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This question is an extension of Fourier series simplification

I know wish to show $$(\frac{1}{\pi})\int_{-\pi}^{\pi}f^2(x) dx = a_{0}^{2}/2 + \sum_{n=1}^{\infty} (a_{n}^{2} + b_{n}^{2})$$ But, we no longer have the help of uniform convergence. In detail, we have $f:[-\pi,\pi] \to \mathbb{R}$ a piecewise smooth function.

We cannot approach the problem as we did in the previous exercise since we relied on uniform convergence of the Fourier series. How then should we approach the problem? I had found https://www.math.ubc.ca/~yhkim/yhkim-home/teaching/Math257/M257-316Notes_Lecture16.pdf but this assumed uniform convergence.

It is quite simple with uniform convergence, but again, we do not have that luxury.

Clearly we have $$\langle f(x),f(x)\rangle = \int_{-\pi}^{\pi}f(x)^2dx$$ but from here?

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  • $\begingroup$ What does "smooth" mean? $\endgroup$ – snar Mar 12 '15 at 23:04
  • $\begingroup$ @snarski mathworld.wolfram.com/SmoothFunction.html $\endgroup$ – Fourier.Me2 Mar 12 '15 at 23:06
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    $\begingroup$ It's Parceval identity (for Fourier series). Can you use it, or do you need to prove it? $\endgroup$ – Tryss Mar 12 '15 at 23:07
  • $\begingroup$ What I wrote about is Parceval's Identity (I thought), so I need to prove it. $\endgroup$ – Fourier.Me2 Mar 12 '15 at 23:13
  • $\begingroup$ This has been proved before on this site (try to search for "Parseval proof"). See for example this or this or this. You might have to search a bit to find a question with the exact conditions you need though. $\endgroup$ – Winther Mar 12 '15 at 23:20

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