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 Lip(α,L) and Lip(α,1/L), where 0 ≤ α ≤ 1 and L = L(x) is a positive, nondecreasing, slowly varying function such that L(x) → ∞ as x → ∞. For example, a 2π-periodic function f is said to belong to the class Lip(α,L) if for all x ∈ , 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 f̃ also belongs to the same generalized Lipschitz class.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm113-1-7,
author = {Ferenc M\'oricz},
title = {Absolutely convergent Fourier series and generalized Lipschitz classes of functions},
journal = {Colloquium Mathematicae},
volume = {111},
year = {2008},
pages = {105-117},
zbl = {1215.42006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm113-1-7}
}
Ferenc Móricz. Absolutely convergent Fourier series and generalized Lipschitz classes of functions. Colloquium Mathematicae, Tome 111 (2008) pp. 105-117. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm113-1-7/