About decay of Fourier coefficients of "almost" $C^2$ functions.

Question

If a function is $C^2$ outside a measure zero set in its domain then do its Fourier coeffients still decay as $O(\frac{1}{k^2})$ ? Assume that the function is continuous but not differentiable on that measure-zero set.

Answer 1

No. For example, consider $f(x)=\sqrt{|x|}$ on the interval $[-\pi, \pi]$. The cosine coefficients are $$ A_n = \frac{2}{\pi} \int_0^{\pi} \sqrt{x}\cos nx\,dx = -\sqrt{\frac{2}{\pi}}\frac{\operatorname{Si}(\sqrt{2n})}{n^{3/2}} $$ according to Wolfram Alpha. Since the sine integral has a nonzero limit at infinity, the size of $A_n$ is $\sim 1/n^{3/2}$.

I also expect that $f(x)=|x|^p$ has cosine coefficients $\sim 1/n^{1+p}$ for $0<p<1$, but do not have a proof.