Characterization of weight-semi-greedy bases

TL;DR

This paper extends the characterization of weight-semi-greedy bases by removing the finite cotype condition and relaxing the Schauder basis requirement using $ ho$-admissibility, advancing greedy approximation theory.

Contribution

It proves that weight-semi-greediness and weight-almost-greediness are equivalent without the finite cotype assumption and introduces $ ho$-admissibility to relax basis conditions.

Findings

Removed the finite cotype condition in the characterization.

Established equivalence of w-semi-greedy and w-almost-greedy bases.

Introduced $ ho$-admissibility to relax basis assumptions.

Abstract

One classical result in greedy approximation theory is that almost-greedy and semi-greedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by S. J. Dilworth, N. J. Kalton and D. Kutzarova in [7] and, recently, the first author in [2] proved that the condition of finite cotype can be removed in this result. In [11], the authors extend the notion of semi-greediness to the context of weights and proved the following: if w is a weight and $\mathcal B$ is a Schauder basis in a Banach space $\mathbb X$ with finite cotype, then w-semi-greediness and w-almost-greediness are equivalent notions. In this paper, we prove the same characterization but removing the condition of finite cotype and, also, we try to relax the condition of Schauder in the characterization of semi-greediness using the $\rho$-admissibility, notion introduced recently in [4].

Equations

$${\mathbb{N}}^m=\{A\subset \mathbb{N}:|A|=m\},{\mathbb{N}}^{<\infty }={\cup }_{m=0}^{\infty }{\mathbb{N}}^m,$$

$$\underset{n\in \Lambda }{\min }|e_n^{\ast }(x)|\ge \underset{n\notin \Lambda }{\max }|e_n^{\ast }(x)|.$$

$${\mathcal{G}}_m[\mathcal{B},\mathbb{X}](x)={\mathcal{G}}_m(x)=\sum _{j=1}^m{e}_{\rho (j)}^{\ast }(x)e_{\rho (j)},$$

$$\Vert x-{\mathcal{G}}_m(x)\Vert \le C\Vert x\Vert ,\forall x\in \mathbb{X},\forall m\in \mathbb{N}.$$

$$\underset{m\to +\infty }{\lim }\Vert x-{\mathcal{G}}_m(x)\Vert =0,\forall x\in \mathbb{X}.$$

$$\Vert x-{\mathcal{G}}_m(x)\Vert \le C{\sigma }_m(x),\forall x\in \mathbb{X},\forall m\in \mathbb{N},$$

$${\sigma }_m(x,\mathcal{B}{)}_{\mathbb{X}}={\sigma }_m(x):=\inf \left\{\Vert x-\sum _{n\in A}{a}_n{e}_n\Vert :a_n\in \mathbb{F},A\subset \mathbb{N},|A|\le m\right\}.$$

$$\Vert x-{\mathcal{G}}_m(x)\Vert \le C\inf \left\{\Vert x-P_A(x)\Vert :|A|=m\right\}.$$

$$\Vert x-{\mathcal{G}}_m(x)\Vert \le C{\tilde{\sigma }}_m(x),\forall m\in \mathbb{N},\forall x\in \mathbb{X},$$

$${\tilde{\sigma }}_m(x,\mathcal{B}{)}_{\mathbb{X}}={\tilde{\sigma }}_m(x):=\inf \left\{\Vert x-P_A(x)\Vert :A\subset \mathbb{N},|A|\le m\right\}.$$

$${\tilde{\sigma }}_{\delta }^w(x,\mathcal{B}{)}_{\mathbb{X}}={\tilde{\sigma }}_{\delta }^w(x):=\inf \{\Vert x-P_A(x)\Vert :A\in {\mathbb{N}}^{<\infty },w(A)\le \delta \}.$$

$$\Vert x-{\mathcal{G}}_m(x)\Vert \le C{\tilde{\sigma }}_{w(A_m(x))}^w(x),\forall x\in \mathbb{X},\forall m\in \mathbb{N}.$$

$$\Vert 1_{\epsilon A}\Vert \le C\Vert 1_{{\epsilon }^{\prime }B}\Vert ,$$

$$C_a\le 8C_q^4{C}_d+C_q+1.$$

$$\max \{C_q,C_d\}\le C_a.$$

$$\max \{C_q,C_{sd}\}\le C_a,C_s\le 2\kappa C_a,$$

$$C_a\le C_q+2C_q{C}_{sd}.$$

$$\Vert x-{\mathcal{C}\mathcal{G}}_m(x)\Vert =\min \left\{\Vert x-\sum _{n\in A_m(x)}{a}_n{e}_n\Vert :a_n\in \mathbb{F}\right\}.$$

$$\Vert x-{\mathcal{C}\mathcal{G}}_m(x)\Vert \le C{\sigma }_m(x),\forall x\in \mathbb{X},\forall m\in \mathbb{N}.$$

$$\Vert x-{\mathcal{C}\mathcal{G}}_m(x)\Vert \le C{\sigma }_{w(A_m(x))}^w(x),\forall x\in \mathbb{X},\forall m\in \mathbb{N},$$

$${\sigma }_{\delta }^w(x,\mathcal{B}{)}_{\mathbb{X}}={\sigma }_{\delta }^w(x):=\inf \left\{\Vert x-\sum _{n\in A}{a}_n{e}_n\Vert :A\in {\mathbb{N}}^{<\infty },w(A)\le \delta ,a_n\in \mathbb{F}\right\}.$$

$$C_q\le C_{sg}{\mathfrak{K}}_b(1+(1+{\mathfrak{K}}_b)C_{sg}+c_2^2),C_s\le {\mathfrak{K}}_b{C}_{sg}((1+{\mathfrak{K}}_b)C_{sg}+c_2^2).$$

$$C_{sg}\le C_q+4C_q{C}_s.$$

$$C_{sg}\le C_q+4C_q^2{C}_{sd}.$$

$$\text{co}(\{1_{\epsilon A}:|\epsilon |=1\})=\left\{\sum _{n\in A}{z}_n{e}_n:|z_n|\le 1\right\},$$

$$T_{\alpha }(z)=\alpha \mathrm{sgn}(z),|z|>\alpha ,T_{\alpha }(z)=z,|z|\le \alpha .$$

$$T_{\alpha }(x):=\sum _{i=1}^{\infty }{T}_{\alpha }(e_i^{\ast }(x))e_i=\alpha 1_{\epsilon {\Gamma }_{\alpha }}+P_{{\Gamma }_{\alpha }^c}(x),$$

$$\Vert T_{\alpha }(x)\Vert \le C_q\Vert x\Vert ,\forall x\in \mathbb{X}.$$

$$\underset{|\epsilon |=1}{\max }\Vert 1_{\epsilon A}\Vert \le C_{sg}{c}_2(1+c_2^2).$$

$$\underset{|\epsilon |=1}{\max }\Vert 1_{\epsilon A}\Vert \le c_2{C}_{sd}.$$

Full text

†† 2000 Mathematics Subject Classification. 46B15, 41A65.

