# Fourier coefficients of an almost-periodic function

From Encyclopedia of Mathematics

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=34331

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