Journal ArticleDOI
The Thresholding Greedy Algorithm, Greedy Bases, and Duality
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Some new conditions that arise naturally in the study of the Thresholding Greedy Algorithm are introduced for bases of Banach spaces and a complete duality theory for greedy bases is obtained.Abstract:
Some new conditions that arise naturally in the study of the Thresholding Greedy Algorithm are introduced for bases of Banach spaces. We relate these conditions to best n-term approximation and we study their duality theory. In particular, we obtain a complete duality theory for greedy bases.read more
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Journal ArticleDOI
On the existence of almost greedy bases in Banach spaces
TL;DR: 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.
Journal ArticleDOI
Best m-term approximation and Lizorkin–Triebel spaces
Markus Hansen,Winfried Sickel +1 more
TL;DR: The asymptotic behaviour of the widths of best m -term approximation with respect to Lizorkin–Triebel as well as Besov spaces is investigated to lead to final assertions in all possible situations.
BookDOI
Greedy approximations with regard to bases
TL;DR: This paper discusses in detail the behavior of the Thresholding Greedy Algorithm (TGA) with regard to a given basis and shows that this general greedy algorithm is very good for a wavelet basis and is not that good for the trigonometric system.
Journal ArticleDOI
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.
Journal ArticleDOI
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.
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.