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

Ferenc Móricz. "Best possible sufficient conditions for the Fourier transform to satisfy the Lipschitz or Zygmund condition." Studia Mathematica 199.2 (2010): 199-205. <http://eudml.org/doc/285599>.

@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 -