Geodesic
Ridgelet transform on tempered distributions
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 3, pp. 431-439 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We prove that ridgelet transform $R:\mathscr{S}(\mathbb{R}^2)\to \mathscr{S} (\mathbb{Y})$ and adjoint ridgelet transform $R^\ast:\mathscr{S}(\mathbb{Y}) \to \mathscr{S}(\mathbb{R}^2)$ are continuous, where $\mathbb{Y}=\mathbb{R}^+\times \mathbb{R}\times [0,2\pi]$. We also define the ridgelet transform $\mathcal{R}$ on the space $\mathscr{S}^\prime(\mathbb{R}^2)$ of tempered distributions on $\mathbb{R}^2$, adjoint ridgelet transform $\mathcal{R}^\ast$ on $\mathscr{S}^\prime(\mathbb{Y})$ and establish that they are linear, continuous with respect to the weak$^\ast$-topology, consistent with $R$, $R^\ast$ respectively, and they satisfy the identity $(\mathcal{R}^\ast \circ \mathcal{R})(u) = u$, $u\in \mathscr{S}^\prime(\mathbb{R}^2)$.
We prove that ridgelet transform $R:\mathscr{S}(\mathbb{R}^2)\to \mathscr{S} (\mathbb{Y})$ and adjoint ridgelet transform $R^\ast:\mathscr{S}(\mathbb{Y}) \to \mathscr{S}(\mathbb{R}^2)$ are continuous, where $\mathbb{Y}=\mathbb{R}^+\times \mathbb{R}\times [0,2\pi]$. We also define the ridgelet transform $\mathcal{R}$ on the space $\mathscr{S}^\prime(\mathbb{R}^2)$ of tempered distributions on $\mathbb{R}^2$, adjoint ridgelet transform $\mathcal{R}^\ast$ on $\mathscr{S}^\prime(\mathbb{Y})$ and establish that they are linear, continuous with respect to the weak$^\ast$-topology, consistent with $R$, $R^\ast$ respectively, and they satisfy the identity $(\mathcal{R}^\ast \circ \mathcal{R})(u) = u$, $u\in \mathscr{S}^\prime(\mathbb{R}^2)$.
Classification : 42C40, 44A15, 65T60
Keywords: ridgelet transform; tempered distributions; wavelets 
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     author = {Roopkumar, R.},
     title = {Ridgelet transform on tempered distributions},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {431--439},
     year = {2010},
     volume = {51},
     number = {3},
     mrnumber = {2741876},
     zbl = {1222.46029},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2010_51_3_a4/}
}
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Roopkumar, R. Ridgelet transform on tempered distributions. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 3, pp. 431-439. http://geodesic.mathdoc.fr/item/CMUC_2010_51_3_a4/