TY - JOUR
T1 - On Double Sine and Cosine Transforms, Lipschitz and Zygmund Classes
AU - Vanda Fülöp , 
AU - Mόricz , Ferenc
JO - Analysis in Theory and Applications
VL - 4
SP - 351
EP - 364
PY - 2011
DA - 2011/11
SN - 27
DO - http://doi.org/10.1007/s10496-011-0351-9
UR - https://global-sci.org/intro/article_detail/ata/4607.html
KW - double sine and cosine Fourier transform, Lipschitz class Lip$(\alpha, \beta)$, $0< \alpha, \beta \leq 1$, Zygmund class Zyg$(\alpha, \beta)$, $0 < \alpha,\beta  \leq 2$.
AB - <p style="text-align: justify;">We consider complex-valued functions $f \in &nbsp;L^1(\mathbf{R}^2_+)$, 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 $\hat{f}_{cc}$ &nbsp;belong to one of the two-dimensional Lipschitz classes $Lip(\alpha,\beta )$ for some $0 &lt; \alpha,\beta \leq 1$; or to one of the Zygmund classes Zyg$(\alpha,\beta )$ for some $0 &lt; \alpha,\beta &nbsp;\leq 2$. These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions $f \in L^1(\mathbf{R}^2_+)$.</p>