Nonlinear Approximation and the Space BV(R²)
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| Venue: | AMER. J. MATH |
| Citations: | 7 - 1 self |
BibTeX
@ARTICLE{Cohen_nonlinearapproximation,
author = {Albert Cohen and Ronald DeVore and Pencho Petrushev and Hong Xu},
title = {Nonlinear Approximation and the Space BV(R²) },
journal = {AMER. J. MATH},
year = {},
volume = {121},
pages = {587--628}
}
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Abstract
Given a function f 2 L 2 (Q), Q := [0; 1)² and a real number t ? 0, let U(f; t) := inf g2BV(Q) kf \Gamma gk 2 L 2 (I) + t VQ (g); where the infimum is taken over all functions g 2 BV of bounded variation on I. This and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation, as well as in digital image processing. Techniques for finding minimizers g for U(f; t) based on variational calculus and nonlinear partial differential equations have been put forward by several authors ([DMS], [LOR], [MS], [CL]). The main disadvantage of these approaches is that they are numerically intensive. On the other hand, it is well-known that more elementary methods based on wavelet shrinkage solve related extremal problems, for example, the above problem with BV replaced by the Besov space B 1 1 (L 1 (I)) (see e.g. [CDLL]). However, since BV has no simple description in terms of wavelet coefficients, it is not clear that minimi...
