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Displaying similar documents to “Inequalities connected with binary derivation, integrated Lipschitz conditions and Fourier series”

Best possible sufficient conditions for the Fourier transform to satisfy the Lipschitz or Zygmund condition

Ferenc Móricz (2010)

Studia Mathematica

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We consider complex-valued functions f ∈ L¹(ℝ), and prove sufficient conditions in terms of f to ensure that the Fourier transform f̂ belongs to one of the Lipschitz classes Lip(α) and lip(α) for some 0 < α ≤ 1, or to one of the Zygmund classes zyg(α) and zyg(α) for some 0 < α ≤ 2. These sufficient conditions are best possible in the sense that they are also necessary in the case of real-valued functions f for which either xf(x) ≥ 0 or f(x) ≥ 0 almost everywhere.

Absolutely convergent Fourier series and generalized Lipschitz classes of functions

Ferenc Móricz (2008)

Colloquium Mathematicae

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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 ) | ≤ C h α L ( 1 / h ) for all...