Difference between revisions of "Fourier coefficients of an almost-periodic function"
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| − | The coefficients | + | {{TEX|done}} |
| + | 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 | 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 | + | 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 | + | which is non-zero only for the countable set of values $\lambda=\lambda_n$. |
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====References==== | ====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> | + | <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]] | ||
Latest revision as of 21:53, 7 November 2014
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=16385
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=16385
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article