Sine, cosine transforms and classical function classes

We study the continuity and smoothness properties of functions f ∈ L ¹(0, ∞)) whose sine transforms $ \hat f_s $ and cosine tranforms $ \hat f_c $ belong to L ¹(0,∞)). We give best possible sufficient conditions in terms of $ \hat f_s $ and $ \hat f_c $ to ensure that f belongs 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. The conditions given by us are not only sufficient, but also necessary in the case when the sine and cosine transforms are nonnegative. Our theorems are extensions of the corresponding theorems by Boas from sine and cosine series to sine and cosine transforms.