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Advanced Fourier Analysis

  • Eugen Skudrzyk

Abstract

The Fourier transform of a function f(x) exists if f(x) is absolutely integrable, i. e., if
$$ \int\limits_{ - \infty }^\infty {\left| {f\left( x \right)} \right|} dx $$
(1)
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→± ∞.

Keywords

Spectral Density Step Function Spectral Function Time Function Imaginary Axis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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