Volume 27, Issue 4
On Double Sine and Cosine Transforms, Lipschitz and Zygmund Classes

Anal. Theory Appl., 27 (2011), pp. 351-364.

Published online: 2011-11

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• 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 $f \in L^1(\mathbf{R}^2_+)$.

42B10, 26A16, 26B35

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@Article{ATA-27-351, author = {Vanda Fülöp , and Mόricz , Ferenc}, title = {On Double Sine and Cosine Transforms, Lipschitz and Zygmund Classes}, journal = {Analysis in Theory and Applications}, year = {2011}, volume = {27}, number = {4}, pages = {351--364}, 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 $f \in L^1(\mathbf{R}^2_+)$.

}, issn = {1573-8175}, doi = {https://doi.org/10.1007/s10496-011-0351-9}, url = {http://global-sci.org/intro/article_detail/ata/4607.html} }
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 -

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 $f \in L^1(\mathbf{R}^2_+)$.

Vanda Fülöp & Ferenc Mόricz. (1970). On Double Sine and Cosine Transforms, Lipschitz and Zygmund Classes. Analysis in Theory and Applications. 27 (4). 351-364. doi:10.1007/s10496-011-0351-9
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