained by formal computations are meaningless. For instance, a sinusoidal vibration that is switched on at a certain instant but goes on forever has no Fourier spectrum, nor does a step function have a Fourier spectrum because these functions do not satisfy the convergence criterion for t→± ∞.
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Skudrzyk, E. (1971). Advanced Fourier Analysis. In: The Foundations of Acoustics. Springer, Vienna. https://doi.org/10.1007/978-3-7091-8255-0_6
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