# Absolutely convergent Fourier series and generalized Lipschitz classes of functions

Colloquium Mathematicae (2008)

• Volume: 113, Issue: 1, page 105-117
• ISSN: 0010-1354

top

## Abstract

top
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 $|f\left(x+h\right)-f\left(x\right)|\le C{h}^{\alpha }L\left(1/h\right)$ 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.

## How to cite

top

Ferenc Móricz. "Absolutely convergent Fourier series and generalized Lipschitz classes of functions." Colloquium Mathematicae 113.1 (2008): 105-117. <http://eudml.org/doc/286073>.

@article{FerencMóricz2008,
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 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 $|f(x+h) - f(x)| ≤ Ch^\{α\}L(1/h)$ 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.},
author = {Ferenc Móricz},
journal = {Colloquium Mathematicae},
keywords = {Fourier series; absolute convergence; modulus of continuity; slowly varying functions; generalized Lipschitz classes; conjugate series; conjugate functions},
language = {eng},
number = {1},
pages = {105-117},
title = {Absolutely convergent Fourier series and generalized Lipschitz classes of functions},
url = {http://eudml.org/doc/286073},
volume = {113},
year = {2008},
}

TY - JOUR
AU - Ferenc Móricz
TI - Absolutely convergent Fourier series and generalized Lipschitz classes of functions
JO - Colloquium Mathematicae
PY - 2008
VL - 113
IS - 1
SP - 105
EP - 117
AB - 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 $|f(x+h) - f(x)| ≤ Ch^{α}L(1/h)$ 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.
LA - eng
KW - Fourier series; absolute convergence; modulus of continuity; slowly varying functions; generalized Lipschitz classes; conjugate series; conjugate functions
UR - http://eudml.org/doc/286073
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.