Lebesgue inequalities for Chebyshev thresholding greedy algorithms

Abstract

We establish estimates for the Lebesgue parameters of the Chebyshev weak thresholding greedy algorithm in the case of general bases in Banach spaces. These generalize and slightly improve earlier results in Dilworth et al. (Rev Mat Complut 28(2):393–409, 2015), and are complemented with examples showing the optimality of the bounds. Our results also clarify certain bounds recently announced in Shao and Ye (J Inequal Appl 2018(1):102, 2018), and answer some questions left open in that paper.

1 Introduction

Let $\mathbb {X}$ be a Banach space over $\mathbb {K}= \mathbb {R}$ or $\mathbb {C}$, let $\mathbb {X}^*$ be its dual space, and consider a system $\lbrace {\mathbf {e}}_n, {{\mathbf {e}}^*_n}\rbrace _{n=1}^\infty \subset \mathbb {X}\times \mathbb {X}^*$ with the following properties:

• (a) $0<\inf _n \lbrace \Vert {\mathbf {e}}_n\Vert , \Vert {{\mathbf {e}}^*_n}\Vert \rbrace \le \sup _n\lbrace \Vert {\mathbf {e}}_n\Vert , \Vert {{\mathbf {e}}^*_n}\Vert \rbrace <\infty$

• (b) ${{\mathbf {e}}^*_n}({\mathbf {e}}_m) = \delta _{n,m}$, for all $n,m\ge 1$

• (c) $\mathbb {X}= \overline{\text{ span }\,\lbrace {\mathbf {e}}_n : n\in \mathbb {N}\rbrace }$

• (d) $\mathbb {X}^* = \overline{\text{ span }\,\lbrace {{\mathbf {e}}^*_n}: n\in \mathbb {N}\rbrace }^{w^*}$.

Under these conditions $\mathscr {B}=\lbrace {\mathbf {e}}_n\rbrace _{n=1}^\infty$ is called a seminormalized Markushevich basis for $\mathbb {X}$ (or M-basis for short), with dual system $\lbrace {{\mathbf {e}}^*_n}\rbrace _{n=1}^\infty$. Sometimes we shall consider the following special cases

• (e) $\mathscr {B}$ is a Schauder basis if $K_b:=\sup _N \Vert S_N\Vert <\infty$, where $S_Nx:=\sum _{n=1}^N {\mathbf {e}}^*_n(x){\mathbf {e}}_n$ is the N-th partial sum operator

• (f) $\mathscr {B}$ is a Cesàro basis if $\sup _N \Vert F_N\Vert <\infty$, where $F_N:= \frac{1}{N}\sum _{n=1}^N S_n$ is the N-th (C,1)-Cesàro operator. In this case we use the constant

  $$\begin{aligned} \beta =\;\max \left\{ \sup _N\Vert F_N\Vert ,\,\sup _N\Vert I-F_N\Vert \right\} . \end{aligned}$$

  (1.1)

For the latter terminology, see e.g. [21, Def. III.11.1]. With every $x\in \mathbb {X}$, we shall associate the formal series $x\sim \sum _{n=1}^{\infty } {{\mathbf {e}}^*_n}(x){\mathbf {e}}_n$, where a)-c) imply that $\lim _{n}{{\mathbf {e}}^*_n}(x)=0$. As usual, we denote $\mathrm{supp}x=\lbrace n\in \mathbb {N} : {{\mathbf {e}}^*_n}(x)\ne 0\rbrace$.

We recall standard notions about (weak) greedy algorithms; see e.g. the texts [23, 25] for details and historical background. Fix $t\in (0,1]$. We say that A is a t-greedy set for x of order m, denoted $A\in G(x, m, t)$, if $\vert A\vert =m$ and

$$\begin{aligned} \min _{n\in A}\vert {{\mathbf {e}}^*_n}(x)\vert \ge t\cdot \max _{n\not \in A}\vert {{\mathbf {e}}^*_n}(x)\vert . \end{aligned}$$

(1.2)

A t-greedy operator of order m is any mapping $\mathscr {G}_m^t: \mathbb {X}\rightarrow \mathbb {X}$ which at each $x\in \mathbb {X}$ takes the form

$$\begin{aligned} \mathscr {G}_m^t(x)=\sum _{n\in A}{{\mathbf {e}}^*_n}(x){\mathbf {e}}_n,\quad \text {for some set}\quad A=A(x,\mathscr {G}^t_m)\in G(x, m, t). \end{aligned}$$

