Weak thresholding greedy algorithms in Banach spaces
We consider weak thresholding greedy algorithms with respect to Markushevich bases in general Banach spaces. We find sufficient conditions for the equivalence of boundedness and convergence of the approximants. We also show that if there is a weak thresholding algorithm for the system which gives th...
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| Published in: | Journal of functional analysis Vol. 263; no. 12; pp. 3900 - 3921 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
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Elsevier Inc
15.12.2012
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| ISSN: | 0022-1236, 1096-0783 |
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| Abstract | We consider weak thresholding greedy algorithms with respect to Markushevich bases in general Banach spaces. We find sufficient conditions for the equivalence of boundedness and convergence of the approximants. We also show that if there is a weak thresholding algorithm for the system which gives the best n-term approximation up to a multiplicative constant, then the system is already “greedy”. Similar results are proved for “almost greedy” and “semi-greedy” systems. |
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| AbstractList | We consider weak thresholding greedy algorithms with respect to Markushevich bases in general Banach spaces. We find sufficient conditions for the equivalence of boundedness and convergence of the approximants. We also show that if there is a weak thresholding algorithm for the system which gives the best n-term approximation up to a multiplicative constant, then the system is already “greedy”. Similar results are proved for “almost greedy” and “semi-greedy” systems. |
| Author | Schlumprecht, Th Dilworth, S.J. Kutzarova, Denka Wojtaszczyk, P. |
| Author_xml | – sequence: 1 givenname: S.J. surname: Dilworth fullname: Dilworth, S.J. email: dilworth@math.sc.edu organization: Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA – sequence: 2 givenname: Denka surname: Kutzarova fullname: Kutzarova, Denka email: denka@math.uiuc.edu organization: Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria – sequence: 3 givenname: Th surname: Schlumprecht fullname: Schlumprecht, Th email: thomas.schlumprecht@math.tamu.edu organization: Department of Mathematics, Texas A&M University, College Station, TX 78712, USA – sequence: 4 givenname: P. surname: Wojtaszczyk fullname: Wojtaszczyk, P. email: wojtaszczyk@mimuw.edu.pl organization: Institute of Applied Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland |
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| Cites_doi | 10.1007/s003659900090 10.1006/jath.2001.3641 10.1023/A:1018900431309 10.1007/s00365-002-0525-y 10.4064/sm159-1-4 10.4064/sm-58-1-45-90 10.1016/j.jat.2008.08.017 10.1016/j.jat.2005.09.017 10.1006/jath.2000.3512 |
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| Issue | 12 |
| Keywords | Thresholding greedy algorithm Banach spaces Weak thresholding Greedy approximation |
| Language | English |
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| References | Wojtaszczyk (br0150) 2000; 107 Wojtaszczyk (br0160) 2003 Temlyakov (br0140) 2011 Dilworth, Kutzarova, Kalton, Temlyakov (br0030) 2003; 19 Temlyakov (br0110) 1998; 14 Dilworth, Kalton, Kutzarova (br0020) 2003; 159 Temlyakov (br0130) 2007 Dilworth, Kutzarova, Wojtaszczyk (br0040) 2002; 114 Gogyan (br0070) 2009; 161 John Elton, Weakly null normalized sequences in Banach spaces, PhD thesis, Yale University, 1978. Albiac, Wojtaszczyk (br0010) 2006; 138 Dilworth, Odell, Schlumprecht, Zsák (br0050) 2009; 35 Konyagin, Temlyakov (br0090) 2002; 28 Maurey, Pisier (br0100) 1976; 58 Konyagin, Temlyakov (br0080) 1999; 5 Temlyakov (br0120) 1998; 8 10.1016/j.jfa.2012.09.011_br0060 Dilworth (10.1016/j.jfa.2012.09.011_br0020) 2003; 159 Wojtaszczyk (10.1016/j.jfa.2012.09.011_br0160) 2003 Konyagin (10.1016/j.jfa.2012.09.011_br0090) 2002; 28 Temlyakov (10.1016/j.jfa.2012.09.011_br0110) 1998; 14 Albiac (10.1016/j.jfa.2012.09.011_br0010) 2006; 138 Dilworth (10.1016/j.jfa.2012.09.011_br0040) 2002; 114 Temlyakov (10.1016/j.jfa.2012.09.011_br0130) 2007 Dilworth (10.1016/j.jfa.2012.09.011_br0030) 2003; 19 Gogyan (10.1016/j.jfa.2012.09.011_br0070) 2009; 161 Temlyakov (10.1016/j.jfa.2012.09.011_br0140) 2011 Konyagin (10.1016/j.jfa.2012.09.011_br0080) 1999; 5 Temlyakov (10.1016/j.jfa.2012.09.011_br0120) 1998; 8 Dilworth (10.1016/j.jfa.2012.09.011_br0050) 2009; 35 Maurey (10.1016/j.jfa.2012.09.011_br0100) 1976; 58 Wojtaszczyk (10.1016/j.jfa.2012.09.011_br0150) 2000; 107 |
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| SubjectTerms | Banach spaces Greedy approximation Thresholding greedy algorithm Weak thresholding |
| Title | Weak thresholding greedy algorithms in Banach spaces |
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