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Keywords:
ridgelet transform; tempered distributions; wavelets
ridgelet transform; tempered distributions; wavelets
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 $(\mathcal{R}^\ast \circ \mathcal{R})(u) = u$, $u\in \mathscr{S}^\prime(\mathbb{R}^2)$.
References:
[1] Candès E.J.: Harmonic analysis of neural networks. Appl. Comput. Harmon. Anal. 6 (1999), 197–218. DOI 10.1006/acha.1998.0248 | MR 1676767
[2] Constantine G.M., Savits T.H.: A multivariate Faa di Bruno formula with applications. Trans. Amer. Math. Soc. 348 (1996), 503–520. DOI 10.1090/S0002-9947-96-01501-2 | MR 1325915 | Zbl 0846.05003
[3] Deans S.R.: The Radon Transform and Some of its Applications. John Wiley & Sons, New York, 1983. MR 0709591 | Zbl 1121.44004
[4] Holschneider M.: Wavelets. An Analysis Tool. Clarendon Press, New York, 1995. MR 1367088 | Zbl 0952.42016
[5] Pathak R.S.: The wavelet transform of distributions. Tohoku Math. J. 56 (2004), 411–421. DOI 10.2748/tmj/1113246676 | MR 2075775 | Zbl 1078.42029
[6] Roopkumar R.: Ridgelet transform on square integrable Boehmains. Bull. Korean Math. Soc. 46 (2009), 835–844. DOI 10.4134/BKMS.2009.46.5.835 | MR 2554278
[7] Rudin W.: Functional Analysis. McGraw-Hill, New York, 1973. MR 0365062 | Zbl 0867.46001
[8] Starck J.L., Candès E.J., Donoho D.: The Curvelet Transform for Image Denoising. IEEE Trans. Image Process. 11 (2002), 670–684. DOI 10.1109/TIP.2002.1014998 | MR 1929403
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