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Difference between revisions of "Fourier coefficients of an almost-periodic function"

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The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041020/f0410201.png" /> of the Fourier series (cf. [[Fourier series of an almost-periodic function|Fourier series of an almost-periodic function]]) corresponding to the given almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041020/f0410202.png" />:
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The coefficients $a_n$ of the Fourier series (cf. [[Fourier series of an almost-periodic function|Fourier series of an almost-periodic function]]) corresponding to the given almost-periodic function $f$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041020/f0410203.png" /></td> </tr></table>
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$$f(x)\sim\sum_na_n e^{i\lambda_n}x,$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041020/f0410204.png" /></td> </tr></table>
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$$a_n=M\{f(x)e^{-i\lambda_nx}\}=\lim_{T\to\infty}\frac1T\int\limits_0^Tf(x)e^{-i\lambda_nx}dx.$$
  
The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041020/f0410205.png" /> are completely determined by the theorem on the existence of the mean value
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The coefficients $a_n$ are completely determined by the theorem on the existence of the mean value
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041020/f0410206.png" /></td> </tr></table>
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$$a(\lambda)=M\{f(x)e^{-i\lambda x}\},$$
  
which is non-zero only for the countable set of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041020/f0410207.png" />.
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which is non-zero only for the countable set of values $\lambda=\lambda_n$.
  
  

Revision as of 04:06, 1 August 2014

The coefficients $a_n$ of the Fourier series (cf. Fourier series of an almost-periodic function) corresponding to the given almost-periodic function $f$:

$$f(x)\sim\sum_na_n e^{i\lambda_n}x,$$

where

$$a_n=M\{f(x)e^{-i\lambda_nx}\}=\lim_{T\to\infty}\frac1T\int\limits_0^Tf(x)e^{-i\lambda_nx}dx.$$

The coefficients $a_n$ are completely determined by the theorem on the existence of the mean value

$$a(\lambda)=M\{f(x)e^{-i\lambda x}\},$$

which is non-zero only for the countable set of values $\lambda=\lambda_n$.


Comments

References

[a1] A.S. Besicovitch, "Almost periodic functions" , Cambridge Univ. Press (1932) pp. Chapt. I
[a2] N. Wiener, "The Fourier integral and certain of its applications" , Dover, reprint (1933) pp. Chapt. II
How to Cite This Entry:
Fourier coefficients of an almost-periodic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier_coefficients_of_an_almost-periodic_function&oldid=32626
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article