Quasi-greedy bases in ℓ (0 < p < 1) are democratic

Bidemocratic Bases and Their Connections with Other Greedy-Type Bases

“…In the case when X spans the whole space X, this theorem says that any truncation quasi-greedy basis of a GT-space is democratic with fundamental function equivalent to (m) ∞ m=1 (see [5,Theorem 4.3]). As a matter of fact, the proof of [5, Theorem 4.3] as well as the proofs of Lemmas 2.2 and 4.2 of [5] on which it relies, work for basic sequences (x n ) ∞ n=1 whose biorthogonal functionals extend to functionals (x * n ) ∞ n=1 defined on the whole space X in such a way that condition (3.7) holds.…”

Sparse approximation using new greedy-like bases in superreflexive spaces

“…Based on this, one might feel tempted to conjecture that, in spite of the fact that truncation quasi-greedy bases are a weaker form of quasi-greediness, the efficiency of the greedy algorithm for the former kind of bases is the same as the efficiency we would get for the latter. There are recent results of a more qualitative nature that substantiate this guess, such as [5, Theorem 9.14], [10,Theorem 4.3], [5, Proposition 10.17(iii)], [3,Corollary 4.5] and [2, Corollary 2.6], all of which are generalizations to truncation quasi-greedy bases of results previously obtained for quasi-greedy bases. These results improve [1, Theorem 3.1], [19,Theorem 4.2], [18,Proposition 4.4], [17,Corollary 8.6] and [18,Theorem 5.4], respectively.…”

Democracy of quasi-greedy bases in -Banach spaces with applications to the efficiency of the Thresholding Greedy Algorithm in the Hardy spaces

“…We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a -Banach space for are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space for are democratic while, in contrast, no quasi-greedy basis of for is, solving thus a problem that was raised in [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided. …”