We write $\mathbb {G}_m^t$ for the set of all t-greedy operators of order m. The approximation scheme which assigns a sequence $\{\mathscr {G}_m^t(x)\}_{m=1}^\infty$ to each vector $x\in \mathbb {X}$ is called a Weak Thresholding Greedy Algorithm (WTGA), see [16, 24]. When $t=1$ one just says Thresholding Greedy Algorithm (TGA), and drops the super-index t, that is $\mathscr {G}_m^1 = \mathscr {G}_m$, etc.

It is standard to quantify the efficiency of these algorithms, among all possible m-term approximations, in terms of Lebesgue-type inequalities. That is, for each $m=1, 2,\ldots$, we look for the smallest constant ${\mathbf {L}}_m^t$ such that

$$\begin{aligned} \Vert x-\mathscr {G}_m^t(x)\Vert \le {\mathbf {L}}_m^t\sigma _m(x),\quad \forall \; x\in \mathbb {X}, \quad \forall \; \mathscr {G}_m^t\in \mathbb {G}_m^t, \end{aligned}$$

(1.3)

where

$$\begin{aligned} \sigma _m(x):=\inf \Big \lbrace \Big \Vert x-\sum _{n\in B}b_n {\mathbf {e}}_n\Big \Vert {\,\,\,:\,\,\,}b_n\in \mathbb {K},\quad \vert B\vert \le m\Big \rbrace . \end{aligned}$$

We call the number ${\mathbf {L}}_m^t$ the Lebesgue parameter associated with the WTGA, and we just write ${\mathbf {L}}_m$ when $t=1$. We refer to [25, Chapter 3] for a survey on such inequalities, and to [1, 5, 6, 10, 12, 26] for recent results. It is known that ${\mathbf {L}}_m^t=O(1)$ holds for a fixed t if and only if it holds for all $t\in (0,1]$, and if and only if $\mathscr {B}$ is unconditional and democratic; see [15] and [23, Thm. 1.39]. In this special case $\mathscr {B}$ is called a greedy basis.

In this paper we shall be interested in Chebyshev thresholding greedy algorithms. These were introduced by Dilworth et al. [8, §3], as an enhancement of the TGA. Here, we use the weak version considered in [10]. Namely, for fixed $t\in (0,1]$ we say that ${\mathfrak {CG}}_m^t:\mathbb {X}\rightarrow \mathbb {X}$ is a Chebyshev t-greedy operator of order m if for every $x\in \mathbb {X}$ there is a set $A=A(x,{\mathfrak {CG}}_m^t)\in G(x, m, t)$ such that $\mathrm{supp}{\mathfrak {CG}}_m^t(x)\subset A$ and moreover

$$\begin{aligned} \Vert x-{\mathfrak {CG}}_m^t(x)\Vert = \min \left\{ \left\| x-\sum _{n\in A}a_n{\mathbf {e}}_n\right\| {\,\,\,:\,\,\,}a_n\in \mathbb {K}\right\} . \end{aligned}$$

Finally, we define the weak Chebyshevian Lebesgue parameter ${\mathbf {L}}_m^{{\mathrm{ch}}, t}$ as the smallest constant such that

$$\begin{aligned} \Vert x-{\mathfrak {CG}}_m^t(x)\Vert \le {\mathbf {L}}_m^{{\mathrm{ch}}, t}\sigma _m(x),\quad \forall \; x\in \mathbb {X},\quad \forall \;{\mathfrak {CG}}_m^t\in {\mathbb {G}_m^{{\mathrm{ch}},t}}\,, \end{aligned}$$

where ${\mathbb {G}_m^{{\mathrm{ch}},t}}$ is the collection of all Chebyshev t-greedy operators of order m. As before, when $t=1$ we shall omit the index t, that is $\mathbf {L}_m^{\mathrm{ch}}:=\mathbf {L}_m^{\mathrm{ch},1}$.

When $\mathbf {L}_m^{\mathrm{ch}}=O(1)$ the system $\mathscr {B}$ is called semi-greedy; see [8]. We remark that the first author recently established that a Schauder basis $\mathscr {B}$ is semi-greedy if and only if is quasi-greedy and democratic; see [3].

