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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Greedy wavelet projections are bounded on BV
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by Paweł Bechler, Ronald DeVore, Anna Kamont, Guergana Petrova and Przemysław Wojtaszczyk PDF
Trans. Amer. Math. Soc. 359 (2007), 619-635 Request permission


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:
  • Ronald DeVore
  • Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
  • Email:
  • Anna Kamont
  • Affiliation: Institute of Mathematics, Polish Academy of Sciences, Branch in Gdansk, ul. Abrahama 18, 81-825 Sopot, Poland
  • Email:
  • Guergana Petrova
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
  • Email:
  • Przemysław Wojtaszczyk
  • Affiliation: Institute of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland
  • MR Author ID: 192029
  • Email:
  • 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.
  • © Copyright 2006 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 619-635
  • MSC (2000): Primary 42C40, 46B70, 26B35, 42B25
  • DOI:
  • MathSciNet review: 2255189