On the existence of almost greedy bases in Banach spaces
TLDR
In this article, the authors consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the thresholding greedy algorithm and show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation.Abstract:
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 c0 is the only L∞ space to have a quasi-greedy basis. The Banach spaces which contain almost greedy basic sequences are characterized.read more
Citations
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Book
Dirichlet Series and Holomorphic Functions in High Dimensions
TL;DR: In this paper, the convergence of Dirichlet series and holomorphic functions in high dimensions has been studied in the context of functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory.
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On Lebesgue-type inequalities for greedy approximation
TL;DR: In this paper, the efficiency of greedy algorithms with respect to redundant dictionaries in Hilbert spaces was studied and upper estimates for the errors of the pure greedy algorithm and the orthogonal greedy algorithm in terms of the best m-term approximations were obtained.
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On the Structure of the Spreading Models of a Banach Space
TL;DR: In this paper, the authors studied the structure of the set of spreading models of a Banach space X and proved that for any countable set C, there is a spreading model generated by a weakly null sequence which dominates each element of C. In particular, they gave an example of a reflexive X so that all spreading models contained l 1 but none of them is isomorphic to l 1.
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Lebesgue-Type Inequalities for Quasi-greedy Bases
TL;DR: In particular, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N-term approximation ∥x−GNx∥ and the best Nterm approximation σN(x) is controlled by max{μ(N),kN}, where μ(N) and kN are well-known constants that quantify the democracy and conditionality of the basis.
Journal ArticleDOI
Greedy approximation for biorthogonal systems in quasi-Banach spaces
TL;DR: In this paper, a systematic study of the thresholding greedy algorithm for general biorthogonal systems in quasi-Banach spaces from a functional-analytic point of view is presented.
References
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Book
Classical Banach spaces
Joram Lindenstrauss,L. Tzafriri +1 more
TL;DR: Springer-Verlag is reissuing a selected few of these highly successful books in a new, inexpensive sofcover edition to make them easily accessible to younger generations of students and researchers.
MonographDOI
Factorization of Linear Operators and Geometry of Banach Spaces
TL;DR: In this paper, the authors present operators and basic applications for factorization through a Hilbert space Type and cotype, and apply them to Grothendieck's conjecture and the volume ratio method.
Journal ArticleDOI
A Characterization of Banach Spaces Containing l1
TL;DR: In this article, it was shown that a Banach space contains a subspace isomorphic to l 1 if and only if it has a bounded sequence with no weak-Cauchy subsequence.