In this paper we shall be interested in quantitative bounds of ${\mathbf {L}}_m^{{\mathrm{ch}}, t}$ in terms of the quasi-greedy and democracy parameters of a general M-basis $\mathscr {B}$. Earlier bounds were obtained by Dilworth et al. [10] when $\mathscr {B}$ is a quasi-greedy basis, and very recently, some improvements were also announced by Shao and Ye [19, Theorem 3.5]. Unfortunately, various arguments in the last paper seem not to be correct, so one of our goals here is to give precise statements and proofs for the results in [19], and also settle some of the questions which are left open there.

To state our results, we recall the definitions of the involved parameters. Given a finite set $A\subset \mathbb {N}$, we shall use the following standard notation for the indicator sums:

$$\begin{aligned} \mathbf{1}_A = \sum _{n\in A}{\mathbf {e}}_n{\quad \text{ and }\quad }\mathbf{1}_{\varepsilon A}=\sum _{n\in A}\varepsilon _n {\mathbf {e}}_n,\quad {\varepsilon }\in \Upsilon \end{aligned}$$

where $\Upsilon$ is the set of all $\varepsilon = \lbrace \varepsilon _n\rbrace _n\subset \mathbb {K}$ with $\vert \varepsilon _n\vert =1$. Similarly, we write

$$\begin{aligned} P_A(x)=\sum _{n\in A}{{\mathbf {e}}^*_n}(x){\mathbf {e}}_n. \end{aligned}$$

The relevant parameters for this paper are the following:

• Conditionality parameters:

  $$\begin{aligned} k_m := \sup _{\vert A\vert \le m}\Vert P_A\Vert {\quad \text{ and }\quad }k_m^c=\sup _{\vert A\vert \le m}\Vert I-P_A\Vert . \end{aligned}$$

• Quasi-greedy parameters:

  $$\begin{aligned} g_m := \sup _{\mathscr {G}_k\in \mathbb {G}_k,\,k\le m}\,\Vert \mathscr {G}_k\Vert {\quad \text{ and }\quad }g_m^c := \sup _{\mathscr {G}_k\in \mathbb {G}_k,\,k\le m}\,\Vert I-\mathscr {G}_k\Vert . \end{aligned}$$

  Below we shall also use the variant

  $$\begin{aligned} {\tilde{g}}_m:=\sup _{\begin{array}{c} \mathscr {G}'<\mathscr {G}\\ \mathscr {G}\in \mathbb {G}_k,\;k\le m \end{array}}\Vert \mathscr {G}-\mathscr {G}'\Vert , \end{aligned}$$

  where $\mathscr {G}'<\mathscr {G}$ means that $A(x,\mathscr {G}')\subset A(x,\mathscr {G})$ for all x; see [5].

• Super-democracy parameters:

  $$\begin{aligned} \tilde{\mu }_m = \sup _{\underset{\vert \varepsilon \vert =\vert \eta \vert =1}{\vert A\vert =\vert B\vert \le m}}\dfrac{\Vert \mathbf{1}_{\varepsilon A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }{\quad \text{ and }\quad }\tilde{\mu }_m^d=\sup _{\underset{\vert \varepsilon \vert =\vert \eta \vert =1}{\vert A\vert =\vert B\vert \le m, \; A\cap B=\emptyset }}\dfrac{\Vert \mathbf{1}_{\varepsilon A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }. \end{aligned}$$

• Quasi-greedy parameters for constant coefficients (see [5, (3.11)])

  $$\begin{aligned} {\gamma }_m=\sup _{\underset{B\subset A,\;\vert A\vert \le m}{\vert \varepsilon \vert =1}}\dfrac{\Vert \mathbf{1}_{\varepsilon B}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }. \end{aligned}$$

Note that ${\gamma }_m\le g_m\le {\tilde{g}}_m\le 2g_m$, but in general ${\gamma }_m$ may be much smaller than $g_m$; see e.g. [5, §5.5]. Likewise, in §5 below we show that ${\tilde{\mu }}^d_m$ may be much smaller than ${\tilde{\mu }}_m$, except for Schauder bases, where both quantities turn out to be equivalent; see Theorem 5.2.

