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Displaying similar documents to “Factorization of Lipschitz Functioins and Absolute Convergence of Vilenkin-Fourier Series.”

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...

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

Ferenc Móricz (2010)

Studia Mathematica

Similarity:

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.