Given that $f \in L^2(\mathbb{T})$ and the sequence of Fourier coefficients $(\hat{f_n})\in l^1(\mathbb{Z})$, must $f$ be continuous?

Question

Note that $\mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \}$. This detail wouldn't fit in the title.

This is a previous exam question I am practicing with and I'm at a loss! Any advice on how to think about this problem would be much appreciated.

Answer 1

If $f\in L^2$, then $\sum_{n=-N}^{N}\hat{f}(n)e^{inx}$ converges to $f$ in $L^2$. Assuming that $\hat{f} \in \ell^1$, then the series also converges absolutely and uniformly, which means that $f$ is equal a.e. to a continuous function.