Our first result is a general upper bound, which improves and extends [19, Theorem 2.4].

Theorem 1.1

Let $\mathscr {B}$ be an M-basis in $\mathbb {X}$, and let $\mathfrak {K}=\sup _{n,j}\Vert {{\mathbf {e}}^*_n}\Vert \Vert {\mathbf {e}}_j\Vert$. Then,

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\le \, 1\,+\,\left( 1+\tfrac{1}{t}\right) \,\mathfrak {K}\,m\,, \quad \forall \;m\in \mathbb {N},\;\; t\in (0,1]. \end{aligned}$$

(1.4)

Moreover, there exists a pair $(\mathbb {X},\mathscr {B})$ where the equality is attained for all m and t.

The second result is a slight generalization of [10, Theorem 4.1], and gives a correct version of [19, Theorem 3.5].

Theorem 1.2

Let $\mathscr {B}$ be an M-basis in $\mathbb {X}$. Then, for all $m\ge 1$ and $t\in (0,1]$,

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\;\le \;g_{2m}^c\,+\,\frac{2}{t}\,\min \big \{\,{\tilde{g}}_m{\tilde{\mu }}_m\,,\;{\gamma }_{2m}{\tilde{g}}_{2m}{\tilde{\mu }}^d_m\, \big \}. \end{aligned}$$

(1.5)

Our next result concerns lower bounds for ${\mathbf {L}}_m^{{\mathrm{ch}}, t}$, for which we need to introduce weaker versions of the democracy parameters with an additional separation condition. For two finite sets $A,B\subset \mathbb {N}$ and $c\ge 1$, the notation $A>cB$ will stand for $\min A >c\max B$.

• Given an integer $c\ge 2$, we define

  $$\begin{aligned} \vartheta _{m,c} := \sup \left\{ \dfrac{\Vert \mathbf{1}_{\varepsilon A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }{\,\,\,:\,\,\,}\vert \varepsilon \vert =\vert \eta \vert =1,\;|A|=|B|\le m\; \text{ with } A>cB \text{ or } B>cA \right\} . \end{aligned}$$

  (1.6)

Theorem 1.3

If $\mathscr {B}$ is a Cesàro basis in $\mathbb {X}$ with constant $\beta$, then for every $c\ge 2$

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\ge \frac{1}{t\beta ^2}\frac{c-1}{c+1}\vartheta _{m,c},\quad \forall \;m\in \mathbb {N},\;t\in (0,1]. \end{aligned}$$

We shall also establish, in Theorem 3.10 below, a similar lower bound valid for more general M-bases (not necessarily of Cesàro type), in terms of a new parameter $\theta _m$ which is invariant under rearrangements of $\mathscr {B}$.

Remark 1.4

One may compare the bounds for ${\mathbf {L}}_m^{\mathrm{ch}}$ above with those for ${\mathbf {L}}_m$ given in [5]

$$\begin{aligned} (1)\;{\mathbf {L}}_m\le 1+3\mathfrak {K}m,\qquad (2)\; {\mathbf {L}}_m\le k^c_{2m}+{\tilde{g}}_m{\tilde{\mu }}_m,\quad {\quad \text{ and }\quad }\;(3)\; {\mathbf {L}}_m\ge {\tilde{\mu }}_m^d, \end{aligned}$$

which illustrate a slightly better behavior of the Chebishev TGA. Observe that one also has the trivial inequalities

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\le {\mathbf {L}}_m^t\le k^c_m\,{\mathbf {L}}_m^{{\mathrm{ch}}, t}. \end{aligned}$$

Indeed, ${\mathbf {L}}_m^{{\mathrm{ch}}, t}\le {\mathbf {L}}_m^t$ is direct by definition, while ${\mathbf {L}}_m^t \le k_m^c {\mathbf {L}}_m^{{\mathrm{ch}}, t}$ can be proved as follows: take $x\in \mathbb {X}$ and let $A=\mathrm{supp}\mathscr {G}^t_m(x)$. Pick a Chebyshev greedy operator ${\mathfrak {CG}}^t_m$ such that $\mathrm{supp}{\mathfrak {CG}}^t_m (x)=A$. Then

