Rate of decay of Fourier coefficients vs smoothness

[Notation. $a_n \lesssim b_n$ means: there exists some positive constant $C$ s.t. $a_n \le C b_n$.]

A rough'n'ready argument would be:

let

$$c_n=\frac{1}{2 \pi} \int_{-\pi}^{\pi}f(y)e^{-i n y}\, dy$$

and write

$$f(x)=\sum_{n\in \mathbb{Z}} c_n e^{i n x}\quad (1)$$

The decay condition on $c_n$ implies uniform convergence of this series:

$$\lvert c_n e^{i n x} \rvert \le \lvert n\rvert^{-k}\lvert n^k c_n \rvert \lesssim \lvert n\rvert^{-k}$$

and $\sum_{n \in \mathbb{Z}} \lvert n\rvert^{-k}$ is a convergent numerical series. Now differentiate (1) termwise: you get

$$\sum_{n \in \mathbb{Z}}i n c_n e^{i n x}$$

which is again a uniformly convergent series:

$$\lvert i n c_n \rvert \lesssim \lvert n\rvert^{1-k}.$$

So (1) is a uniformly convergent series whose term-by-term derivative is uniformly convergent. This implies that $f$ is differentiable and

$$f'(x)=\sum_{n \in \mathbb{Z}}i n c_n e^{i n x}$$

so that, again by uniform convergence, $f'$ is also continuous.