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Greedy wavelet projections are bounded on BV

Authors:	Pawel Bechler Ronald DeVore Anna Kamont Guergana Petrova Przemyslaw Wojtaszczyk

Institution:	Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-950 Warsaw, Poland ; Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208 ; Institute of Mathematics, Polish Academy of Sciences, Branch in Gdansk, ul. Abrahama 18, 81-825 Sopot, Poland ; Department of Mathematics, Texas A&M University, College Station, Texas 77843 ; Institute of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland

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 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 . 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 a constant independent of . This answers in the affirmative a conjecture of Meyer (2001).

Keywords:	$N$-term approximation greedy approximation functions of bounded variation thresholding bounded projections

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