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Transactions of the American Mathematical Society

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Greedy wavelet projections are bounded on BV


Authors: Paweł Bechler, Ronald DeVore, Anna Kamont, Guergana Petrova and Przemysław Wojtaszczyk
Journal: Trans. Amer. Math. Soc. 359 (2007), 619-635
MSC (2000): Primary 42C40, 46B70, 26B35, 42B25
DOI: https://doi.org/10.1090/S0002-9947-06-03903-1
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., $|\psi _\lambda |_{\mathrm {BV}(\mathbb {R}^d)}=1$, $\lambda \in \Delta$. 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 $|c_\lambda (f)|$ 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 $|\mathcal {G}_N(f)|_{\mathrm {BV}(\mathbb {R}^d)}\le C|f|_{\mathrm {BV}(\mathbb {R}^d)}$ with $C$ a constant independent of $f$. This answers in the affirmative a conjecture of Meyer (2001).


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Additional Information

Paweł Bechler
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-950 Warsaw, Poland
Email: pbechler@impan.gov.pl

Ronald DeVore
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: devore@math.sc.edu

Anna Kamont
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Branch in Gdansk, ul. Abrahama 18, 81-825 Sopot, Poland
Email: A.Kamont@impan.gda.pl

Guergana Petrova
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: gpetrova@math.tamu.edu

Przemysław Wojtaszczyk
Affiliation: Institute of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland
MR Author ID: 192029
Email: pwojt@mimuw.edu.pl

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