Difference between revisions of "Fourier indices of an almost-periodic function"
From Encyclopedia of Mathematics
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| − | The real numbers | + | {{TEX|done}} |
| + | 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 | + | 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=32872
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=32872
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article