View source for Fourier coefficients of an almost-periodic function

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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 $f$:

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

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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.S. Besicovitch,   "Almost periodic functions" , Cambridge Univ. Press  (1932)  pp. Chapt. I</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Wiener,   "The Fourier integral and certain of its applications" , Dover, reprint  (1933)  pp. Chapt. II</TD></TR>
</table>

[[Category:Harmonic analysis on Euclidean spaces]]