Advanced Fourier Analysis

  • Eugen Skudrzyk


The Fourier transform of a function f(x) exists if f(x) is absolutely integrable, i. e., if
$$ \int\limits_{ - \infty }^\infty {\left| {f\left( x \right)} \right|} dx $$
is convergent and if f(x) satisfies the Dirichlet conditions1:
  1. (a)

    f(x) is single valued

  2. (b)

    f(x) is piecewise continuous

  3. (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→± ∞.


Spectral Density Step Function Spectral Function Time Function Imaginary Axis 
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Copyright information

© Springer-Verlag/Wien 1971

Authors and Affiliations

  • Eugen Skudrzyk
    • 1
  1. 1.Ordnance Research Laboratory and Physics DepartmentThe Pennsylvania State UniversityUniversity ParkUSA

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