Given the Graph of a Fourier Series $\sum c_k e^{2\pi ikx}$ Find the Graphs of $\sum c_{3k} e^{2\pi ikx}$ and $\sum (c_k)^2 e^{2\pi ikx}$ Define a 1-periodic function on $\mathbb{R}$ by:

$f(x) :=$ 
$\left\{\begin{matrix}
1 &amp; if &amp; 0&lt;x&lt;\frac{1}{10}\\ 
0 &amp; if &amp; \frac{1}{10}&lt;x&lt;1
\end{matrix}\right.$

with Fourier series $f(x) = \sum_{-\infty}^{\infty}c_k e^{2\pi ikx}$.

I'm trying to find the graphs of the following two Fourier series:

$\sum_{-\infty}^{\infty}(c_k)^2 e^{2\pi ikx}$ and $\sum_{-\infty}^{\infty}c_{3k} e^{2\pi ikx}$.

Basically my strategy with the others has been to find a way to get these Fourier series into the form $a\sum_{-\infty}^{\infty}c_k e^{2\pi ik(bx)}$ for $a,b\in\mathbb{C}$.  But I haven't been able to do it with these, hopefully someone can help, thanks.
This doesn't require any complicated computations! Look up some tables of identities about the Fourier Series. Which operation would you have to apply to $f$ so that its Fourier series would become $\sum_{-\infty}^{\infty}(c_k)^2 e^{2\pi ikx}$?

For example, we know that derivating in the time domain is equivalent to multiplying the Fourier series coefficients by $i k$ ; so we know that the graph of the Fourier series $\sum_{-\infty}^{\infty} ik (c_k) e^{2\pi ikx}$ is the graph of the derivative of $f$.