Let $(a_n),(b_n)$ be Fourier coefficients of periodic locally integrable function $f: R \rightarrow \R$. Assume that $n^ma_n, n^m b_n \rightarrow 0$ when $n \rightarrow \infty$. By Weierstrass test $f $ is of class $C^{m-2}$. Is maybe $f^{m-2}$ absolutely continuous or differentiable almost everywhere, everywhere, etc.?
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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 $p<\infty$. |
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