Key words and phrases: thresholding greedy algorithm, weight-almost-greedy bases, semi-greedy bases.

The first author was supported by a PhD fellowship of the program ”Ayudas para contratos predoctorales para la formación de doctores 2017” (MINECO, Spain) and the grants MTM-2016-76566-P (MINECO, Spain) and 19368/PI/14 (Fundación Séneca, Región de Murcia, Spain).

1. INTRODUCTION

Let $(\mathbb{X},\|\cdot\|)$ be a Banach space over $\mathbb{F}$ ($\mathbb{F}$ denotes the real field $\mathbb{R}$ or the complex field $\mathbb{C}$) and let $\mathcal{B}=(e_{n})_{n=1}^{\infty}$ be a semi-normalized Markushevich basis of $\mathbb{X}$ with biorthogonal functionals $(e_{n}^{*})_{n=1}^{\infty}$, that is:

• a)

$0<c_{1}:=\inf_{n}\{\|e_{n}\|,\|e_{n}^{*}\|\}\leq\sup_{n}\{\|e_{n}\|,\|e_{n}^{*}\|\}=:c_{2}<\infty$.

• b)

$e_{j}^{*}(e_{i})=\delta_{j,i}$.

• c)

$\mathbb{X}=\overline{span\{e_{i}:i\in\mathbb{N}\}}$.

• d)

If $e_{j}^{*}(x)=0$ for all $j\in\mathbb{N}$, then $x=0$.

We say that $\mathcal{B}$ is a semi-normalized strong Markushevich basis if a)-d) are satisfied and

• e)

$\overline{span\{e_{i}:i\in A\}}=\{x\in\mathbb{X}:e_{j}^{*}(x)=0\;\forall j\not\in A\}$.

Throughout the paper, we will refer to a semi-normalized strong Markushevich basis $\mathcal{B}$ as a basis. Also, we will say that $\mathcal{B}$ is a Schauder basis if $\mathcal{B}$ is a basis in the above sense and if

• f)

$\mathfrak{K}_{b}:=\sup_{m}\|P_{m}\|<\infty$, where $P_{m}(\sum_{j}a_{j}e_{j})=\sum_{j=1}^{m}a_{j}e_{j}$ is the $m$-th partial sum.

As usual $\mathrm{supp\,}(x)=\{n\in\mathbb{N}:e_{n}^{*}(x)\neq 0\}$, given a finite set $A\subset\mathbb{N}$, $|A|$ denotes the cardinality of the set $A$,

[TABLE]

$P_{A}$ is the projection operator, that is, $P_{A}(\sum_{j}a_{j}e_{j})=\sum_{j\in A}a_{j}e_{j}$, $P_{A^{c}}=\text{I}_{\mathbb{X}}-P_{A}$, $\mathbf{1}_{\varepsilon A}=\sum_{n\in A}\varepsilon_{n}e_{n}$ where $\varepsilon=(\varepsilon_{n})_{n}$ is a sign, that is, $|\varepsilon_{n}|=1$ (where $\varepsilon_{n}$ could be real or complex), $\mathbf{1}_{A}=\sum_{n\in A}e_{n}$ and, for $A,B$ finite sets, $A<B$ means that $\max_{i\in A}i<\min_{j\in B}j$.

In the year 1999 ([13]), S. V. Konyagin and V. N. Temlyakov introduced the Thresholding Greedy Algorithm (TGA): given a basis $\mathcal{B}$ in a Banach space and $x\sim\sum_{i=1}^{\infty}e_{i}^{*}(x)e_{i}\in\mathbb{X}$, the collection $(G_{m}(x))_{m=1}^{\infty}$ is a greedy approximation of x, where $G_{m}(x)=\sum_{n\in\Lambda}e_{n}^{*}(x)e_{n}$, and the set $\Lambda$ is any set of cardinality $m$ satisfying the following condition:

[TABLE]

The set $\Lambda$ is called a greedy set.

In general, $(G_{m}(x))_{m}$ can not be unique since we can have some coefficients with the modulus. Hence, we consider the natural ordering existing in $\mathbb{N}$ to solve this fact. Define the natural greedy ordering for $x$ as the map $\rho:\mathbb{N}\longrightarrow\mathbb{N}$ such that $\mathrm{supp\,}(x)\subset\rho(\mathbb{N})$ and so that if $j<k$ then either $|e_{\rho(j)}^{*}(x)|>|e_{\rho(k)}^{*}(x)|$ or $|e_{\rho(j)}^{*}(x)|=|e_{\rho(k)}^{*}(x)|$ and $\rho(j)<\rho(k)$. The $m$-th greedy sum of $x$ is

[TABLE]

and the sequence of maps $(\mathcal{G}_{m})_{m=1}^{\infty}$ is known as the Thresholding Greedy Algorithm associated to $\mathcal{B}$ in $\mathbb{X}$. Of course, we can write $\mathcal{G}_{m}(x)=\sum_{k\in A_{m}(x)}e_{k}^{*}(x)e_{k}$, where $A_{m}(x)=\{\rho(n):n\leq m\}$ is the greedy set of $x$ with cardinality $m$: $\min_{k\in A_{m}(x)}|e_{k}^{*}(x)|\geq\max_{k\notin A_{m}(x)}|e_{k}^{*}(x)|$.

The terminology of the Thresholding Greedy Algorithm can be found, for instance, in [8, 10, 13, 15]. In [13], the authors defined quasi-greedy bases:

Definition 1.1**.**

We say that $\mathcal{B}$ in a Banach space $\mathbb{X}$ is quasi-greedy if there exists a positive constant $C$ such that

[TABLE]

The least constant that verifies (1) is denoted by $C_{q}$ and we say that $\mathcal{B}$ is $C_{q}$-quasi-greedy.

In [15], P. Wojtaszczyk proved that a basis $\mathcal{B}$ is quasi-greedy if and only if

[TABLE]

Then, from the point of view of the approximation, quasi-greediness is the minimum condition that guarantees the convergence of the TGA, but there are others greedy-type bases that we need to attack the problem that we want to study. On the one hand, we have greedy bases, notion introduced by S. V. Konyagin and V. N. Temlyakov in [13]. We say that $\mathcal{B}$ is greedy if there exists a positive constant $C$ such that

[TABLE]

where

[TABLE]

In this paper we focus our attention on almost-greedy bases, notion introduced by S. J. Dilworth, N. J. Kalton, D. Kutzarova and V. N. Temlyakov in [8]. We say that $\mathcal{B}$ is almost-greedy if there exists a positive constant $C$ such that

[TABLE]

Later on, in [1], the authors proved that the notion of almost-greediness is equivalent to

[TABLE]

where $C$ is the same constant than in the definition of almost-greediness and

[TABLE]

