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

Ferenc Móricz¹

¹ Bolyai Institute University of Szeged Aradi vértanúk tere 1 H-6720 Szeged, Hungary

Studia Mathematica, Tome 199 (2010) no. 2, pp. 199-205

Résumé

We consider complex-valued functions $f\in L^1 (\mathbb R)$, and prove sufficient conditions in terms of $f$ to ensure that the Fourier transform $\hat f$ belongs to one of the Lipschitz classes ${\rm Lip} (\alpha)$ and ${\rm lip} (\alpha)$ for some $0 \alpha\le 1$, or to one of the Zygmund classes $\mathop{\rm zyg} (\alpha)$ and ${\rm zyg}(\alpha)$ for some $0\alpha\le 2$. 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 $x f(x) \ge 0$ or $f(x) \ge 0$ almost everywhere.

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DOI : 10.4064/sm199-2-5

Keywords: consider complex valued functions mathbb prove sufficient conditions terms ensure fourier transform hat belongs lipschitz classes lip alpha lip alpha alpha zygmund classes mathop zyg alpha zyg alpha alpha these sufficient conditions best possible sense necessary real valued functions which either almost everywhere