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Ok so I want to prove the above expression, I substituted the complex fourier series for f and using the fact f may be complex-valued, carried on by representing $|f(x)|^2$ as $f(x)f(x)^\ast$ where the ast represents complex conjugation. The problem is when now I substitute the complex fourier series I get three summations, one is the one above and two more by matching the powers of the exponentials. The two extra summations are below:


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    $\begingroup$ it's easier to reply to this if you write down what you got. $\endgroup$ – Thomas Apr 6 '14 at 11:29
  • $\begingroup$ @Thomas sorry edited $\endgroup$ – Raul Apr 6 '14 at 11:36
  • $\begingroup$ I can't understand how come, or in facty where from, you get those "extra summations"...? $\endgroup$ – DonAntonio Apr 6 '14 at 12:05

Assuming the usual (inner product $\;\langle,\rangle\;$, orthonormal basis $\;\{u_i\}\;$ and etc.):

$$\frac1{2L}\int\limits_0^{2L}|f(x)|^2dx=\langle f(x)\,,\,f(x)\rangle=\sum_{k=-\infty}^\infty\sum_{m=-\infty}^\infty c_k\overline{c_m}\langle u_k\,,\,u_m\rangle=\sum_{k=-\infty}^\infty|c_k|^2$$

Of course, we usually have that the $\;u_i$'s are the complex exponentials and etc., and the above follows from the basic orthonormality relations

$$\langle u_k\,,\,u_m\rangle=\delta_{km}:=\begin{cases}1&,\;\;\;k=m\\{}\\0&,\;\;\;k\neq m\end{cases}$$


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