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

@article{FerencMóricz2010,
abstract = {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.},
author = {Ferenc Móricz},
journal = {Studia Mathematica},
keywords = {Fourier transform; best possible sufficient conditions; classical function classes Lip(); lip(); Zyg() and zyg()},
language = {eng},
number = {2},
pages = {199-205},
title = {Best possible sufficient conditions for the Fourier transform to satisfy the Lipschitz or Zygmund condition},
url = {http://eudml.org/doc/285599},
volume = {199},
year = {2010},
}

TY - JOUR
AU - Ferenc Móricz
TI - Best possible sufficient conditions for the Fourier transform to satisfy the Lipschitz or Zygmund condition
JO - Studia Mathematica
PY - 2010
VL - 199
IS - 2
SP - 199
EP - 205
AB - 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.
LA - eng
KW - Fourier transform; best possible sufficient conditions; classical function classes Lip(); lip(); Zyg() and zyg()
UR - http://eudml.org/doc/285599
ER -