Candès, Emmanuel J. (2001) Ridgelets and the representation of mutilated Sobolev functions. SIAM Journal on Mathematical Analysis, 33 (2). pp. 347368. ISSN 00361410. http://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 gHs, 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 nterm 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 

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 9872890 (KDI) and DMS 9505151 and by AFOSR MURI 95P496209610028. 
Subject Keywords:  Sobolev spaces, Fourier transform, singularities, ridgelets, orthonormal ridgelets, nonlinear approximation, sparsity 
Record Number:  CaltechAUTHORS:CANsiamjma01 
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:CANsiamjma01 
Alternative URL:  http://dx.doi.org/10.1137/S003614109936364X 
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:  26 Dec 2012 08:40 
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