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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm199-2-5,
author = {Ferenc M\'oricz},
title = {Best possible sufficient conditions for the Fourier transform to satisfy the Lipschitz or Zygmund condition},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {199-205},
zbl = {1210.42008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm199-2-5}
}
Ferenc Móricz. Best possible sufficient conditions for the Fourier transform to satisfy the Lipschitz or Zygmund condition. Studia Mathematica, Tome 196 (2010) pp. 199-205. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm199-2-5/