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Characterization of 1-greedy bases. (English) Zbl 1087.65048

The authors investigate 1-greedy bases for a real Banach space, i.e., bases for which the greedy algorithm provides the best \(m\)-term approximation. A characterization of 1-greediness is studied. The authors also give a list of open problems that arise from this article.

MSC:

65J05 General theory of numerical analysis in abstract spaces
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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[1] Dilworth, S.J.; Kalton, N.J.; Kutzarova, D.; Temlyakov, V.N., The thresholding greedy algorithm, greedy bases, and duality, Constr. approx., 19, 4, 575-597, (2003) · Zbl 1050.46011
[2] Figiel, T.; Johnson, W.B., A uniformly convex Banach space which contains no \(\ell_p\), Compositio math., 29, 2, 179-190, (1974) · Zbl 0301.46013
[3] Gamlen, J.L.B.; Gaudet, R.J., On subsequences of the Haar system, Israel J. math., 15, 404-413, (1973) · Zbl 0296.46031
[4] Garling, D.J.H., Symmetric bases of locally convex spaces, Studia math., 30, 163-181, (1968) · Zbl 0159.17703
[5] Kalton, N.J.; Leránoz, C.; Wojtaszczyk, P., Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces, Israel J. math., 72, 299-311, (1990) · Zbl 0753.46013
[6] Konyagin, S.V.; Temlyakov, V.N., A remark on greedy approximation in Banach spaces, East J. approx., 5, 365-379, (1999) · Zbl 1084.46509
[7] Megginson, R.E., An introduction to Banach space theory, (1998), Springer New York, Berlin, Heidelberg · Zbl 0910.46008
[8] Temlyakov, V.N., The best \(m\)-term approximation and greedy algorithms, Adv. in comp. math., 8, 249-265, (1998) · Zbl 0905.65063
[9] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0724.46012
[10] Wojtaszczyk, P., Greedy algorithm for general biorthogonal systems, J. approx. theory, 107, 293-314, (2000) · Zbl 0974.65053
[11] P. Wojtaszczyk, Greedy Type Bases in Banach Spaces, Constructive Function Theory, Varna 2002, Darba, Sofia, 2002, pp. 1-20.
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