$$\begin{aligned} \Vert x-\mathscr {G}_m^t(x)\Vert = \Vert (I-P_A)x\Vert =\Vert (I-P_A)(x-\mathfrak {CG}_m^t(x))\Vert \le k_m^c\Vert x-\mathfrak {CG}_m^t(x)\Vert , \end{aligned}$$

so ${\mathbf {L}}_m^t\le k_m^c {\mathbf {L}}_m^{{\mathrm{ch}}, t}$. Hence, when $\mathscr {B}$ is unconditional then ${\mathbf {L}}_m^t \approx {\mathbf {L}}_m^{{\mathrm{ch}}, t}$. However for all conditional quasi-greedy and democratic bases we have ${\mathbf {L}}_m^{\mathrm{ch}}=O(1)$, but ${\mathbf {L}}_m\rightarrow \infty$.

The paper is organized as follows. Section 2 is devoted to preliminary lemmas. In Sect. 3 we prove Theorems 1.1, 1.2 and 1.3, and also establish the more general lower bound in Theorem 3.10, giving various situations in which it applies. Section 4 is devoted to examples illustrating the optimality of the results; in particular, an optimal bound of ${\mathbf {L}}_m^{\mathrm{ch}}$ for the trigonometric system in $L^1(\mathbb {T})$, settling a question left open in [19]. In Sect. 5 we investigate the equivalence between ${\tilde{\mu }}^d_m$ and ${\tilde{\mu }}_m$ and show Theorem 5.2. Finally, in Sect. 6 we study the convergence of ${\mathfrak {CG}}_m (x)$ and $\mathscr {G}_m(x)$ to x, pointing out the role of a strong M-basis assumption for such results.

2 Preliminary results

We recall some basic concepts and results that will be used later in the paper; see [5, 8]. For each ${\alpha }>0$ we define the ${\alpha }$-truncation of a scalar $y\in \mathbb {K}$ as

$$\begin{aligned} T_{\alpha }(y)={\alpha }\, \mathrm {sign }\,y\; \text{ if } \, \vert y\vert \ge {\alpha },{\quad \text{ and }\quad }T_{\alpha }(y)=y\; \text{ if } \, \vert y\vert \le {\alpha }. \end{aligned}$$

We extend $T_{\alpha }$ to an operator in $\mathbb {X}$ by formally assigning $T_{\alpha }(x)\sim \sum _{n=1}^\infty T_{\alpha }({{\mathbf {e}}^*_n}(x)){\mathbf {e}}_n$, that is

$$\begin{aligned} T_{\alpha }(x):= {\alpha }\mathbf{1}_{\varepsilon \Lambda _{\alpha }(x)}+(I-P_{\Lambda _{\alpha }(x)})(x), \end{aligned}$$

where $\Lambda _{\alpha }(x)=\lbrace n : \vert {{\mathbf {e}}^*_n}(x)\vert >{\alpha }\rbrace$ and $\varepsilon =\lbrace \mathrm {sign }\,({{\mathbf {e}}^*_n}(x))\rbrace$. Of course, this operator is well defined since $\Lambda _{\alpha }(x)$ is a finite set. In [5] we can find the following result:

Lemma 2.1

[5, Lemma 2.5] For all ${\alpha }>0$ and $x\in \mathbb {X}$, we have

$$\begin{aligned} \Vert T_{\alpha }(x)\Vert \le g_{\vert \Lambda _{\alpha }(x)\vert }^c\Vert x\Vert . \end{aligned}$$

We also need a well known property from [8, 9], formulated as follows.

Lemma 2.2

[5, Lemma 2.3] If $x\in \mathbb {X}$ and $\varepsilon =\lbrace \mathrm {sign }\,({{\mathbf {e}}^*_n}(x))\rbrace$, then

$$\begin{aligned} \min _{n\in G}\vert {{\mathbf {e}}^*_n}(x)\vert \Vert \mathbf{1}_{\varepsilon G}\Vert \le {\tilde{g}}_{\vert G\vert } \Vert x\Vert ,\quad \forall G\in G(x,m,1). \end{aligned}$$

(2.1)

The following version of (2.1), valid even if G is not greedy, improves [10, Lemma 2.2].

