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• # Artykuł - szczegóły

## Colloquium Mathematicum

2008 | 113 | 1 | 105-117

## Absolutely convergent Fourier series and generalized Lipschitz classes of functions

EN

### Abstrakty

EN
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 ∈ 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 f̃ also belongs to the same generalized Lipschitz class.

105-117

wydano
2008

### Twórcy

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