most recent 30 from http://mathoverflow.net 2013-05-23T05:02:40Z http://mathoverflow.net/feeds/question/47095 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47095/series-of-squared-fourier-coefficients Mermoz 2010-11-23T13:42:39Z 2010-11-23T18:11:55Z

<p>Hi, if the Fourier series development of $g(t)$ (periodic, $C^\infty$) is </p> <p>$$ g(t)=\sum_{-\infty}^{+\infty}a_n e^{in\omega t} $$</p> <p>does the series</p> <p>$$ \sum_{-\infty}^{+\infty}\frac{a_n^2}{n^2}? $$ converges toward something known like average $g^2$ or something like that?</p>

http://mathoverflow.net/questions/47095/series-of-squared-fourier-coefficients/47126#47126 anton 2010-11-23T18:11:55Z 2010-11-23T18:11:55Z

<p>Assume $a_0=0$, which easily can be arranged by adding a constant to $g$. Then the function $$ h(t)=\frac1{i\omega } \sum_{n}\frac{a_n}ne^{in\omega t} $$ is the primitive of $g$. Let $h^* (x)=\overline{h(-x)}$, then the sum you asked for equals the inner product $$ \langle h,h^*\rangle. $$</p>