In [8], the authors proved that a basis is almost-greedy if and only if the basis is quasi-greedy and democratic (that is, there exists a positive constant $C$ such that $\|\mathbf{1}_{A}\|\leq C\|\mathbf{1}_{B}\|$, for any $A,B\in\mathbb{N}^{<\infty}$ and $|A|\leq|B|$). Thanks to a work of G. Kerkyacharian, D. Picard and V. N. Temlyakov ([12]) motivated by the work of A. Cohen, R. A. DeVore and R. Hochmuth ([6]), we consider a generalization of almost-greedy bases. We define a weight $w$ as any collection $w=(w_{i})_{i=1}^{\infty}\in(0,\infty)^{\mathbb{N}}$. If $A\subset\mathbb{N}$, $w(A)=\sum_{i\in A}w_{i}$ denote the $w$-measure of $A$. We define the error

[TABLE]

Definition 1.2** ([11]).**

We say that $\mathcal{B}$ is $w$-almost-greedy if there exists a positive constant $C$ such that

[TABLE]

We denote by $C_{a}$ the least constant that verifies (2) and we say that $\mathcal{B}$ is $C_{a}$-$w$-almost-greedy.

Definition 1.3**.**

We say that $\mathcal{B}$ is $w$-super-democratic if there exists a positive constant $C$ such that

[TABLE]

for any pair of sets $A,B\in\mathbb{N}^{<\infty}$ with $w(A)\leq w(B)$ and for all pair of signs $\varepsilon,\varepsilon^{\prime}$. We denote by $C_{s}$ the least constant that verifies (3) and we say that $\mathcal{B}$ is $C_{s}$-$w$-super-democratic.

If in (3) we add the condition $A\cap B=\emptyset$, we say that $\mathcal{B}$ is $C_{sd}$-$w$-disjoint-super-democratic. If $\varepsilon\equiv\varepsilon^{\prime}\equiv 1$ in (3), we say that $\mathcal{B}$ is $C_{d}$-$w$-democratic.

In [5], the authors gave the following characterization of $w$-almost-greedy bases.

Theorem 1.4** ([5]).**

Assume that $\mathcal{B}$ is a basis in a Banach space $\mathbb{X}$.

• •

If $\mathcal{B}$ is $C_{q}$-quasi-greedy and $C_{d}$-$w$-democratic, then $\mathcal{B}$ is $C_{a}$-$w$-almost-greedy with

[TABLE]

• •

If $\mathcal{B}$ is $C_{a}$-$w$-almost-greedy, then $\mathcal{B}$ is $C_{q}$-quasi-greedy and $C_{d}$-$w$-democratic with

[TABLE]

Here, we reformulate the above theorem using the $w$-disjoint-super-democracy.

Theorem 1.5**.**

Assume that $\mathcal{B}$ is a basis in a Banach space $\mathbb{X}$.

• a)

If $\mathcal{B}$ is $C_{a}$-$w$-almost-greedy, then $\mathcal{B}$ is $C_{q}$-quasi-greedy, $C_{sd}$-$w$-disjoint-super-democratic and $C_{s}$-$w$-super-democratic, with

[TABLE]

where $\kappa=1$ if $\mathbb{F}=\mathbb{R}$ and $\kappa=2$ if $\mathbb{F}=\mathbb{C}$.

• b)

If $\mathcal{B}$ is $C_{q}$-quasi-greedy and $C_{sd}$-$w$-disjoint-super-democratic, then the basis is $C_{a}$-$w$-almost-greedy with

[TABLE]

Remark 1.6**.**

Theorem 1.5 shows that $C_{a}=O(C_{q}C_{sd})$ instead of $C_{a}=O(C_{q}^{4}C_{d})$ as Theorem 1.4 shows. This is an improvement respect to the order of the constants as we can see using the Proposition 6.1 of Section 6.

In [7], S. J. Dilworth, N. J. Kalton and D. Kutzarova study an equivalence of almost-greedy bases from a new point of view to improve the rate of convergence. For this equivalence, the authors introduced the notion of semi-greedy bases. Let $A_{m}(x)$ the greedy set of $x$ of cardinality $m$. Define the $m$-th Chebyshev-greedy sum as any element $\mathcal{CG}_{m}(x)\in span\{e_{i}:i\in A_{m}(x)\}$ such that

[TABLE]

The collection $\{\mathcal{CG}_{m}\}_{m=1}^{\infty}$ is the Thresholding Chebyshev Greedy Algorithm (TCGA). A basis $\mathcal{B}$ is semi-greedy if there exists a positive constant $C$ such that

[TABLE]

The first authors that studied the relation between semi-greediness and almost-greediness were S. J. Dilworth, N. J. Kalton and D. Kutzarova in[7].

Theorem 1.7**.**

[7, Theorem 3.6]** Assume that $\mathcal{B}$ is a Schauder basis in a Banach space $\mathbb{X}$ with finite cotype. Then, $\mathcal{B}$ is semi-greedy if and only if $\mathcal{B}$ is almost-greedy.

Recently, the first author proved in [2] that the condition of finite cotype can be removed.

Theorem 1.8**.**

[2, Theorem 1.10]** A Schauder basis $\mathcal{B}$ in a Banach space $\mathbb{X}$ is semi-greedy if and only if $\mathcal{B}$ is almost-greedy.

Focusing our attention in the weighted case, in [11], the authors extend the definition of semi-greediness.

Definition 1.9** ([11]).**

We say that $\mathcal{B}$ is $w$-semi-greedy if there exists a positive constant $C$ such that

[TABLE]

where

[TABLE]

We denote by $C_{sg}$ the least constant that verifies (4) and we say that $\mathcal{B}$ is $C_{sg}$-$w$-semi-greedy.

Here, we study the equivalence between $w$-semi-greediness and $w$-almost-greediness removing the condition of finite cotype following the spirit of [2] in the world of Schauder bases in Banach spaces.

Remark 1.10**.**

For $w=(1,1,...)$, that is, $w(A)=|A|$, we recover the definitions of super-democracy, disjoint-super-democracy, almost-greediness and semi-greediness.

Theorem 1.11**.**

Assume that $\mathcal{B}$ is a Schauder basis in a Banach space.

• a)

If $\mathcal{B}$ is $C_{sg}$-$w$-semi-greedy, then $\mathcal{B}$ is $C_{q}$-quasi-greedy and $C_{s}$-$w$-super-democratic with

[TABLE]

• b)

If $\mathcal{B}$ is $C_{q}$-quasi-greedy and $C_{s}$-$w$-super-democratic, then $\mathcal{B}$ is $C_{sg}$-$w$-semi-greedy with

[TABLE]

• c)

If $\mathcal{B}$ is $C_{q}$-quasi-greedy and $C_{sd}$-$w$-disjoint-super-democratic, then $\mathcal{B}$ is $C_{sg}$-$w$-semi-greedy with

[TABLE]

Remark 1.12**.**

In [11], the authors proved that $C_{q}$-quasi-greediness and $C_{d}$-$w$-democracy implies $C_{sg}$-$w$-semi-greediness with $C_{sg}=(C_{q}^{3}C_{d})$. Our Theorem 1.11 shows that we can get $C_{sg}=O(C_{q}^{2}C_{sd})$ or $C_{sg}=O(C_{q}C_{s})$. As in the Remark 1.6, the last bound is an improvement respect to the bound $C_{sg}=(C_{q}^{3}C_{d})$ using Proposition 6.1.

