Problem about Fourier series and $L^p$ spaces

Question

Need some help with this problems:

1. Is there $f \in C(\mathbb{T})$ such that $\hat{f}(k) = \dfrac{1}{|k|^{1/2}}$, if $k \neq 0$?

2. Suppose the $f_n \in L^1(\mathbb{T})$, $n = 1,2,...$ and $\| f_n - f \|_{L^1(\mathbb{T})} \xrightarrow{n \to \infty} 0$. Prove that $\hat{f_n} \xrightarrow{u} \hat{f}$ in $\mathbb{Z}$, where $\mathbb{T}$ is the torus.

Notations: $C(\mathbb{T}) = \{ f:\mathbb{T} \to \mathbb{C} : f \hspace{1mm} \mbox{is continuous} \}$;

$\hat{f_n} \xrightarrow{u} \hat{f}$: $\hat{f_n} \to \hat{f}$ uniformly;

$\hat{f}(k)$: coefficients of the Fourier series of $f$, $k \in \mathbb{Z}$

In (1), I don't know how to begin.

In (2), I have been thinking of this: for $\varepsilon > 0$, exists $N \in \mathbb{N}$ such that

$$ \int\,|f_n-f| < \varepsilon $$

for $n > N$, but I don't know how to continue.

Thanks a lot for your help!

Answer 1

Hints: 1. $\sum_k |\hat f (k)|^2 = \text {____?} $

2. $|\hat f (k)| \le (1/2\pi)\int_{-\pi}^{\pi} |f|.$