Riemann Lebesgue lemma

Proposition

If $f\in C^2(\mathbb{T})$ (meaning that $f$ is twice differentiable at every point of $\mathbb{T}$ and $f^{\prime \prime }\in C(\mathbb{T}))$, then $S_{N}f$ converges uniformly to $f$ on $\mathbb{T}$. In fact, for $f\in C^2(\mathbb{T})$, there exists some finite constant $C$ such that $|\hat{f}(n)|\le C{n}^{-2}$ for all $n$ (e.g. $C=\Vert f^{\prime \prime }{\Vert }_{C(\mathbb{T})}$ will do).

Proof

Riemann-Lebesgue lemma

If $f\in L^1(\mathbb{T})$, then $\hat{f}(n)\to 0$ as $n\to \infty .$