Absolutely convergent Fourier series and generalized Lipschitz classes of functions

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 -