Hence, for $w=(1,1,...)$, we recover the result proved in [2] as we say in the following corollary.

Corollary 1.13**.**

Assume that $\mathcal{B}$ is a Schauder basis in a Banach space. The following are equivalent:

• a)

$\mathcal{B}$* is semi-greedy.*

• b)

$\mathcal{B}$* is quasi-greedy and super-democratic.*

• c)

$\mathcal{B}$* is quasi-greedy and disjoint-super-democratic.*

The structure of the paper is the following: in Section 2, we write and study some preliminary results that we will use in the proof of the main results. In Section 3, we improve a recent result proved in [5] establishing the relation between $w$-semi-greediness and the so called Property (C). In Section 4, we prove Theorems 1.5 and 1.11. In Section 5, we relax the condition of Schauder bases in the characterization of semi-greediness and in Section 6, we show the relation between $w$-super-democracy and $w$-disjoint-super-democracy for any weight $w$ and we add some open questions.

2. Preliminary results

To prove the main theorems of this paper, we need the followings results concerning convexity, the truncation operator and some properties of weights.

Lemma 2.1**.**

[3, Lemma 2.7]** For every finite set $A\subset\mathbb{N}$, we have

[TABLE]

where $\text{co}(S)=\{\sum_{j=1}^{n}\alpha_{j}x_{j}:x_{j}\in S,0\leq\alpha_{j}\leq 1,\sum_{j=1}^{n}\alpha_{j}=1,n\in\mathbb{N}\}$.

For each $\alpha>0$, we define the truncation function of $z\in\mathbb{F}$ as

[TABLE]

We can extend $T_{\alpha}$ to an operator in $\mathbb{X}$ by $T_{\alpha}(x)\sim\sum_{i=1}^{\infty}T_{\alpha}(e_{i}^{*}(x))e_{i}$, that is,

[TABLE]

where $\Gamma_{\alpha}=\{n:|e_{n}^{*}(x)|>\alpha\}$ and $\varepsilon_{j}=\mathrm{sgn\,}(e_{j}^{*}(x))$ with $j\in\Gamma_{\alpha}$. Hence, this is a well-defined operator for all $x\in\mathbb{X}$ since $\Gamma_{\alpha}$ is a finite set.

This operator was introduced in [7] to show the equivalence between almost-greediness and semi-greediness and they proved that for quasi-greedy bases, this operator is uniformly bounded. Also, in [3], the authors showed the same result but with a slight improvement of the boundedness constant.

Proposition 2.2**.**

[3, Lemma 2.5]** Assume that $\mathcal{B}$ is $C_{q}$-quasi-greedy basis in a Banach space $\mathbb{X}$. Then, for every $\alpha>0$,

[TABLE]

The last one result that we will use is related to weights.

Proposition 2.3**.**

Let $\mathcal{B}$ be a basis in a Banach space $\mathbb{X}$.

1. i)

Assume that $w(A)\leq\lim\sup_{n\rightarrow\infty}w_{n}$. If $\mathcal{B}$ is $C_{sg}$-$w$-semi-greedy, then

[TABLE]

If in addition $\mathcal{B}$ is Schauder, it is possible to get that $\max_{|\varepsilon|=1}\|\mathbf{1}_{\varepsilon A}\|\leq 2C_{sg}\mathfrak{K}_{b}$.

If $\mathcal{B}$ is $C_{sd}$-$w$-disjoint-super-democratic, then

[TABLE] 2. ii)

If $\mathcal{B}$ is $C_{sg}$-$w$-semi-greedy or $C_{sd}$-$w$-disjoint-super-democratic and $\sup_{n}w_{n}=\infty$ or $\sum_{n}w_{n}<\infty$, then $\mathcal{B}$ is equivalent to the $c_{0}$-basis. 3. iii)

If $\mathcal{B}$ is $C_{sg}$-$w$-semi-greedy or $C_{sd}$-$w$-disjoint-super-democratic and $\inf_{n}w_{n}=0$, $\mathcal{B}$ contains a subsequence equivalent to the $c_{0}$-basis.

Proof.

The case of $w$-disjoint-super-democracy is proved in [5] assuming $w$-super-democracy, but the same proof is also valid for $w$-disjoint-super-democracy. The case of $w$-semi-greediness is proved in [11] but assuming that the basis is Schauder. Here, we show this result for general bases using similar ideas.

• i)

Find $n\in\mathbb{N}\setminus A$ such that $w(A)<w_{n}$. Hence, if we consider the element $x:=\mathbf{1}_{\varepsilon A}+(1+\delta)e_{n}$ with $\delta>0$ and $\varepsilon$ a sign, applying the TCGA with $\{n\}$ the greedy set of $x$,

[TABLE]

Taking limits when $\delta$ goes to [math], $\|\mathbf{1}_{\varepsilon A}\|\leq C_{s}gc_{2}(1+c_{2}^{2})$.

• ii)

If $\sup_{n}w_{n}<\infty$, by a), $\|\mathbf{1}_{\varepsilon A}\|\leq C_{sg}c_{2}(1+c_{2}^{2})$ for any finite set $A$ and for all possible sign $\varepsilon$. Hence, the basis is equivalent to the $c_{0}$-basis. If $\sum_{n}w_{n}<\infty$, we can choose a number $m\in\mathbb{N}$ such that $\sum_{i=m+1}^{\infty}w_{n}<w_{1}$. We can assume that $\min_{i\in A}i\geq m+1$. Hence, with the same procedure as in a), $\|\mathbf{1}_{\varepsilon A}\|\leq C_{sg}c_{2}+C_{sg}c_{2}^{3}$.

• iii)

Choosing a subsequence $(n_{k})_{k}$ such that $\sum_{k=1}^{\infty}w_{n_{k}}<\infty$, we apply the item b) and we have that $(e_{n_{k}})_{k}$ is equivalent to the $c_{0}$-basis.

∎

3. The Property (C)

It is well known that one of the properties that quasi-greediness preserves is the so called Property (C).

Definition 3.1**.**

We say that a basis $\mathcal{B}$ in a Banach space $\mathbb{X}$ has the Property (C) if for any $x\in\mathbb{X}$ and $G$ a greedy set of $x$, there exists a positive constant $C$ such that

[TABLE]

We denote by $C_{u}$ the least constant that verifies the above inequality and we say that $\mathcal{B}$ has the $C_{u}$-Property (C).

Although quasi-greediness implies Property (C), the converse is false as [3, Example 5.5] shows. The following is a known inequality from [8].

