Absolutely convergent Fourier series and generalized Lipschitz classes of functions

Ferenc Móricz¹

¹ Bolyai Institute University of Szeged Aradi vértanúk tere 1 6720 Szeged, Hungary

Colloquium Mathematicum, Tome 113 (2008) no. 1, pp. 105-117

Résumé

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.

Détail

DOI : 10.4064/cm113-1-7

Keywords: investigate order magnitude modulus continuity function absolutely convergent fourier series sufficient conditions terms fourier coefficients order belong generalized lipschitz classes mathop lip alpha mathop lip alpha where alpha positive nondecreasing slowly varying function infty infty example nbsp pi periodic function said belong class mathop lip alpha alpha quad hbox mathbb where constant does depend above sufficient conditions necessary certain subclass fourier coefficients corollary deduce function fourier coefficients subclass belongs these generalized lipschitz classes conjugate function skew tilde belongs generalized lipschitz class