On the degree of continuity and smoothness of sine and cosine Fourier transforms of Lebesgue integrable functions

Abstract

We consider complex-valued functions f∈L1(ℝ+), where ℝ+:=[0,∞), and prove sufficient conditions under which the sine Fourier transform \(\hat{f}_{s}\) and the cosine Fourier transform \(\hat{f}_{c}\) belong 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 if f(x)≧0 almost everywhere.