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...

score 1 · Answer 1 · answered Dec 25 '13 at 10:34

1

Do the computation when $f$ has the form $$\sum_{k=-n}^na_ke^{ikx}, n\in\mathbb N, a_k\in\mathbb R.$$
Use density in $C(\mathbb T)$ of such functions.

answered Dec 25 '13 at 10:34

Davide Giraudo

181,608

are you saying in 1 that every contiunuous periodic function can be described as what you wrote? – jreing Dec 25 '13 at 10:37
Not exactly: a continuous $2\pi$-periodic function can be approximated by such functions, but for example $\sum_n 2^{—n}\sin(2^nx)$ is not of this form. – Davide Giraudo Dec 25 '13 at 10:39
are you saying that the fourier series of f is so close to f (in this case) that we can just interchange them? – jreing Dec 25 '13 at 14:53

score 1 · Accepted Answer · edited Apr 13 '17 at 12:20

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:

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

edited Apr 13 '17 at 12:20

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1

answered Dec 25 '13 at 10:50

Mhenni Benghorbal

48,099

Hi. In the first step, did you mean the sum from -infty to infty? and also, why are you allowed to interchange the sum and the integral? – jreing Dec 25 '13 at 13:35
@user: Yes, it is a typo. – Mhenni Benghorbal Dec 25 '13 at 13:42
and the second question? – jreing Dec 25 '13 at 13:47
and also - I don't really understand where you start from. Do you start from the right side of the original equation? And if so- why are you allowed to replace f(x) with the sum? – jreing Dec 25 '13 at 13:52
@user: because your function admits Fourier series expansion. – Mhenni Benghorbal Dec 25 '13 at 14:19
what do you mean by admits? – jreing Dec 25 '13 at 14:25
It has a Fourier .... – Mhenni Benghorbal Dec 25 '13 at 14:29
Do you mean that for each function that has a fourier series we are allowed to assume they're equal and replace them with each other? – jreing Dec 25 '13 at 14:33
We've studied that for a continuous function, we can assume that the fourier series converges to the function only if it converges uniformly. But here we don't know that it converges uniformly.. (or do we?) – jreing Dec 25 '13 at 14:37

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

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