Displaying similar documents to “Fourier Transforms of Lipschitz Functions and Fourier Multipliers on Compact Groups.”

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

Ferenc Móricz (2010)

Studia Mathematica


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.