Find the sums $S_1$ and $S_2$ $$ S_1=\sum_{k=1}^\infty\frac{\cos^2 kx}{k^2}\\ S_2=\sum_{k=1}^\infty\frac{\sin^2 kx}{k^2} $$ using the following expansion: $$ I_{[a,b]}(x)=\begin{cases} 1,\ a\leqslant x\leqslant b\\ 0\ \text{otherwise} \end{cases};\ \ \ \ [a,b]\subset[-\pi,\pi]\\ I_{[a,b]}(x)=\frac{b-a}{2\pi}+\frac{1}{\pi}\sum_{n=1}^\infty\left(\frac{2}{n}\sin\frac{n(b-a)}{2}\cos\frac{n(b+a-2x)}{2}\right) $$

I found another solution to this problem (not using Fourier series of Indicator function), and here is my answer: $$ \begin{aligned} &S_1=\frac{\pi^2}{6}+\frac{x^2}{2}-\frac{\pi x}{2}\\ &S_2=-\frac{x^2}{2}+\frac{\pi x}{2} \end{aligned} $$ I did it by finding the sum of $\sum_{k=1}^\infty\frac{\cos 2kx}{k^2}$ as a subtask. But now I have to somehow apply that Indicator function. I've been told that the solution should be straightforward. However, I haven't succeeded in finding it so far.

  • $\begingroup$ What is the significance of k in the first 2 summations? Did you mean to put n instead? $\endgroup$ – Aniket Gupta Feb 21 at 18:33
  • $\begingroup$ Oops, yes, I made a typo. $\endgroup$ – Bonrey Feb 21 at 19:34
  • $\begingroup$ I suggest using the identity $$\sin(\alpha+\beta)+\sin(\alpha-\beta)=2\sin\alpha\sin\beta, $$ to see that the indicator function in question is $$\chi_{[a,b]}(x)=\frac{b-a}{2\pi}+f(x-a)-f(x-b),$$ where $$f(x)=\frac1\pi\sum_{n\ge1}\frac{\sin nx}{n}.$$ $\endgroup$ – clathratus Feb 21 at 22:35
  • $\begingroup$ But the identity is wrong. It should be $\sin(\alpha+\beta)+\sin(\alpha-\beta)=2\sin\alpha\cos\beta$ $\endgroup$ – Bonrey Feb 22 at 7:39

See my answer at Calculate $\sum \limits_{n = 1}^{\infty} \frac{\cos 2n}{n^2}$. You can regard your series as sums of the squares of coefficients of Fourier series. In your expansion, choose $b+a=0$ so that the cosine becomes $\cos nx$, and use Parseval’s theorem to replace the sum of the squares of the coefficients by the square of the integral of the (squared) indicator function. This gives you $S_2$, and then $S_1=\frac{\pi^2}6-S_2$ since $\cos^2+\sin^2=1$. (Here $\frac{b-a}2$ and $n$ in the expansion play the role of $x$ and $k$, respectively, in the series.)

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  • $\begingroup$ Nice!${}{}{}{}{}{}{}$ $\endgroup$ – mjw Feb 25 at 22:16
  • $\begingroup$ I finally solved this problem. Thank you! $\endgroup$ – Bonrey Feb 26 at 11:57

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