Let $f:[0,2\pi] \rightarrow \mathbb{R}$ be $C^k$ for some $k >0$. Prove that
$|\widehat{f}(n)|n|^k|$
is bounded above by some constant independent of $n$.
To do this, we've been given the Riemann-Lebesgue lemma and Bessel's inequality. What I tried was using integration by parts to express the Fourier coefficients of $f^{(k)}(n)$ in terms of $n^k \widehat{f}(n) n^k$:
$\widehat{f^{(k)}}(n) = \frac{1}{2\pi} \sum_{j=1}^k (in)^{j-1} \left( f^{(k-j)}(2 \pi)-f^{(k-j)}(0) \right)+ \widehat{f}(n)(in)^k.$
So, if this is right, all I need is to bound the sum. Bessel's inequality tells me the LHS is bounded, so the result will follow. Am I on the right track? How can I get Riemann-Lebesgue on it?