Wiener’s Generalized Harmonic Analysis

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.

N. Wiener, Generalized Harmonic Analysis, Acta Math. 55 (1930), 117–258.