(untitled) (Analysis in Theory and Applications, 2011)
We consider complex-valued functions f L^1(R^2_+), where R_+ := [0,∞), and prove sufficient conditions under which the double sine Fourier transform f_ss and the double cosine Fourier transform f_cc belong to one of the two-dimensional Lipschitz classes Lip(α,β ) for some 0 < α,β ≤ 1; or to one of the Zygmund classes Zyg(α,β ) for some 0 < α,β ≤ 2. These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions f L^1(R^2_+).