Absolutely convergent Fourier integrals and classical function spaces

We study the continuity and smoothness properties of functions whose Fourier transforms. belong to , and give sufficient conditions in terms of 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. This research was supported by the Hungarian National Foundation for Scientific Research under Grant T 046 192.