# Article

 Title: Ridgelet transform on tempered distributions (English) Author: Roopkumar, R. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 51 Issue: 3 Year: 2010 Pages: 431-439 Summary lang: English . Category: math . Summary: 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 $(\mathcal{R}^\ast \circ \mathcal{R})(u) = u$, $u\in \mathscr{S}^\prime(\mathbb{R}^2)$. (English) Keyword: ridgelet transform Keyword: tempered distributions Keyword: wavelets MSC: 42C40 MSC: 44A15 MSC: 65T60 idZBL: Zbl 1222.46029 idMR: MR2741876 . Date available: 2010-09-02T14:14:55Z Last updated: 2013-09-22 Stable URL: http://hdl.handle.net/10338.dmlcz/140719 . Reference: [1] Candès E.J.: Harmonic analysis of neural networks.Appl. Comput. Harmon. Anal. 6 (1999), 197–218. MR 1676767, 10.1006/acha.1998.0248 Reference: [2] Constantine G.M., Savits T.H.: A multivariate Faa di Bruno formula with applications.Trans. Amer. Math. Soc. 348 (1996), 503–520. Zbl 0846.05003, MR 1325915, 10.1090/S0002-9947-96-01501-2 Reference: [3] Deans S.R.: The Radon Transform and Some of its Applications.John Wiley & Sons, New York, 1983. Zbl 1121.44004, MR 0709591 Reference: [4] Holschneider M.: Wavelets. An Analysis Tool.Clarendon Press, New York, 1995. Zbl 0952.42016, MR 1367088 Reference: [5] Pathak R.S.: The wavelet transform of distributions.Tohoku Math. J. 56 (2004), 411–421. Zbl 1078.42029, MR 2075775, 10.2748/tmj/1113246676 Reference: [6] Roopkumar R.: Ridgelet transform on square integrable Boehmains.Bull. Korean Math. Soc. 46 (2009), 835–844. MR 2554278, 10.4134/BKMS.2009.46.5.835 Reference: [7] Rudin W.: Functional Analysis.McGraw-Hill, New York, 1973. Zbl 0867.46001, MR 0365062 Reference: [8] Starck J.L., Candès E.J., Donoho D.: The Curvelet Transform for Image Denoising.IEEE Trans. Image Process. 11 (2002), 670–684. MR 1929403, 10.1109/TIP.2002.1014998 .

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