Let $f$ be 1-periodic, $m$ times continuously differentiable function on $\mathbb{R}$
1- Show that: $$ \sup_{n \in \mathbb{Z}} |n|^m |\hat{f}(n)| < \infty$$ where $\hat{f}(n)$ is the Fourier transform of $f$.
2- Show that
$$\sum_{n \in \mathbb{Z}} |\hat{f}(n)|< \infty$$
Could someone please suggest how to approach proving these statements?
Say, $m=1$, integration by parts gives \begin{align*} \widehat{f}(n)=\int_{\mathbb{T}}f(x)e^{-2\pi inx}dx=\dfrac{1}{2\pi in}\int_{\mathbb{T}}f'(x)e^{-2\pi inx}dx, \end{align*} so $|n\widehat{f}(n)|\leq\|f'\|_{L^{\infty}(\mathbb{T})}$.
For higher $m$, perform several times integration by parts trick.
If $m\geq 2$, then \begin{align*} \sum_{n}|\widehat{f}(n)|\leq\left(\sup_{n}|n|^{m}|\widehat{f}(n)|\right)\cdot\sum_{n}\dfrac{1}{|n|^{m}}<\infty. \end{align*}
If $m=1$, then \begin{align*} \sum_{n}|\widehat{f}(n)|=\dfrac{1}{2\pi}\sum_{n}\dfrac{1}{|n|}\cdot|\widehat{f'}(n)|\leq\dfrac{1}{2\pi}\left(\sum_{n}\dfrac{1}{|n|^{2}}\right)^{1/2}\left(\sum_{n}|\widehat{f'}(n)|^{2}\right)^{1/2}. \end{align*} But \begin{align*} \left(\sum_{n}|\widehat{f'}(n)|^{2}\right)^{1/2}=\|f'\|_{L^{2}(\mathbb{T})}<\infty. \end{align*}