Growth of Fourier coefficients of piecewise linear function

Question

Suppose $f$ is a periodic continuous piecewise linear function. What can be said about the growth (or decay, rather) of the Fourier coefficients $\hat f(n)$ as $n\to\infty$, other than the Riemann-Lebesgue Lemma result that these coefficients go to zero?

Answer 1

In the link I shared in the comments section you can see that the following holds:

1. If $f$ is absolutely continuous, $$ \left| \widehat f(n) \right| \le {K \over |n|}. $$
2. If $f$ is a $BV$ function, $$ \left|\widehat f(n)\right|\le {\|f\|_{BV}\over 2\pi|n|}. $$
3. If $f \in C^p$, $$ \left|\widehat{f}(n)\right|\le {\| f^{(p)}\|_1\over |n|^p}. $$
4. If $f\in C^p$ and $f^{(p)}$ has modulus of continuity $\omega$, $$ \left|\widehat{f}(n)\right|\le {\omega(2\pi/n)\over |n|^p}. $$
5. If $f$ is Hölder, (i.e. $f \in C^{0,\alpha}$), $$ \left|\widehat{f}(n)\right|\le {K\over |n|^\alpha}. $$

A good reference to start with is Fourier analysis by Stein and Shakarchi.