Absolutely convergent Fourier series and generalized Lipschitz classes of functions
Volume 113 / 2008
Abstract
We investigate the order of magnitude of the modulus of continuity of a function $f$ with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that $f$ belong to one of the generalized Lipschitz classes $\mathop{\rm Lip}(\alpha, L)$ and $\mathop{\rm Lip}(\alpha, 1/L)$, where $0\le \alpha\le 1$ and $L=L(x)$ is a positive, nondecreasing, slowly varying function such that $L(x)\to \infty$ as $x\to \infty$. For example, a $2\pi$-periodic function $f$ is said to belong to the class $\mathop{\rm Lip} (\alpha, L)$ if $$ |f(x+h) - f(x)| \le C h^\alpha L({1/h}) \quad\ \hbox{for all } x\in {\mathbb T} ,\, h>0, $$ where the constant $C$ does not depend on $x$ and $h$. The above sufficient conditions are also necessary in the case of a certain subclass of Fourier coefficients. As a corollary, we deduce that if a function $f$ with Fourier coefficients in this subclass belongs to one of these generalized Lipschitz classes, then the conjugate function $\skew4\tilde f$ also belongs to the same generalized Lipschitz class.