Lemma 2.3

Let $x\in \mathbb {X}$ and $\varepsilon =\lbrace \mathrm {sign }\,({{\mathbf {e}}^*_n}(x))\rbrace$. For every set finite $A\subset \mathbb {N}$, if ${\alpha }=\min _{n\in A}|{\mathbf {e}}^*_n(x)|$, then

$$\begin{aligned} {\alpha }\Vert \mathbf{1}_{\varepsilon A}\Vert \,\le \,{\gamma }_{|A\cup {\Lambda }_{\alpha }(x)|}\,{\tilde{g}}_{|A\cup {\Lambda }_{\alpha }(x)|} \Vert x\Vert , \end{aligned}$$

(2.2)

where $\Lambda _{\alpha }(x)=\lbrace n : \vert {{\mathbf {e}}^*_n}(x)\vert >{\alpha }\rbrace$.

Proof

Call $G=A\cup {\Lambda }_{\alpha }(x)$, and notice that it is a greedy set for x. Then,

$$\begin{aligned} {\alpha }\Vert \mathbf{1}_{\varepsilon A}\Vert \,\le \,{\alpha }\,{\gamma }_{|G|}\Vert \mathbf{1}_{{\varepsilon }G}\Vert \,\le \,{\gamma }_{|G|}\,{\tilde{g}}_{|G|}\, \Vert x\Vert , \end{aligned}$$

using (2.1) in the last step.

Remark 2.4

The following is a variant of (2.2) with a different constant

$$\begin{aligned} \min _{n\in A}|{\mathbf {e}}^*_n(x)|\; \Vert \mathbf{1}_{\varepsilon A}\Vert \,\le \,k_{|A|}\, \Vert x\Vert . \end{aligned}$$

(2.3)

A similar proof as the one in Lemma 2.3 can be seen in [4, Proposition 2.5].

Finally, we need the following elementary result, which follows directly from the convexity of the norm; see e.g [25, p. 108] (or [5, Lemma 2.7] if $\mathbb {K}=\mathbb {C}$).

Lemma 2.5

For all finite sets $A\subset \mathbb {N}$ and scalars $a_n\in \mathbb {K}$ it holds

$$\begin{aligned} \left\| \sum _{n\in A} a_n {\mathbf {e}}_n\right\| \le \,\max _{n\in A}|a_n|\,\sup _{|{\varepsilon }|=1}\big \Vert \mathbf{1}_{{\varepsilon }A}\big \Vert . \end{aligned}$$

3 Proof of the main results

3.1 Proof of Theorem 1.1

Let $x\in \mathbb {X}$ and ${\mathfrak {CG}}^t_m\in {\mathbb {G}_m^{{\mathrm{ch}},t}}$ be a fixed Chebyshev t-greedy operator. Let $A=A(x,{\mathfrak {CG}}^t_m)\in G(x,m,t)$. Pick any $z=\sum _{n\in B}b_n{\mathbf {e}}_n$ such that $\vert B\vert = m$. By definition of the Chebyshev operators,

$$\begin{aligned} \Vert x-\mathfrak {CG}_m^t(x)\Vert \le \Vert x-P_{A\cap B}(x)\Vert \le \Vert P_{B{\setminus } A}(x)\Vert + \Vert x-P_B(x)\Vert . \end{aligned}$$

On the one hand, using (1.2),

$$\begin{aligned} \Vert P_{B{\setminus } A}(x)\Vert \le \sup _n\Vert {\mathbf {e}}_n\Vert \sum _{j\in B{\setminus } A}\vert \mathbf {e}_j^*(x)\vert \le \frac{1}{t}\sup _n\Vert {\mathbf {e}}_n\Vert \sum _{j\in A{\setminus } B}\vert \mathbf {e}_j^*(x-z)\vert \le \frac{1}{t}\mathfrak {K}m\Vert x-z\Vert . \end{aligned}$$

On the other hand, using the inequality (3.9) of [5],

$$\begin{aligned} \Vert x-P_B(x)\Vert =\Vert (I-P_B)(x-z)\Vert \le k_m^c\Vert x-z\Vert \le (1+\mathfrak {K}m)\Vert x-z\Vert . \end{aligned}$$

Hence, ${\mathbf {L}}_m^{{\mathrm{ch}}, t}\le 1+\left( 1+\frac{1}{t}\right) \mathfrak {K}m$. Finally, the fact that the equality in (1.4) can be attained is witnessed by Examples 4.1 and 4.2 below.

