Let $f: [0,2\pi]\to \mathbb C$ be an continuously differentiable function with $f^{(j)}(0)=f^{(j)}(2\pi)$ for $j=0,...,m-2$ and let the fourier transformation $\hat f(k)= \int^{2\pi}_{0}f(x)e^{ikx}\,dx$, $(k\in \mathbb R)$
Show that a $C>0$ exists such that $\forall k \in \mathbb N$ such that
$$|\hat f(k)|\leq \frac{C}{|k|^{m}}$$
Normally, I would attempt to explain my thought process but given that we have just started with fourier transformations I really am in the dark. My only suggestion would be to prove via induction?
Hint: By definition we have $$\widehat{f}(k)= \int_0^{2\pi} f(x) \mathrm{e}^{\mathrm{i} kx} \mathrm{d} x.$$ Now apply integration by parts. What is the antiderivate of $\mathrm{e}^{\mathrm{i} kx}$?
Added Solution: As in the comment indicated, we can apply $(m-1)$-times partial integration to obtain $$\widehat{f}(k) = \big( - \frac{1}{\mathrm{i} k} \big)^{m-1} \int_0^{2\pi} f^{m-1}(x) \mathrm{e}^{\mathrm{i}x k} \, \mathrm{d} x.$$ Note that this formula may fail to be true for the next step, because $f^{(m-1)}(0) \neq f^{(m-1)}(2\pi)$, i.e. the boundary values may not vanish. However, what happens if we again integrate by parts?