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Wiener’s Generalized Harmonic Analysis

  • Eugen Skudrzyk

Abstract

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
$$ \bar S\left( {x,T} \right) = \int\limits_{ - T}^T {u\left( x \right){e^{ - xx}}} dx $$
(1)
diverges if T→∞. But the stochastic integral
$$ \left[ {\bar{Z}\left( x \right)} \right]_{{x'}}^{{x}} = {\kern 1pt} \,\mathop{{\lim }}\limits_{{T \to \infty }} \smallint _{{x'}}^{{x}}\bar{S}\left( {x,T} \right)dx/2\pi $$
(2)
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:
$$ \left[ {\bar{Z}\left( x \right)} \right]_{{x'}}^{{x}} = \int\limits_{{ - \infty }}^{\infty } {u\left( x \right)} {\text{ }}\left( {\frac{{{{e}^{{ - jxx - }}}\,{{e}^{{ - jx'x}}}}}{{ - jx}}} \right)\,dx $$
(3)
so that
$$ d\bar Z\left( x \right) = \left[ {\bar Z\left( x \right)} \right]_x^{x + dx} = \int\limits_{ - \infty }^\infty {u\left( x \right)} {e^{ - jxx}}\frac{{\left( {{e^{ - jdxx}} - 1} \right)}}{{ - jx}}dx $$
(4)
If \( \bar S\left( {x,T = \infty } \right) \) existed, we would have
$$ d\bar Z\left( x \right) = \bar S\left( x \right)dx/2\pi $$
(5)
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.

Keywords

Real Axis Statistical Function Fourier Coefficient Inverse Relation Correlation Method 
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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References

  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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