Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Greedy wavelet projections are bounded on BV

Authors: Pawel Bechler, Ronald DeVore, Anna Kamont, Guergana Petrova and Przemyslaw Wojtaszczyk
Journal: Trans. Amer. Math. Soc. 359 (2007), 619-635
MSC (2000): Primary 42C40, 46B70, 26B35, 42B25
Published electronically: August 16, 2006
MathSciNet review: 2255189
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Abstract: Let $ \mathrm{BV}=\mathrm{BV}(\mathbb{R}^d)$ be the space of functions of bounded variation on $ \mathbb{R}^d$ with $ d\ge 2$. Let $ \psi_\lambda$, $ \lambda\in\Delta$, be a wavelet system of compactly supported functions normalized in $ \mathrm{BV}$, i.e., $ \vert\psi_\lambda\vert _{\mathrm{BV}(\mathbb{R}^d)}=1$, $ \lambda\in\Delta$. Each $ f\in \mathrm{BV}$ has a unique wavelet expansion $ \sum_{\lambda\in\Delta} c_\lambda(f)\psi_\lambda$ with convergence in $ L_1(\mathbb{R}^d)$. If $ \Lambda_N(f)$ is the set of $ N$ indicies $ \lambda\in\Delta$ for which $ \vert c_\lambda(f)\vert$ are largest (with ties handled in an arbitrary way), then $ \mathcal{G}_N(f):=\sum_{\lambda\in\Lambda_N(f)}c_\lambda(f)\psi_\lambda$ is called a greedy approximation to $ f$. It is shown that $ \vert\mathcal{G}_N(f)\vert _{\mathrm{BV}(\mathbb{R}^d)}\le C\vert f\vert _{\mathrm{BV}(\mathbb{R}^d)}$ with $ C$ a constant independent of $ f$. This answers in the affirmative a conjecture of Meyer (2001).

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Additional Information

Pawel Bechler
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-950 Warsaw, Poland

Ronald DeVore
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Anna Kamont
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Branch in Gdansk, ul. Abrahama 18, 81-825 Sopot, Poland

Guergana Petrova
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Przemyslaw Wojtaszczyk
Affiliation: Institute of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland

Keywords: $N$-term approximation, greedy approximation, functions of bounded variation, thresholding, bounded projections
Received by editor(s): November 4, 2003
Received by editor(s) in revised form: November 15, 2004
Published electronically: August 16, 2006
Additional Notes: This work was supported in part by the NRC New Investigators Twinning Program 2003-2004 as well as the Office of Naval Research Contract N00014-03-1-0051, the Air Force of Scientific Research Contracts UFEIES0302005USC, the NSF Grant DMS-0296020 and DAAD 19-02-1-0028, the Foundation for Polish Science and KBN grant 5P03A 03620 located at the Institute of Mathematics of the Polish Academy of Sciences.
Article copyright: © Copyright 2006 American Mathematical Society