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

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The real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041040/f0410401.png" /> in the Fourier series corresponding to the given almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041040/f0410402.png" />:
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The real numbers $\lambda_n$ in the Fourier series 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/f041040/f0410403.png" /></td> </tr></table>
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$$f(x)\sim\sum_na_ne^{i\lambda_nx},$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041040/f0410404.png" /> are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041040/f0410405.png" /> (cf. [[Fourier coefficients of an almost-periodic function|Fourier coefficients of an almost-periodic function]]; [[Fourier series of an almost-periodic function|Fourier series of an almost-periodic function]]). The set of Fourier indices of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041040/f0410406.png" /> is called its spectrum. In contrast to the periodic case, the spectrum of an almost-periodic function can have finite limit points and can even be everywhere dense. Therefore, the behaviour of the Fourier series of an almost-periodic function depends in an essential way on the arithmetic structure of its spectrum.
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where the $a_n$ are the Fourier coefficients of $f$ (cf. [[Fourier coefficients of an almost-periodic function|Fourier coefficients of an almost-periodic function]]; [[Fourier series of an almost-periodic function|Fourier series of an almost-periodic function]]). The set of Fourier indices of a function $f$ is called its spectrum. In contrast to the periodic case, the spectrum of an almost-periodic function can have finite limit points and can even be everywhere dense. Therefore, the behaviour of the Fourier series of an almost-periodic function depends in an essential way on the arithmetic structure of its spectrum.

Latest revision as of 16:45, 12 August 2014

The real numbers $\lambda_n$ in the Fourier series corresponding to the given almost-periodic function $f$:

$$f(x)\sim\sum_na_ne^{i\lambda_nx},$$

where the $a_n$ are the Fourier coefficients of $f$ (cf. Fourier coefficients of an almost-periodic function; Fourier series of an almost-periodic function). The set of Fourier indices of a function $f$ is called its spectrum. In contrast to the periodic case, the spectrum of an almost-periodic function can have finite limit points and can even be everywhere dense. Therefore, the behaviour of the Fourier series of an almost-periodic function depends in an essential way on the arithmetic structure of its spectrum.

How to Cite This Entry:
Fourier indices of an almost-periodic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier_indices_of_an_almost-periodic_function&oldid=13037
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article