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 $C^{m2}$. Is $f^{(m2)}$ absolutely continuous or differentiable almost everywhere, everywhere, etc.?
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Under these assumptions the function is in the Sobolev space $H^{m1/2\epsilon}$ for any $\epsilon>0$. This implies in particular that $f^{(m1)}$ is in $L^p$ for any $p<\infty$. 

