Let $(a\_n)$, $(b\_n)$ be the Fourier coefficients of a periodic, locally integrable function $f: \mathbb{R} \rightarrow \mathbb{R}$. Assume that $n^m a\_n, n^m b\_n \rightarrow 0$ when $n \rightarrow \infty$. By the Weierstrass test, $f$ is of class $C^{m-2}$. Is $f^{(m-2)}$ absolutely continuous or differentiable almost everywhere, everywhere, etc.?



Function with Fourier coefficient of order $o(n^{-m})$
Under these assumptions the function is in the Sobolev space $H^{m-1/2-\epsilon}$ for any $\epsilon>0$. This implies in particular that $f^{(m-1)}$ is in $L^p$ for any $p<\infty$.


