Series of squared Fourier coefficients - MathOverflow 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 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 Answer by anton for Series of squared Fourier coefficients 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>