3.2 Proof of Theorem 1.2

The scheme of the proof follows the lines in [8, Theorem 3.2] and [10, Theorem 4.1], with some additional simplifications introduced in [5].

Given $x\in \mathbb {X}$ and ${\mathfrak {CG}}^t_m\in {\mathbb {G}_m^{{\mathrm{ch}},t}}$, let $A= A(x,{\mathfrak {CG}}^t_m)\in G(x,m,t)$. Pick any $z=\sum _{n\in B}b_n{\mathbf {e}}_n$ such that $\vert B\vert = m$. By definition of the Chebyshev operators,

$$\begin{aligned} \Vert x-{\mathfrak {CG}}^t_m(x)\Vert \le \Vert x-p\Vert ,\quad \text{ for } \text{ any } p=\sum _{n\in A}a_n{\mathbf {e}}_n. \end{aligned}$$

(3.1)

We make the selection of p suggested in [8]. Namely, if ${\alpha }=\max _{n\notin A}|{\mathbf {e}}^*_n(x)|$, we let

$$\begin{aligned} p=P_A(x)- P_A\big (T_{\alpha }(x-z)\big ). \end{aligned}$$

It is easily verified that

$$\begin{aligned} x-p= & {} (I-P_A)\big (x-T_{\alpha }(x-z)\big )+T_{\alpha }(x-z) \nonumber \\= & {} P_{B{\setminus } A}\big (x-T_{\alpha }(x-z)\big )+T_{\alpha }(x-z). \end{aligned}$$

(3.2)

Since ${\Lambda }_{\alpha }(x-z)=\{n{\,\,\,:\,\,\,}|{\mathbf {e}}^*_n(x-z)|>{\alpha }\}\subset A\cup B$, then Lemma 2.1 gives

$$\begin{aligned} \big \Vert T_{\alpha }(x-z)\big \Vert \le g^c_{2m}\Vert x-z\Vert . \end{aligned}$$

(3.3)

Next we treat the first term in (3.2). Observe that $\max _{n\in B{\setminus } A}\vert {{\mathbf {e}}^*_n}(x-T_\alpha (x-z))\vert \le 2\alpha$, so Lemma 2.5 gives

$$\begin{aligned} \big \Vert P_{B{\setminus } A}\big (x-T_{\alpha }(x-z)\big )\big \Vert\le & {} 2{\alpha }\,\sup _{|{\varepsilon }|=1}\big \Vert \mathbf{1}_{{\varepsilon }(B{\setminus } A)}\big \Vert \nonumber \\\le & {} \frac{2}{t}\,\min _{n\in A{\setminus } B}|{\mathbf {e}}^*_n(x-z)|\,\sup _{|{\varepsilon }|=1}\big \Vert \mathbf{1}_{{\varepsilon }(B{\setminus } A)}\big \Vert =(*). \end{aligned}$$

(3.4)

At this point we have two possible approaches. Let $\eta _n=\mathrm {sign }\,[e^*_n(x-z)]$. In the first approach we pick a greedy set ${\Gamma }\in G(x-z,|A{\setminus } B|,1)$, and control (3.4) by

$$\begin{aligned} (*) \,\le \,\frac{2}{t}\,\min _{n\in {\Gamma }}|{\mathbf {e}}^*_n(x-z)|\,\,{\tilde{\mu }}_m\,\big \Vert \mathbf{1}_{\eta {\Gamma }}\big \Vert \, \le \, \frac{2}{t}\,{\tilde{\mu }}_m\,{\tilde{g}}_{m}\Vert x-z\Vert , \end{aligned}$$

(3.5)

using Lemma 2.2 in the last step. In the second approach, we argue as follows

$$\begin{aligned} (*) \,\le \,\frac{2}{t}\,\min _{n\in A{\setminus } B}|{\mathbf {e}}^*_n(x-z)|\,\,{\tilde{\mu }}^d_m\,\big \Vert \mathbf{1}_{\eta (A{\setminus } B)}\big \Vert \, \le \, \frac{2}{t}\,{\gamma }_{2m}\,{\tilde{g}}_{2m}\,{\tilde{\mu }}_m^d\,\Vert x-z\Vert , \end{aligned}$$

(3.6)

using in the last step Lemma 2.3 and the fact that, if ${\delta }=\min _{A{\setminus } B}|{\mathbf {e}}^*_n(x-z)|$, then the set $(A{\setminus } B)\cup \{n{\,\,\,:\,\,\,}|{\mathbf {e}}^*_n(x-z)|>{\delta }\}\subset A\cup B$ and hence has cardinality $\le 2m$.

