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) corresponding to the given almost-periodic function $f$:

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



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 $\lambda=\lambda_n$.



[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:
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article