Building highly conditional almost greedy and quasi-greedy bases in Banach spaces

Abstract

It is known that for a conditional quasi-greedy basis B in a Banach space X, the associated sequence (k m [B]) ∞ m=1 of its conditionality constants verifies the estimate k m [B] = O(log m) and that if the reverse inequality log m = O(k m [B]) holds then X is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate k m [B] = O(log m) 1−ǫ for some ǫ > 0. However, in the existing literature one finds very few instances of spaces possessing quasigreedy basis with conditionality constants "as large as possible." Our goal in this article is to fill this gap. To that end we enhance and exploit a technique developed by Dilworth et al. in [16] and craft a wealth of new examples of both non-superreflexive classical Banach spaces having quasi-greedy bases B with k m [B] = O(log m) and superreflexive classical Banach spaces having for every ǫ > 0 quasi-greedy bases B with k m [B] = O(log m) 1−ǫ . Moreover, in most cases those bases will be almost greedy. 2010 Mathematics Subject Classification. 46B15, 41A65.

Greedy approximation for biorthogonal systems in quasi-Banach spaces

“…We emphasize that (9.8) is considered by some authors as a condition which ensures in a certain sense the optimality of the compression algorithms with respect to the basis (see [40]). We refer the reader to [9,20,33] for the uses of this type of embeddings in the study of non-linear approximation in Banach spaces with respect to bases.…”

“…Proposition 11.15 (cf. [9,Proposition 4.21]). Let (X n ) ∞ n=1 be a sequence of finitedimensional quasi-Banach spaces, and let…”

Non-superreflexivity of Garling sequence spaces and applications to the existence of special types of conditional bases

“…One way to specify a special property on conditional bases is precisely by quantifying their conditionality. In order to do that we consider the sequences has shown to be in some settings a more accurate tool for studying conditional bases (see also [3] m, for m ∈ N. Conversely, it is known (see [8,Theorem 3.5]) that X is not superreflexive if and only if there is a basic sequence B ′ in X with m L m [B ′ , X] for m ∈ N. Hence it is natural to wonder if, being non-superreflexive, g(w, p) will possess not only a basic sequence but a basis of the whole space with this property. The answer is positive as we next show.…”

“…e j is a basis for c 0 such that span(s j : 1 ≤ j ≤ n) = ℓ n ∞ for all n ∈ N with L m [S] ≈ m for m ∈ N (see, e.g., [3,Lemma 4.9]). Hence,…”

“…Almost greedy bases enjoy the property of being democratic (see [14,Theorem 3.3]), i.e., there is a sequence (λ m ) ∞ m=1 such that for any finite subset A of N, When a basis B of a Banach space X is almost greedy, the size of the members of the sequence (k m [B, X]) ∞ m=1 is controlled by a slowly growing function to the extent that (see [ [17] and has been given continuity through several papers and authors (see [3,4,8,11,18]).…”