Prove Parseval Identity for $f \in C(\Bbb T) 2\pi$ periodic continuous functions

Question

Question: Prove Parseval Identity for $f \in C(\Bbb T) $ $2\pi$ periodic continuous functions

$$ \frac{1}{2 \pi} \int_{-\pi}^\pi |f(x)|^2 dx =\sum_{n=-\infty}^\infty |\hat f(n)|^2 $$

Thoughts:

We are somehow supposed to use the fact that exists $g\in C^1(\Bbb T)$ s.t. $f$ and $g$ are very close. Tried defining a function $h=f-g$, and then the identity is true for the $g$ part but not for the $f$ part...

Answer 1

Here is how,

$$ \int_{-\pi}^{\pi}\left|\sum_{n=-\infty}^{\infty}\hat{f}(n)e^{inx}\right|^{2}dx = \sum_{n=-\infty}^{\infty}\hat{f}(n)\sum_{m=-\infty}^{\infty}\overline {\hat{f}(m)} \int_{-\pi}^{\pi} e^{i(n-m)x}dx .$$

Now, see here for details and how to finish the problem.

Note:

$$ \left|\sum_{n=-\infty}^{\infty}\hat{f}(n)e^{inx}\right|^{2}= \sum_{n=-\infty}^{\infty}\hat{f}(n)e^{ix}\overline {\sum_{m=-\infty}^{\infty} {\hat{f}(m)}e^{ix}}. $$

Answer 2

1. Do the computation when $f$ has the form $$\sum_{k=-n}^na_ke^{ikx}, n\in\mathbb N, a_k\in\mathbb R.$$

2. Use density in $C(\mathbb T)$ of such functions.