On Double Sine and Cosine Transforms, Lipschitz and Zygmund Classes

Abstract

We consider complex-valued functions $f \in  L^1(\mathbf{R}^2_+)$, where $\mathbf{R}_+ := [0,\infty)$, and prove sufficient conditions under which the double sine Fourier transform $\hat{f}_{ss}$ and the double cosine Fourier transform $\hat{f}_{cc}$  belong to one of the two-dimensional Lipschitz classes $Lip(\alpha,\beta )$ for some $0 < \alpha,\beta \leq 1$; or to one of the Zygmund classes Zyg$(\alpha,\beta )$ for some $0 < \alpha,\beta  \leq 2$. These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions $f \in L^1(\mathbf{R}^2_+)$.