Counterexamples in isometric theory of symmetric and greedy bases (original) (raw)
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Our aim is to investigate the properties of existence and uniqueness of greedy bases in Banach spaces. We show the non-existence of greedy basis in some Nakano spaces and Orlicz sequence spaces and produce the first-known examples of non-trivial spaces (i.e., different from c_0c_0c0, ell1\ell_1ell1, and ell2\ell_2ell_2) with a unique greedy basis.
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Journal of Approximation Theory, 2016
The purpose of this article is to undertake an in-depth study of the properties of existence and uniqueness of greedy bases in Banach spaces. We show that greedy bases fail to exist for a range of neo-classical spaces within the family of Nakano and Orlicz sequence spaces and find the first-known cases of non-trivial spaces (i.e., different from c 0 , ℓ 1 , and ℓ 2) with a unique greedy basis. The variety and nature of those examples evince that a complete classification of Banach spaces with a unique greedy basis cannot be expected.
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Bulletin of the Malaysian Mathematical Sciences Society, 2023
Partially greedy bases in Banach spaces were introduced by Dilworth et al. as a strictly weaker notion than the (almost) greedy bases. In this paper, we study two natural ways to strengthen the definition of partial greediness. The first way produces what we call the consecutive almost greedy property, which turns out to be equivalent to the almost greedy property. Meanwhile, the second way reproduces the PG property for Schauder bases but a strictly stronger property for general bases.
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We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of almost greedy bases begun in [3]. We show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space X has a basis and contains a complemented subspace with a symmetric basis and finite cotype then X has an almost greedy basis. We show that c 0 is the only L ∞ space to have a quasi-greedy basis. The Banach spaces which contain almost greedy basic sequences are characterized. Contents 8. Quasi-greedy bases in L ∞ spaces 31 References 37
Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces
Israel Journal of Mathematics, 1996
Let X be a Banach space with an unconditional finite-dimensional Schauder decomposition (E n). We consider the general problem of characterizing conditions under which one can construct an unconditional basis for X by forming an unconditional basis for each E n. For example, we show that if sup dim E n < ∞ and X has Gordon-Lewis local unconditional structure then X has an unconditional basis of this type. We also give an example of a non-Hilbertian space X with the property that whenever Y is a closed subspace of X with a UFDD (E n) such that sup dim E n < ∞ then Y has an unconditional basis, showing that a recent result of Komorowski and Tomczak-Jaegermann cannot be improved.
2020
We introduce and study the notion of weak semi-greedy systems -which is inspired in the concepts of semi-greedy and Branch semi-greedy systems and weak thresholding sets-, and prove that in the context Markushevich bases in infinite dimensional Banach spaces, the notions of \textit{ semi-greedy, branch semi-greedy, weak semi-greedy, and almost greedy} Markushevich bases are all equivalent. This completes and extends some results from \cite{Berna2019}, \cite{Dilworth2003b}, and \cite{Dilworth2012}. We also exhibit an example of a semi-greedy system that is neither almost greedy nor a Markushevich basis, showing that the Markushevich condition cannot be dropped from the equivalence result. In some cases, we obtain improved upper bounds for the corresponding constants of the systems.
An Example of an Almost Greedy Basis in L-1(0,1)
2010
We give an explicit construction of an almost greedy basis of L 1 (0, 1), complementing the results on existence of such a basis. The basis is described in terms of the Haar basis. We construct a quasi-greedy basis in a Banach space which is isomorphic to L 1 (0, 1), and then we calculate an isomorphic image of a quasi-greedy basis.
Weaker forms of unconditionality of bases in greedy approximation
2021
From the abstract perspective of Banach spaces, the theory of (nonlinear) greedy approximation using bases sprang from the seminal characterization of greedy bases by Konyagin and Temlyakov in 1999 as those bases that are simultaneously unconditional and democratic [16]. These two properties are, a priori, independent of each other and we find examples of unconditional bases which are not democratic and the other way around already in the very early stages of the theory (see, e.g., [7, Example 10.4.4]). However, the geometry of some spaces X can make these properties intertwine, to the extent that the unconditional semi-normalized bases in X end up being democratic (hence greedy). This is the case of unconditional bases in Hilbert spaces, and also in the spaces l1 and c0 for instance (see [12, Theorem 4.1], [21, Theorem 3] and [10, Corollary 8.6]).
Renormings and symmetry properties of 1-greedy bases
Journal of Approximation Theory, 2011
We continue the study of 1-greedy bases initiated by F. Albiac and P. Wojtaszczyk [1]. We answer several open problems they raised concerning symmetry properties of 1-greedy bases and the improving of the greedy constant by renorming. We show that 1-greedy bases need not be symmetric nor subsymmetric. We also prove that one cannot in general make a greedy basis 1-greedy as demonstrated for the Haar basis of dyadic Hardy space H 1 (R) and for the unit vector basis of Tsirelson space. On the other hand, we give a renorming of L p (1 < p < ∞) that makes the Haar basis 1-unconditional and 1-democratic. Other results in this paper clarify the relationship between various basis constant that arise in the context of greedy bases.