Assume $f\in L^1(\mathbb T)$ and $\hat f(n)=O(|n|^{-k})$. Show that $f$ is $m$-times differentiable with $f^{(m)}\in L^2(\mathbb T)$ provided $k-m>\frac{1}{2}$.
Since $\hat f(n)=O(|n|^{-k})$, we know that $\widehat{f^{(m-1)}}(n)=O(|n|^{m-k-1})$ which implies $f^{(m-1)}$ is continous. Now it suffices to show that $f^{(m-1)}$ is differentiable. What we have now is $f^{(m-1)}(t)=\sum_{n=-\infty}^\infty\widehat{f^{(m-1)}}(n)e^{int} $. If we take the derivative formlly, then we have $ f^{(m)}(t)\sim i\sum_{n=-\infty}^\infty n\widehat{f^{(m-1)}}(n)e^{int} $. So we are interested in proving the uniform convergence of $\sum_{n=-\infty}^\infty n\widehat{f^{(m-1)}}(n)e^{int}$ on $\mathbb T$. But I don't have a good idea to use $\widehat{f^{(m-1)}}(n)=O(|n|^{m-k-1})$ with $k-m>\frac 12$. Any suggestion?