CaltechAUTHORS
  A Caltech Library Service

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. http://resolver.caltech.edu/CaltechAUTHORS:CANsiamjma01

[img]
Preview
PDF
See Usage Policy.

229Kb

Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:CANsiamjma01

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
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
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

Repository Staff Only: item control page