Showing $\lim_{n \to\infty} \hat{g}(n) \to 0$

Question

Let $g \in L^{1}(\mathbb{T})$. Show that $|\hat{g}(n)| \leqslant ||g||_{1}$ for all $n \in\mathbb{Z}$ and $\displaystyle \lim_{|n|\to\infty} \hat{g}(n)\to 0$.

Answer 1

The fact that $|\widehat{f}(n)|\leqslant \lVert f\rVert_{L^1}$ can be obtained by bounding the integrand of the integral which defines $\widehat{f}(n)$.

We can assume that $g$ is a simple function, and by linearity that $g$ is the characteristic function of a Borel set. As Lebesgue measure is finite on $\mathcal T$, we an approximate it by $\chi_F$, where $F$ is closed. So we can express $g$ as the limit in $L^1$ of continuous functions. There is a theorem which allows us to do it when $g$ is a polynomial.