Solution: Prove Sum of Fourier Coefficients of a C2π Function is Finite

[P3X-018]

[SOLVED] Fourier coefficients

Homework Statement

For $f \in C^{2\pi}\cap C^1[-\pi,\pi]$, I have to show that

$$\sum_{n\in\mathbb{Z}}|c_n(f)| < \infty$$

where c_n(f) is the Fourier coefficient of f;

$$c_n(f) = (f, e_n) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(t)e^{-int}\,dt$$

$f \in C^{2\pi}$ means f continuous and that $f(-\pi) = f(\pi)$.
Hint: Use Cauchy-Schwartz (CS) inequality.

($e_n = e^{int}$)

The Attempt at a Solution

I just can't seem to use CS in a useful way, I keep running into dead ends:

It is easily shown that $c_n(f') = inc_n(f)$. So by using this and splitting the sum up in 2 parts and using that $e_{-n} = \bar{e}_{n}$, I get

$$\sum_{n\in\mathbb{Z}}|c_n(f)| = c_0(f) + \sum_{1}^{\infty}\frac{|(f',e_n)| - |(f',\bar{e}_n)|}{n}$$

Using triangle inequality I can get

$$|(f',e_n)| - |(f',\bar{e}_n)| \leq |(f', e_n-\bar{e}_n)| =2|(f', \sin(nt))|$$.

Even here CS won't be useful. Is there a different an easier approach?

 

[morphism]

What theorems do you know? Try looking at Parseval and its proof.

 

[P3X-018]

I kept looking at the wrong inner product to use CS on, that is that inner product of $L_2$, but I should look at the inner product of $\ell_2$ space. So by using that (replacing the minus sign with + in the afore mentioned equation for c_n(f))

$$\sum_{n\in\mathbb{Z}}|c_n(f)| \leq c_0(f) + \sum_{1}^{\infty}\frac{|(f',e_n)| + |(f',\bar{e}_n)|}{n}$$

and then using that for $|(f',\bar{e}_n)|/n$ (and the same for $|(f',e_n)|/n$) we get the summation by CS on l_2 innerproduct to be

$$\sum_{1}^{\infty}\frac{1}{n}|(f',e_{-n})| \leq \left(\sum_{1}^{\infty}\frac{1}{n}\right)^{1/2}\left(\sum_{1}^{\infty} |(f',e_{-n})|^2 \right)^{1/2}$$

Were the $C^2 = \sum \frac{1}{n^2}$ is convergent. Before I can use Parseval's theorem, I need to extend the last sum to go from -infinity to +infinity so (1 more inequality), so I get (f', e_n) instead of (f',e_{-n}),

$$\sum_{1}^{\infty}\frac{1}{n}|(f',e_{-n})| \leq C\left(\sum_{-\infty}^{\infty} |(f',e_{n})|^2 \right)^{1/2} = C\|f'\| < \infty$$

Is this correct, or have I done some 'illegal' steps? If I haven't then the problem is solved.