Lemma 3.2**.**

[8, Lemma 2.2]** If $\mathcal{B}$ is a $C_{q}$-quasi-greedy basis in $\mathbb{X}$, then, for all $x\in\mathbb{X}$ and for all greedy set $G$ of $x$, we have

[TABLE]

where $\varepsilon=\{\mathrm{sgn\,}(e_{j}^{*}(x))\}$.

Remark 3.3**.**

Since any quasi-greedy basis is unconditional for constant coefficients (see [15]), that is, $\|\mathbf{1}_{\varepsilon A}\|\approx\|\mathbf{1}_{A}\|$ for any sign $\varepsilon$ and any finite set $A$, from Lemma 3.2 we can deduce that any quasi-greedy basis has the Property (C).

[5, Proposition 4.10] shows that any Schauder and $w$-semi-greedy basis satisfies the Property (C) assuming that $0<\inf_{n}w_{n}\leq\sup_{n}w_{n}<\infty$. Here, we extend this result proving that $w$-semi-greediness implies the Property (C) for any weight $w$. Before that, we study the following lemma.

Lemma 3.4**.**

Assume that $\mathcal{B}$ is a Schauder basis with constant $\mathfrak{K}_{b}$ and $C_{sg}$-$w$-semi-greedy in a Banach space $\mathbb{X}$. Then, if $F$ is a set such that $F>\mathrm{supp\,}(x)$ and $w(F)\leq w(G)$ for some greedy set $G$, hence

[TABLE]

Proof.

Take $x\in\mathbb{X}$, $G$ a greedy set of $x$ and a set $F$ and a sign $\eta$ as in the statement of the lemma. Define the following element $y:=\min_{i\in G}|e_{i}^{*}(x)|\mathbf{1}_{\eta F}+P_{G^{c}}(x)+\sum_{i\in G}(e_{i}^{*}(x)+\delta\varepsilon_{i})e_{i}$, where $\delta>0$ and $\varepsilon\equiv\{\mathrm{sgn\,}(e_{j}^{*}(x))\}$. Then, for the element $y$, the set $G$ is a greedy set. Hence, applying the TCGA,

[TABLE]

Taking limits when $\delta$ goes to [math], we obtain the result. ∎

Proposition 3.5**.**

Let $\mathcal{B}$ be a Schauder basis with constant $\mathfrak{K}_{b}$ in a Banach space $\mathbb{X}$. If $\mathcal{B}$ is $C_{sg}$-$w$-semi-greedy, then $\mathcal{B}$ has the $C_{u}$-Property (C) with $C_{u}\leq\mathfrak{K}_{b}C_{sg}((1+\mathfrak{K}_{b})C_{sg}+c_{2}^{2})$.

Proof.

Take $x\in\mathbb{X}$ and let $G$ be a greedy set of $x$, $\alpha=\min_{i\in G}|e_{i}^{*}(x)|$ and $|\varepsilon|=1$. We consider different cases (these cases are inspired by [11]).

Case 1: $\sum_{n=1}^{\infty}w_{n}=\infty$ and $\sup_{n}w_{n}<\infty$.

Case 1.1: If $w(G)>\lim\sup_{n\rightarrow\infty}w_{n}$, since $\sum_{n}w_{n}=\infty$, we can choose $E$ and $n_{0}\in\mathbb{N}$ with $E>\mathrm{supp\,}(x)$ and $n_{0}>\max E$ such that

[TABLE]

Define then the element $y:=\alpha\mathbf{1}_{\varepsilon G}+(\alpha+\delta)\mathbf{1}_{F}$, where $\delta>0$ and $F=E\cup\{n_{0}\}$. Then, a greedy set of $y$ is $F$ and hence, applying the TCGA,

[TABLE]

Taking limits when $\delta$ goes to [math],

[TABLE]

Now, it is only necessary to estimate $\|\alpha\mathbf{1}_{E}\|$. For that, we only have to apply Lemma 3.4 and then, we obtain that

[TABLE]

Using (6) and (7), we obtain the result in this case.

Case 1.2: Now, if $w(G)\leq\lim\sup_{n\rightarrow\infty}w_{n}$, using Proposition 2.3,

[TABLE]

Hence,

[TABLE]

Case 2: If $\sum_{n}w_{n}<\infty$ or $\sup_{n}w_{n}=\infty$, using Proposition 2.3, $\mathcal{B}$ is equivalent to the $c_{0}$-basis and the result is trivial.

∎

4. Proof of the main results

Proof of Theorem 1.5.

: First, we prove the item a). Assume that $\mathcal{B}$ is $C_{a}$-$w$-almost-greedy and take two sets $A,B\in\mathbb{N}^{<\infty}$ with $w(A)\leq w(B)$ and two signs $\varepsilon,\varepsilon^{\prime}$. First, we show that

[TABLE]

Define the element $x:=\mathbf{1}_{A^{\prime}\setminus B}+\mathbf{1}_{A^{\prime}\cap B}+(1+\delta)\mathbf{1}_{B\setminus A^{\prime}}$, with $\delta>0$. Since $w(A^{\prime})\leq w(A)\leq w(B)$, then $w(A^{\prime}\setminus B)\leq w(B\setminus A^{\prime})$. Thus,

[TABLE]

Hence, taking the limits when $\delta$ goes to [math], we prove (8). Taking into account that (8) does not change if we apply the estimate for $\{\varepsilon^{\prime}_{n}e_{n}\}_{n}$ for any $|\varepsilon^{\prime}|=1$, we can assume that $\varepsilon^{\prime}\equiv 1$.

Now, to conclude the result, we realize that $\mathbf{1}_{\varepsilon A}\in 2S$ for the real case and $\mathbf{1}_{\varepsilon A}\in 4S$ for the complex case, where

[TABLE]

Hence, applying this remark in (8) and [9, Lemma 6.4],

[TABLE]

This completes the $w$-super-democracy. To show the $w$-disjoint-super-democracy, take $A,B,\varepsilon,\varepsilon^{\prime}$ as in the beginning with $A\cap B=\emptyset$. Define the element $x:=\mathbf{1}_{\varepsilon A}+(1+\delta)\mathbf{1}_{\varepsilon^{\prime}B}$ with $\delta>0$. Hence, the set $B$ is the greedy set of $x$ with cardinality $m:=|B|$. Thus,

[TABLE]

Taking limits when $\delta$ goes to [math], we obtain that $\mathcal{B}$ is $C_{sd}$-$w$-disjoint-super-democratic with $C_{sd}\leq C_{a}$.

