The main purpose of this paper is to study weight-semi-greedy Markushevich bases, and in particular, find conditions under which such bases are weight-almost greedy. In this context, we prove that, for a large class of weights, the two notions are equivalent. We also show that all weight semi-greedy bases are truncation quasi-greedy and weight-superdemocratic. In all of the above cases, we also bring to the context of weights the weak greedy and Chebyshev greedy algorithms—which are frequently studied in the literature on greedy approximation. In the course of our work, a new property arises naturally and its relation with squeeze symmetric and bidemocratic bases is given. In addition, we study some parameters involving the weak thresholding and Chebyshevian greedy algorithms. Finally, we give examples of conditional bases with some of the weighted greedy-type conditions we study.

Let $\mathcal {F}$ be a hereditary collection of finite subsets of $\mathbb {N}$. In this paper, we introduce and characterize $\mathcal {F}$-(almost) greedy bases. Given such a family $\mathcal {F}$, a basis $(e_n)_n$ for a Banach space X is called $\mathcal {F}$-greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$, $m \in \mathbb {N}$, and $G_m(x)$, we have

$$ \begin{align*} \|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}. \end{align*} $$

$G_m(x)$

$\mathbb {K}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

Furthermore, we provide several examples of bases that are nontrivially $\mathcal {F}$-greedy. For a countable ordinal $\alpha $, we consider the case $\mathcal {F}=\mathcal {S}_{\alpha }$, where $\mathcal {S}_{\alpha }$ is the Schreier family of order $\alpha $. We show that for each $\alpha $, there is a basis that is $\mathcal {S}_{\alpha }$-greedy but is not $\mathcal {S}_{\alpha +1}$-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha < \beta $,

$$ \begin{align*} \mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_{\alpha}\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_{\beta}\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}. \end{align*} $$

We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0< p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p({\mathbb {D}})$ for $0< p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p({\mathbb {D}}^d)$ for $d\ge 2$ 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.

The purpose of this paper is to quantify the size of the Lebesgue constants $(\boldsymbol {L}_m)_{m=1}^{\infty }$ associated with the thresholding greedy algorithm in terms of a new generation of parameters that modulate accurately some features of a general basis. This fine tuning of constants allows us to provide an answer to the question raised by Temlyakov in 2011 to find a natural sequence of greedy-type parameters for arbitrary bases in Banach (or quasi-Banach) spaces which combined linearly with the sequence of unconditionality parameters $(\boldsymbol {k}_m)_{m=1}^{\infty }$ determines the growth of $(\boldsymbol {L}_m)_{m=1}^{\infty }$ . Multiple theoretical applications and computational examples complement our study.

We prove that, given $2< p<\infty$, the Fourier coefficients of functions in $L_2(\mathbb {T}, |t|^{1-2/p}\,{\rm d}t)$ belong to $\ell _p$, and that, given $1< p<2$, the Fourier series of sequences in $\ell _p$ belong to $L_2(\mathbb {T}, \vert {t}\vert ^{2/p-1}\,{\rm d}t)$. Then, we apply these results to the study of conditional Schauder bases and conditional almost greedy bases in Banach spaces. Specifically, we prove that, for every $1< p<\infty$ and every $0\le \alpha <1$, there is a Schauder basis of $\ell _p$ whose conditionality constants grow as $(m^{\alpha })_{m=1}^{\infty }$, and there is an almost greedy basis of $\ell _p$ whose conditionality constants grow as $((\log m)^{\alpha })_{m=2}^{\infty }$.

Let $\unicode[STIX]{x1D6FA}$ be a member of a certain class of convex ellipsoids of finite/infinite type in $\mathbb{C}^{2}$. In this paper, we prove that every holomorphic function in $L^{p}(\unicode[STIX]{x1D6FA})$ can be approximated by holomorphic functions on $\bar{\unicode[STIX]{x1D6FA}}$ in $L^{p}(\unicode[STIX]{x1D6FA})$-norm, for $1\leq p<\infty$. For the case $p=\infty$, the continuity up to the boundary is additionally required. The proof is based on $L^{p}$ bounds in the additive Cousin problem.

We consider two problems concerning Kolmogorov widths of compacts in Banach spaces. The first problem is devoted to relations between the asymptotic behavior of the sequence of n-widths of a compact and of its projections onto a subspace of codimension one. The second problem is devoted to comparison of the sequence of n-widths of a compact in a Banach space 𝒴 and of the sequence of n-widths of the same compact in other Banach spaces containing 𝒴 as a subspace.

We establish the existence and approximation of solutions to the operator inclusion y ∈ Ty for deterministic and random cases for a nonexpansive and *-nonexpansive multivalued mapping T defined on a closed bounded (not necessarily convex) subset C of a Banach space. We also prover random fixed points and approximation results for*-nonexpansive random operators defined on an unbounded subject C of a uniformly convex Banach space.

We prove that if X is a complex strictly monotone sequence space with 1-unconditional basis, Y ⊆ X has no bands isometric to ℓ22 and Y is the range of norm-one projection from X, then Y is a closed linear span a family of mutually disjoint vectors in X.

We completely characterize 1-complemented subspaces and norm-one projections in complex spaces ℓp(ℓq) for 1 ≤ p,q > ∞.

Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are 1-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space X is not isomorphic to ℓp for some 1 ≤ p,q > ∞ then the only subspaces of X which are 1-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.