Greedy Algorithm for General Biorthogonal Systems

By Przemyslaw Wojtaszczyk. 2000

213 citations recorded by the source corpus.

We consider biorthogonal systems in quasi-Banach spaces such that the greedy algorithm converges for each [email protected]?X (quasi-greedy systems). We construct quasi-greedy conditional bases in a wide range of Banach spaces. We also compare the greedy algorithm for the multidimensional Haar system with the optimal m-term approximation for this system. This substantiates a conjecture by Temlyakov.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/45f2814e237ee4ee16a51fd35c5cb9d1f25064d6

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In Applied Mathematics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Greedy approximation (2008, methodology): “The conjecture was proved in Wojtaszczyk (2000).”
Some remarks on non-linear approximation with Schauder bases (2001, methodology): “The notion of a quasi-greedy basis was introduced in [6] and studied further in [11].”
Lebesgue inequalities for the greedy algorithm in general bases (2016, background): “As suggested in [24, 25], greedy algorithms can be considered in this generality, by formally associating with every x ∈ X the series x ∼ ∞n=1 en(x)en.”
Lebesgue-Type Inequalities for Quasi-greedy Bases (2011, methodology): “For instance, μ(N) is defined in [22], and hl, hr appear explicitly in [11].”
Nonlinear Approximation with Dictionaries I. Direct Estimates (2004, unknown): “We refer to [28, 42] for the notions of (quasi)-greedy bases and to [11] for the notion of almost-greedy bases.”
Greedy Approximation with Regard to Bases and General Minimal Systems (2002, background): “The proof is based on the following known lemma (see [6]) that is essentially due to Wojtaszczyk [32].”
Greedy approximation for biorthogonal systems in quasi-Banach spaces (2019, methodology): “The idea of studying greedy bases and related greedy algorithms from a more abstract point of view seems to have originated with the work of Dilworth, Kutzarova, and Temlyakov [36], and the work of Wojtaszczyk [73], who characterized quasi-greedy bases for biorthogonal systems in the setting of quas”
On the existence of almost greedy bases in Banach spaces (2003, background): “The following two lemmas are essentially due to Wojtaszczyk [22].”
Characterization of 1-almost greedy bases (2015, methodology): “As Wojtaszczyk pointed out in [7], the choice of the greedy ordering for each x ∈ X with which to construct the greedy operators (Gm) ∞ m=1 plays no relevant role in the theory.”
The Thresholding Greedy Algorithm, Greedy Bases, and Duality (2003, background): “P. WOJTASZCZYK (2000): Greedy algorithm for general biorthogonal systems.”