The proof of quasi-greediness is trivial since in the definition of $w$-almost-greediness we can take $A=\emptyset$.

b) Assume now that $\mathcal{B}$ is $C_{q}$-quasi-greedy and $C_{sd}$-$w$-disjoint-super-democratic. Take $m\in\mathbb{N}$, $\mathcal{G}_{m}(x)=P_{A}(x)$ and $B$ such that $w(B)\leq w(A)$ and $\|x-P_{B}(x)\|<\tilde{\sigma}_{w(A)}^{w}(x)+\delta$ with $\delta>0$. We have the following decomposition:

[TABLE]

On the one hand, since $A\setminus B$ is a greedy set of $x-P_{B}(x)$,

[TABLE]

On the other hand, since $w(B\setminus A)\leq w(A\setminus B)$, using Lemma 2.1 and $w$-disjoint-super-democracy with $\varepsilon\equiv\{\mathrm{sgn\,}(e_{j}^{*}(x))\}$,

[TABLE]

Now, by Lemma 3.2, using that $A\setminus B$ is a greedy set of $x-P_{B}(x)$ and $\min_{j\in A\setminus B}|e_{j}^{*}(x)|=\min_{j\in A\setminus B}|e_{j}^{*}(x-P_{B}(x))|$,

[TABLE]

By (9) and (10), we obtain that $\mathcal{B}$ is $C_{a}$-$w$-almost-greedy with $C_{a}\leq C_{q}+2C_{q}C_{sd}$.

∎

Proof of Theorem 1.11.

: Assume that $\mathcal{B}$ is $C_{sg}$-$w$-semi-greedy. To exhibit the $w$-super-democracy and quasi-greediness we consider the different cases that we have considered in the Proposition 3.5.

Case 1: $\sum_{n=1}^{\infty}w_{n}=\infty$ and $\sup_{n}w_{n}<\infty$.

• •

To prove quasi-greediness, take $x\in\mathbb{X}$ such that $|\mathrm{supp\,}(x)|<\infty$, and without loss of generality we can assume that $\max_{j}|e_{j}^{*}(x)|\leq 1$, $m\in\mathbb{N}$ and consider that $w(A_{m}(x))>\lim\sup_{n\rightarrow\infty}w_{n}$. Since $\sum_{n}w_{n}=\infty$, we can choose $E$ and $n_{0}\in\mathbb{N}$ with $E>\mathrm{supp\,}(x)$ and $n_{0}>\max_{i\in E}i$ such that

[TABLE]

Set $F:=E\cup\{n_{0}\}$ and $\alpha=\min_{j\in A_{m}(x)}|e_{j}^{*}(x)|$. Define the element

[TABLE]

with $\delta>0$. Hence, the greedy set of $y$ is $F$ and then, if the scalars $(a_{n})_{n}$ are given by the TCGA,

[TABLE]

Taking limits when $\delta$ goes to [math],

[TABLE]

Of course, $\alpha\|e_{n_{0}}\|\leq c_{2}^{2}\|x\|$, so we only have to estimate $\|\alpha\mathbf{1}_{E}\|$. For that, using Lemma 3.4,

[TABLE]

Then, we have that the basis is quasi-greedy for elements with finite support with

[TABLE]

Define now $C_{1}=C_{sg}\mathfrak{K}_{b}(1+(1+\mathfrak{K}_{b})C_{sg}+c_{2}^{2})$. To show the quasi-greediness for any $x\in\mathbb{X}$, we need the following result (see [14, Lemma 2.2]): if $x\in\mathbb{X}$ and $A_{m}(x)$ is the greedy set of cardinality $m$ of $x$, then for any $\varepsilon>0$ there exists $y\in\mathbb{X}$ with $|\mathrm{supp\,}(y)|<\infty$ such that $\|x-y\|<\varepsilon$ and $A_{m}(x)=A_{m}(y)$. Using that, we proceed as follows:

[TABLE]

Taking now limits when $\varepsilon$ goes to [math], we obtain that $\mathcal{B}$ is $C_{q}$-quasi-greedy with $C_{q}\leq C_{1}$.

Now, consider that $w(A_{m}(x))\leq\lim\sup_{n\rightarrow\infty}w_{n}$. Using Proposition 2.3,

[TABLE]

Then, using convexity,

[TABLE]

Hence, $\mathcal{B}$ is quasi-greedy with $C_{q}\leq 2c_{2}C_{sg}\mathfrak{K}_{b}+1$.

• •

To show the $w$-super-democracy in this case, take $A,B\in\mathbb{N}^{<\infty}$ such that $w(A)\leq w(B)$ and two signs $\varepsilon,\varepsilon^{\prime}$. If $w(B)>\lim\sup_{n\rightarrow\infty}w_{n}$, we tan take the set $F$ as before, that is, $F=E\cup\{n_{0}\}$ such that $w(E)\leq w(B)<w(F)$, $n_{0}>\max E$ and $E>A\cup B$. Then, taking the element $x:=\mathbf{1}_{\varepsilon A}+(1+\delta)\mathbf{1}_{F}$, with $\delta>0$, the greedy set of $x$ is $F$. Using the scalars $(a_{i})_{i\in F}$ given by the TCGA, we have that

[TABLE]

Taking $\delta\searrow 0$, $\|\mathbf{1}_{\varepsilon A}\|\leq\mathfrak{K}_{b}C_{sg}(\|\mathbf{1}_{E}\|+c_{2}^{2}\|\mathbf{1}_{\eta B}\|)$. Now, as $E>B$, taking the element $x:=(1+\delta)\mathbf{1}_{\varepsilon^{\prime}B}+\mathbf{1}_{E}$, using the same ideas that before with $w(B)\geq w(E)$, we obtain that

[TABLE]

Hence, the basis is $w$-super-democratic with constant $C_{s}\leq 2\mathfrak{K}_{b}^{2}C_{sg}^{2}+\mathfrak{K}_{b}C_{sg}c_{2}^{2}$.

If $w(B)\leq\lim\sup_{n\rightarrow\infty}w_{n}$, using Proposition 2.3,

[TABLE]

Case 2: If $\sum_{n}w_{n}<\infty$ or $\sup_{n}w_{n}=\infty$, using Proposition 2.3, $\mathcal{B}$ is equivalent to the canonical basis of $c_{0}$ and the result is trivial.

The item a) is proved. Now, we show b). Assume that $\mathcal{B}$ is $C_{q}$-quasi-greedy and $C_{sd}$-$w$-disjoint-super-democratic. Take $m\in\mathbb{N}$, $\mathrm{supp\,}(\mathcal{G}_{m}(x))=A_{m}(x)$ and $z=\sum_{n\in B}a_{n}e_{n}$ such that $\|x-z\|<\sigma_{w(A_{m}(x))}^{w}(x)+\delta$ with $\delta>0$. If $\alpha=\max_{j\not\in A_{m}(x)}|e_{j}^{*}(x)|$, we take the element $\nu$ as is defined in [7]:

[TABLE]

where $y_{i}=e_{i}^{*}(x)-a_{i}$. Of course, $\nu$ satisfies that $\mathrm{supp\,}(x-\nu)\subset A_{m}(x)$ and we will prove that $\|\nu\|\leq(C_{q}+4C_{q}C_{sd})\|x-z\|$.

