A remark on approximation with polynomials and greedy bases

We investigate properties of the $m$-th error of approximation by polynomials with constant coefficients $\mathcal{D}_{m}(x)$ and with modulus-constant coefficients $\mathcal{D}_{m}^{\ast}(x)$ introduced by Bern\'a and Blasco (2016) to study greedy bases in Banach spaces. We characterize when $\liminf_{m}{\mathcal{D}_{m}(x)}$ and $\liminf_{m}{\mathcal{D}_{m}^*(x)}$ are equivalent to $\| x\|$ in terms of the democracy and superdemocracy functions, and provide sufficient conditions ensuring that $\lim_{m}{\mathcal{D}_{m}^*(x)} = \lim_{m}{\mathcal{D}_{m}(x)} = \| x\|$, extending previous very particular results

Extensions of greedy-like bases for sequences with gaps

In [17], T. Oikhberg introduced and studied variants of the greedy and weak greedy algorithms for sequences with gaps. In this paper, we continue the study of these algorithms, extending the notions of some greedy-like bases and of several properties generally studied in connection with them to the context of sequences with gaps. A key classification of these sequences distinguishes between bounded gaps and arbitrarily large ones. We establish several equivalences for sequences in the first of these classes, and provide examples showing that they do not hold for sequences in the second one.Comment: Fixed errors in Lemma 4.5 and Proposition 4.6. Corrected information on the funding of the first autho

Greedy approximation for sequences with gaps

In this paper, we establish new advances in the theory started by T. Oikhberg in [15] where the author joins greedy approximation theory with the use of sequences with gaps. Concretely, we address and partially answer three open questions related to quasi-greedy bases for sequences with gaps posed in [15, Section 6].Comment: 18 pages; typos correcte