Candès, Emmanuel J. (2001) Ridgelets and the representation of mutilated Sobolev functions. SIAM Journal on Mathematical Analysis, 33 (2). pp. 347-368. ISSN 0036-1410. doi:10.1137/S003614109936364X. https://resolver.caltech.edu/CaltechAUTHORS:CANsiamjma01
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Abstract
We show that ridgelets, a system introduced in [E. J. Candes, Appl. Comput. Harmon. Anal., 6(1999), pp. 197–218], are optimal to represent smooth multivariate functions that may exhibit linear singularities. For instance, let {u · x − b > 0} be an arbitrary hyperplane and consider the singular function f(x) = 1{u·x−b>0}g(x), where g is compactly supported with finite Sobolev L2 norm ||g||Hs, s > 0. The ridgelet coefficient sequence of such an object is as sparse as if f were without singularity, allowing optimal partial reconstructions. For instance, the n-term approximation obtained by keeping the terms corresponding to the n largest coefficients in the ridgelet series achieves a rate of approximation of order n−s/d; the presence of the singularity does not spoil the quality of the ridgelet approximation. This is unlike all systems currently in use, especially Fourier or wavelet representations.
Item Type: | Article | ||||||
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Additional Information: | © 2001 Society for Industrial and Applied Mathematics. Received by the editors November 3, 1999; accepted for publication (in revised form) December 16, 2000; published electronically July 19, 2001. This research was supported by National Science Foundation grants DMS 98-72890 (KDI) and DMS 95-05151 and by AFOSR MURI 95-P49620-96-1-0028. | ||||||
Subject Keywords: | Sobolev spaces, Fourier transform, singularities, ridgelets, orthonormal ridgelets, nonlinear approximation, sparsity | ||||||
Issue or Number: | 2 | ||||||
DOI: | 10.1137/S003614109936364X | ||||||
Record Number: | CaltechAUTHORS:CANsiamjma01 | ||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:CANsiamjma01 | ||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||
ID Code: | 559 | ||||||
Collection: | CaltechAUTHORS | ||||||
Deposited By: | Tony Diaz | ||||||
Deposited On: | 18 Aug 2005 | ||||||
Last Modified: | 08 Nov 2021 19:03 |
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