One the one hand, using Proposition 2.2,

[TABLE]

On the other hand, since $|e_{i}^{*}(x)-T_{\alpha}(y_{i})|\leq 2\alpha$ for all $i\in B\setminus A_{m}(x)$, using $C_{sd}$-$w$-disjoint-super-democracy with $\eta\equiv\{\mathrm{sgn\,}(e_{j}^{*}(x-z))\}$, $w(B)\leq w(A_{m}(x))$ and Lemma 2.1,

[TABLE]

Now, if we take $\Lambda:=\{n:|e_{n}^{*}(x-z)|\geq\min_{i\in A_{m}(x)\setminus B}|e_{i}^{*}(x)|,n\not\in A_{m}(x)\setminus B\}$, $C=\Lambda\cup(A_{m}(x)\setminus B)$ is a greedy set of $x-z$. Using quasi-greediness,

[TABLE]

Finally, using this fact and Proposition 3.2,

[TABLE]

Hence, the basis $\mathcal{B}$ is $C_{sg}$-$w$-semi-greedy with $C_{sg}\leq C_{q}+4C_{sd}C_{q}^{2}$ and b) is finished. Now, we proved the item c). For that, we only have to estimate the inequality (4) in a different way. Consider that the basis is $C_{q}$-quasi-greedy and $C_{s}$-$w$-super-democratic and take the sets $A_{m}(x),B,C$ and $\eta=\{\mathrm{sgn\,}(e_{j}^{*}(x-z))\}$ as in b). It is clear that $w(B\setminus A_{m}(x))\leq w(A_{m}(x)\setminus B)\leq w(C)$, hence, applying the $w$-super-democracy, Lemma 2.1 and Proposition 3.2 in (4),

[TABLE]

Thus, the basis is $C_{sg}$-$w$-semi-greedy with $C_{sg}\leq C_{q}+4C_{q}C_{s}$. This completes the proof.

∎

5. $\rho$-admissibility and semi-greediness

In the most papers where we study some characterization about greedy-type bases, the more general stage involves only the condition of (strong) Markushevich bases. Then, a natural question is if we can remove the condition of Schauder basis in Theorem 1.11. This question is so closed to the Question 1 established in [2]. Here, we present a weaker condition than Schauder to give a characterization of semi-greediness<, that is, the version of Theorem 1.11 for the weight $w=(1,1,...)$. For that purpose, we consider the following definition that we can find in [4].

Definition 5.1**.**

For $\rho\geq 1$, we say that $(e_{n})_{n=1}^{\infty}$ is $\rho$-admissible if the following holds: for each finite set $A\subset\mathbb{N}$, there exists $n_{0}=n_{0}(A)$ such that, for all sets $B$ with $\min_{i\in B}i\geq n_{0}$ and $|B|\leq|A|$,

[TABLE]

Of course, this condition is satisfied for Schauder bases, but, in fact, it is satisfied for a more general bases. We remind some classical definitions:

• •

$(e_{n})_{n=1}^{\infty}$ is weakly null if

[TABLE]

• •

Given $Y\subset\mathbb{X}^{*}$, $(e_{n})_{n=1}^{\infty}$ is $Y$-null if

[TABLE]

• •

Given $\kappa\in(0,1]$, a set $Y\subset\mathbb{X}^{*}$ is $\kappa$-norming whenever

[TABLE]

In [4], we can find the following result.

Proposition 5.2**.**

Let $\{e_{n},e_{n}^{*}\}_{n=1}^{\infty}$ be a biorthogonal system in $\mathbb{X}x\mathbb{X}^{*}$. Suppose that the sequence $\{\tilde{e}_{n}:=\|e_{n}^{*}\|e_{n}\}_{n=1}^{\infty}\subset\mathbb{X}$ is $Y$-null, for some subset $Y\subset\mathbb{X}^{*}$ wich is $\kappa$-norming. Then, $\{e_{n}\}_{n=1}^{\infty}$ is $\rho$-admissible for every $\rho>1/k$.

Some examples of bases that are not Schauder satisfying the above proposition can be found in Section 3 of [4]. One of them is the trigonometric system in $C([0,1])$. Our contribution in this case is the following theorem.

Theorem 5.3**.**

Assume that $\mathcal{B}$ is a $\rho$-admissible basis. Then, $\mathcal{B}$ is semi-greedy if and only if $\mathcal{B}$ is quasi-greedy and disjoint-super-democratic.

Proof.

We only have to show that semi-greediness implies quasi-greediness and disjoint-super-democracy. The ideas that we use are the same than in Theorem 1.11 (and [2, Theorem 1.10]) applying the condition of $\rho$-admissibility.

First, we show super-democracy. Take two sets $A$ and $B$ such that $|A|\leq|B|$, $A\cap B=\emptyset$ and two signs $\varepsilon,\eta$. Since the basis is $\rho$-admissible, we can find a set $F$ such that $|F|=|A\cup B|$ and $F>A\cup B$. Now, select a set $C\subset F$ such that $|C|=|A|$. Hence,

[TABLE]

Consider the element $x:=\mathbf{1}_{\varepsilon A}+(1+\delta)\mathbf{1}_{C}$, with $\delta>0$. Using the TCGA,

[TABLE]

Using semi-greediness and (15) with $\alpha_{n}=\varepsilon_{n}$ if $n\in A$, $\alpha_{n}=0$ if $n\in B$ and $\alpha_{n}=a_{n}$ if $n\in C$,

[TABLE]

Taking limits when $\delta$ goes to [math], we obtain that

[TABLE]

Now, consider $y:=(1+\delta)\mathbf{1}_{\eta B}+\mathbf{1}_{C}$ with $\delta>0$,

[TABLE]

As before, using semi-greediness and (15) with $\alpha_{n}=0$ if $n\in A$, $\alpha_{n}=b_{n}$ if $n\in B$ and $\alpha_{n}=1$ if $n\in C$,

[TABLE]

Taking limits when $\delta$ goes to [math] and (16), we obtain that

[TABLE]

Thus, the basis is $C_{sd}$-disjoint-super-democratic with $C_{sd}\leq C_{sg}^{2}\rho(1+\rho)$.

