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Assume that a function $f: R\rightarrow R$ is $2 \pi$ -periodic and integrable on $[ -\pi,\pi] $. Let $(a_n)$, $(b_n)$ are its Fourier coefficients and $n^2 a_n, n^2 b_n \rightarrow 0$. Then by Weierstrass test $f$ is continuous. What we can say yet about $f$ (differentiability, continuously differentiability, Lipschitz condition, etc.) ?

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    http://books.google.it/books?id=Z76N_Ab5pp8C&pg=PA66&lpg=PA66&dq=regularity+decay+fourier+coefficients&source=bl&ots=qUQQem3ydX&sig=BtFAPtIjOGYnRnRLFBxet47QMPg&hl=it&sa=X&ei=ag-lT8KlCovesgaT-ZznBA&ved=0CF8Q6AEwBQ#v=onepage&q=regularity%20decay%20fourier%20coefficients&f=false2012-05-05

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