show that $f\in C^{l}(\mathbb{T})$. given $\sum_n |\hat f(n)||n|^{l} <\infty$

Question

Show that if $f \in\ C(\mathbb{T})$ and further $\sum_n |\hat f(n)||n|^{l} <\infty$ for some positive integer $l$.

show that $f\in C^{l}(\mathbb{T})$.

Hint is given differentiation under integral .

but i could not get the hint. please help. thanks in advanced

Answer 1

Here is how I understand the hint:

The Fourier series of $f$ is $$f(x) = \sum_n \hat f(n) e^{-2\pi i n x}$$

Now if you use derivative on the right hand side you have $\sum_n (-2\pi i n )\hat f(n) e^{2\pi i n x}$ which is, by assumption converge to $f'(x)$ (This is not that trivial, but it's true).

Now do the same with the $l$'th derivative.