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

摘要

We consider complex-valued functions f∈L 1(ℝ₊), where ℝ₊:=[0,∞), and prove sufficient conditions under which the sine Fourier transform [^(f)]_shat{f}_{s} and the cosine Fourier transform [^(f)]_chat{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.