Ridgelets and the representation of mutilated Sobolev functions
- Creators
- Candès, Emmanuel J.
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.
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.Files
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Additional details
- Eprint ID
- 559
- Resolver ID
- CaltechAUTHORS:CANsiamjma01
- Created
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2005-08-18Created from EPrint's datestamp field
- Updated
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2021-11-08Created from EPrint's last_modified field