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**[SOLVED] Fourier coefficients**

## Homework Statement

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

[tex] \sum_{n\in\mathbb{Z}}|c_n(f)| < \infty [/tex]

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

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

[itex] f \in C^{2\pi} [/itex] means f continuous and that [itex] f(-\pi) = f(\pi)[/itex].

Hint: Use Cauchy-Schwartz (CS) inequality.

([itex] e_n = e^{int} [/itex])

## 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 [itex] c_n(f') = inc_n(f) [/itex]. So by using this and splitting the sum up in 2 parts and using that [itex] e_{-n} = \bar{e}_{n} [/itex], I get

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

Using triangle inequality I can get

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

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

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