Where $ \mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}.$
My initial thoughts are that if I can show that $ \frac{1}{2\pi}(f*F_{N}) \rightarrow f$ uniformly then I can use a previous result to show that $f \in C^{0}(\mathbb{T})$.
However showing this seems quite difficult. Is this the correct way to proceed or is there a simpler method?
As $\sum_{n\in\mathbb Z}|\hat f(n)|<\infty$, we know that the Fourier series $\sum_{n\in\mathbb Z}\hat f(n)e^{int}, t\in\mathbb T$ converges absolutely and uniformly on $\mathbb T$. However, the partial sum $\sum_{|n|\le N}\hat f(n)e^{int}$ of the Fourier series is continouus on $\mathbb T$ for all $N\in\mathbb Z_{\ge 0}$. By the Uniform limit theorem, we conclude that the limit function $f\in C^0(\mathbb T)$.