# Absolutely convergent Fourier series and generalized Lipschitz classes of functions

Colloquium Mathematicae (2008)

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

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topFerenc 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 -

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