Embeddings and Lebesgue-Type Inequalities for the Greedy Algorithm in Banach Spaces

Abstract

We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of non necessarily quasi-greedy bases.

Greedy approximation for biorthogonal systems in quasi-Banach spaces

“…Delving deeper into the construction of the space, the authors of [22] proved that B e is not a quasi-greedy basis for KT[ 1,q , u]. This result can also be derived from our next general theorem.…”

“…The example in [58, §3.3] is the case KT[ 2 ], while KT[ p ], 1 < p < ∞, was later considered in [47]. The case in which X is a Lorentz space was studied in [22]. We will refer to this method for building quasi-Banach spaces as the KT-method.…”

“…The examples of non-quasi-greedy bases obtained in [22,Lemma 8.13] can be alternatively constructed combining Theorem 11.34 with Proposition 11.35. Next we exhibit an example, constructed using the KT-method from a space for which the operator T η is unbounded, of a non-quasi-greedy basis that can be squeezed between 1 and 1,∞ .…”

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

“…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]).…”

A remark on approximation with polynomials and greedy bases

“…). In[4, Section 8.2] it is shown that h * l (m) ≈ m. On the other hand, in the same reference it is proved that the dual space X * with the corresponding dual basis B * satisfies h *…”