Now, we prove quasi-greediness. Since our basis is strong Markushevich, it is enough to consider $x\in\mathbb{X}$ with finite support $A=\mathrm{supp\,}(x)$ as we have said in Theorem 1.11. Using the $\rho$-admissibility, we can find a set $C$ such that $|C|=|A|$, $C>A$ and

[TABLE]

Take $m\in\mathbb{N}$ and $\delta>0$. Define the element $y:=(x-\mathcal{G}_{m}(x))+(\alpha+\delta)\mathbf{1}_{F}$, where $F\subset C$ with $|F|=m$, $\alpha=\min_{j\in A_{m}(x)}|e_{j}^{*}(x)|$ and $A_{m}(x)$ is the greedy set of $x$ with cardinality $m$. Then, using the TCGA,

[TABLE]

Using semi-greediness and (17) with $\alpha_{n}=0$ if $n\in A_{m}(x)$, $\alpha_{n}=e_{n}^{*}(x)$ if $n\in A\setminus A_{m}(x)$, $\alpha_{n}=a_{n}$ if $n\in F$ and $\alpha_{n}=0$ if $n\in C\setminus F$,

[TABLE]

Taking limits when $\delta$ goes to [math],

[TABLE]

Now, take $\eta\equiv\{\mathrm{sgn\,}(e_{i}^{*}(x))\}$ and define $z:=\sum_{i\in A_{m}(x)}(e_{i}^{*}(x)+\delta\eta_{i})e_{i}+P_{(A_{m}(x))^{c}}(x)+\alpha\mathbf{1}_{F}$ for $\delta>0$. Thus, by TCGA,

[TABLE]

Again, using semi-greediness and (17) with $\alpha_{n}=b_{n}$ if $n\in A_{m}(x)$, $\alpha_{n}=e_{n}^{*}(x)$ if $n\in A\setminus A_{m}(x)$, $\alpha_{n}=\alpha$ if $n\in F$ and $\alpha_{n}=0$ if $n\in C\setminus F$,

[TABLE]

Taking limits when $\delta$ goes to [math],

[TABLE]

By (18) and (19), $\mathcal{B}$ is $C_{q}$-quasi-greedy with $C_{q}\leq C\rho(1+(1+\rho)C_{sg})$. ∎

Remark 5.4**.**

We have studied the characterization of semi-greediness using the $\rho$-admissibility. But, at the moment, we don’t know if it is possible to prove the same characterization for $w$-semi-greediness since the condition of the $\rho$-admissibility talks about the cardinality over the sets and not over the weights.

6. Final comments

In this last section we will discuss two questions. The first one is to show that Remark 1.6 is an improvement respect to the bound of Theorem 1.4. To proved that, we establish the following result that is the weighted version of [3, Lemma 3.5].

Proposition 6.1**.**

Assume that $\mathcal{B}$ is a basis in a Banach space $\mathbb{X}$. If $\mathcal{B}$ is $C_{d}$-$w$-democratic and $C_{q}$-quasi-greedy, then $\mathcal{B}$ is $C_{s}$-$w$-super-democratic with $C_{s}\leq 4\kappa^{2}C_{q}C_{d}$, where $\kappa=1$ if $\mathbb{F}=\mathbb{R}$ and $\kappa=2$ if $\mathbb{F}=\mathbb{C}$.

Proof.

First, we prove the result for the real case. Consider $A,B$ two sets with $w(A)\leq w(B)$ and two signs $\varepsilon,\eta$. If we denote by $A^{\pm}=\{n\in A:\varepsilon_{n}=\pm 1\}$, using the democracy with $w(A^{\pm})\leq w(A)\leq w(B)$ and quasi-greediness,

[TABLE]

Now, we decompose $B$ as the set $A$, that is, $B^{\pm}=\{n\in B:\eta_{n}=\pm 1\}$. Hence, using quasi-greediness,

[TABLE]

Then, by (20) and (21), the basis is $C_{s}$-$w$-super-democratic with $C_{s}\leq 4C_{q}C_{d}$. For the complex case, we can proceed using [9, Lemma 6.4] as in Theorem 1.5 to conclude that $\mathcal{B}$ is $C_{s}$-$w$-super-democratic with $C_{s}\leq 4\kappa^{2}C_{q}C_{d}$. ∎

The second question that we study is related to $w$-super-democracy and $w$-disjoint-super-democracy. We know that, if $w=(1,1,...)$, that is, $w(A)=|A|$, a basis $\mathcal{B}$ is super-democratic if and only if $\mathcal{B}$ is disjoint-super-democratic. Quantitatively,

• •

If $\mathcal{B}$ is $C_{s}$-super-democratic, then $\mathcal{B}$ is $C_{sd}$-disjoint-super-democratic with $C_{sd}\leq C_{s}$.

• •

If $\mathcal{B}$ is $C_{sd}$-disjoint-super-democratic, then $\mathcal{B}$ is $C_{s}$-super-democratic with $C_{s}\leq C_{sd}^{2}$.

This result is trivial. Indeed, if the basis is super-democratic, then it is automatically disjoint-super-democratic. For the converse, if we consider that $\mathcal{B}$ is $C_{sd}$-disjoint-super-democratic and take $|A|\leq|B|$ and $C$ such that $C>(A\cup B)$ with $|A|=|C|$,

[TABLE]

Now, we ask the same equivalence for general weights. The result is the following:

Proposition 6.2**.**

Assume that $\mathcal{B}$ is a basis in a Banach space $\mathbb{X}$.

If $\mathcal{B}$ is $C_{s}$-$w$-super-democratic, then $\mathcal{B}$ is $C_{sd}$-$w$-disjoint-super-democratic with $C_{sd}\leq C_{s}$.

If $\mathcal{B}$ is $C_{sd}$-$w$-disjoint-super-democratic, then $\mathcal{B}$ is $C_{s}$-super-democratic with $C_{s}\leq C_{sd}(1+c_{2}^{2}C_{sd})$.

Proof.

Only the item b) requires a proof. We proceed as in the proof of Theorem 1.11. Take $A$ and $B$ such that $w(A)\leq w(B)$.

Case 1: $\sum_{n=1}^{\infty}w_{n}=\infty$ and $\sup_{n}w_{n}<\infty$.

Case 1.1: Assume that $\lim\sup_{n\rightarrow\infty}w_{n}<w(B)$. Since $\sum_{n}w_{n}=\infty$, we can take $E$ and $n_{0}$ such that $n_{0}>E>A\cup B$ such that

[TABLE]

In this case, since $A\cap(E\cup\{n_{0}\})=\emptyset$,

[TABLE]

On the other hand, due to $w(E)\leq w(B)$ and $E\cap B=\emptyset$,

[TABLE]

Using (22) and (23), we obtain that $\mathcal{B}$ is $C_{s}$-super-democratic with $C_{s}\leq C_{sd}^{2}(1+c_{2}^{2})$.

Case 1.2: $w(B)\leq\lim\sup_{n\rightarrow\infty}w_{n}$. Using the item i) of Proposition 2.3, we obtain that

[TABLE]

Case 2: If $\sup_{n}w_{n}=\infty$, the basis is equivalent to the $c_{0}$-basis and the result is trivial.

The proof is over. ∎

Question: Recently, in [4], the authors proved that for Schauder bases and $w=(1,1,...)$, the constants of super-democracy and disjoint-super-democracy are of the same order up to the basis constant, that is, $C_{sd}\leq C_{s}\leq 2(\mathfrak{K}_{b}+1)C_{sd}+\kappa\mathfrak{K}_{b}$, where $\kappa=\sup_{n}\|e_{n}\|\|e_{n}^{*}\|$. Is it possible to show the same result for general weights?