Wiener’s Generalized Harmonic Analysis

  • Eugen Skudrzyk


The Fourier coefficients of statistical functions are usually rapidly oscillating functions. Integrals over the Fourier coefficients, therefore, stay finite. For instance, if u is a statistical function, its Fourier coefficient
$$ \bar S\left( {x,T} \right) = \int\limits_{ - T}^T {u\left( x \right){e^{ - xx}}} dx $$
diverges if T→∞. But the stochastic integral
$$ \left[ {\bar{Z}\left( x \right)} \right]_{{x'}}^{{x}} = {\kern 1pt} \,\mathop{{\lim }}\limits_{{T \to \infty }} \smallint _{{x'}}^{{x}}\bar{S}\left( {x,T} \right)dx/2\pi $$
exists1 as T→∞, \( \bar S\left( {x,T} \right) \) may become infinite for certain values of ϰ, but the ares \( \bar S\left( x \right)dx \) under the infinite peak stays finite. We thus have:
$$ \left[ {\bar{Z}\left( x \right)} \right]_{{x'}}^{{x}} = \int\limits_{{ - \infty }}^{\infty } {u\left( x \right)} {\text{ }}\left( {\frac{{{{e}^{{ - jxx - }}}\,{{e}^{{ - jx'x}}}}}{{ - jx}}} \right)\,dx $$
so that
$$ d\bar Z\left( x \right) = \left[ {\bar Z\left( x \right)} \right]_x^{x + dx} = \int\limits_{ - \infty }^\infty {u\left( x \right)} {e^{ - jxx}}\frac{{\left( {{e^{ - jdxx}} - 1} \right)}}{{ - jx}}dx $$
If \( \bar S\left( {x,T = \infty } \right) \) existed, we would have
$$ d\bar Z\left( x \right) = \bar S\left( x \right)dx/2\pi $$
However, \( \bar S\left( x \right) \) does not exist for a statistical function that does not vanish as T→∞, and the right-hand side is senseless.


Real Axis Statistical Function Fourier Coefficient Inverse Relation Correlation Method 
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  1. Batchelor, G. K.: The theory of homogeneous turbulence. Cambridge University Press. 1953.Google Scholar
  2. Wiener, N.: Generalized harmonic analysis. Acta Math. 55 (1930) 117–258; The Fourier integral and certain of its applications. New York, N. Y.: Dover Publication. 1933.Google Scholar

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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