Absolutely convergent Fourier integrals and classical function spaces

Abstract.

We study the continuity and smoothness properties of functions $f \in L^{1}({\mathbb{R}})$ whose Fourier transforms. $\hat {f}$ belong to $L^{1}({\mathbb{R}})$, and give sufficient conditions in terms of $\hat {f}$ to ensure that f belongs either 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 also necessary under an additional positivity assumption. Our theorems extend known results from periodic to nonperiodic functions.