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Ridgelets and the representation of mutilated Sobolev functions

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.

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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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Candès, Emmanuel J.0000-0001-9234-924X
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
Record Number:CaltechAUTHORS:CANsiamjma01
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:559
Deposited By: Tony Diaz
Deposited On:18 Aug 2005
Last Modified:08 Nov 2021 19:03

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