Let $f(x):\mathbb{R}\to\mathbb{R}$ be a 1-periodic function, which is in $L^1((0,1))$. I have seen that if $f$ is $C^k$ then the Fourier coefficients can be bounded by $C/n^k$, for some constant $C$.
But if we suppose, as in the title that $f$ is $C^1$ and not $C^2$, can we bound $\hat{f}(n)$ from below in an interesting way?
For example, perhaps we can find two constants $c, C$ such that
$$c/n\leq \hat{f}(n)\leq C/n $$
Or if this is false, a counter example would be finding $C^1$ and not $C^2$ function(s), such that the Fourier coefficient decrease arbitrarily fast.
Thanks in advance