Abstract. We study the differentiability properties of a function $f$ with absolutely convergent Fourier series and the smoothness property of the $r$th derivative $f^{(r)}$, where $r$ is a given natural number. We give best possible sufficient conditions in terms of the Fourier coefficients of $f$ to ensure that $f^{(r)}$ belongs either to one of the Lipschitz classes $\mathop{\rm Lip}(\alpha )$ and ${\rm lip}(\alpha )$ for some $0<\alpha < 1$, or to one of the Zygmund classes $\mathop{\rm Zyg}(1)$ and ${\rm zyg}(1)$. These sufficient conditions are also necessary in the cases when the Fourier coefficients $c_k$ of $f$ are real numbers such that either $k c_k\ge0$ for all $k$ or $c_k \ge0$ for all $k$.

AMS Subject Classification (1991): 42A32; 26A16, 26A24

Keyword(s): absolutely convergent Fourier series, Lipschitz classes and Zygmund classes of functions, formal differentiation of Fourier series

Received March 4, 2008. (Registered under 6067/2009.)