We can now combine the estimates displayed in (3.1)–(3.6) and obtain

$$\begin{aligned} \Vert x-{\mathfrak {CG}}^t_m(x)\Vert \le \,\big [g^c_{2m}+\frac{2}{t}\,\min \left\{ \,{\tilde{g}}_m{\tilde{\mu }}_m\,,\;{\gamma }_{2m}{\tilde{g}}_{2m}{\tilde{\mu }}^d_m\,\big \}\right] \,\Vert x-z\Vert , \end{aligned}$$

which after taking the infimum over all z establishes Theorem 1.2. $\square$

Remark 3.1

In [19, Theorem 3.5] a stronger inequality is stated (for $t=1$), namely

$$\begin{aligned} {\mathbf {L}}_m^{\mathrm{ch}}\le g^c_{2m}+2{\tilde{g}}_m{\tilde{\mu }}^d_m. \end{aligned}$$

(3.7)

The proof, however, seems to contain a gap, and a missing factor $k_m^c$ should also appear in the last summand. Nevertheless, it is still fair to ask whether the inequality (3.7) asserted in [19] may be true with a different proof.

Remark 3.2

Using Remark 2.4 in place of Lemma 2.3 in (3.6) above leads to an alternative and slightly simpler estimate

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\;\le \;g_{2m}^c\,+\,\frac{2}{t}\,k_m{\tilde{\mu }}_m^d\,. \end{aligned}$$

(3.8)

However, this would not be as efficient as (1.5) when $\mathscr {B}$ is quasi-greedy and conditional.

Remark 3.3

When $\mathscr {B}$ is quasi-greedy with constant $\mathbf {q}=\sup _m g_m<\infty$, then Theorem 1.2 implies the following

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\le \mathbf {q}+ 4t^{-1}\, \mathbf {q}^2 \,\tilde{\mu }^d_m. \end{aligned}$$

This is a slight improvement with respect to [10, Theorem 4.1].

3.3 Proof of Theorem 1.3

Recall that $S_N=\sum _{n=1}^N {\mathbf {e}}^*_n(\cdot ){\mathbf {e}}_n$ and

$$\begin{aligned} F_N(x)=\frac{1}{N}\sum _{n=1}^NS_n(x)=\sum _{n=1}^N\left( 1-\frac{n-1}{N}\right) {\mathbf {e}}^*_n(x){\mathbf {e}}_n. \end{aligned}$$

For $M>N$ we define the operators (of de la Vallée-Poussin type)

$$\begin{aligned} V_{N,M}(x)= & {} \frac{M}{M-N}\,F_{M}(x)-\frac{N}{M-N}F_{N}(x)\nonumber \\= & {} \sum _{n=1}^N{\mathbf {e}}^*_n(x){\mathbf {e}}_n\,+\,\sum _{n=N+1}^M\left( 1-\frac{n-N-1}{M-N}\right) \,{\mathbf {e}}^*_n(x){\mathbf {e}}_n. \end{aligned}$$

(3.9)

In particular, observe that, for $\beta$ as in (1.1) we have

$$\begin{aligned} \max \big \{\Vert V_{N,M}\Vert , \Vert I-V_{N,M}\Vert \big \}\,\le \, \frac{M+N}{M-N}\,\beta . \end{aligned}$$

(3.10)

We next prove that, if $c\ge 2$, then for all $A,B\subset \mathbb {N}$ such that $B>cA$ with $|A|=|B|\le m$ it holds

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\ge \frac{1}{t\beta }\,\frac{c-1}{c+1}\,\frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert },\quad \forall \;|{\varepsilon }|=|\eta |=1. \end{aligned}$$

(3.11)

Pick any set $C>B$ such that $|B\cup C|=m$, and let

$$\begin{aligned} x=\mathbf{1}_{{\varepsilon }A}+t\mathbf{1}_{\eta B}+t\mathbf{1}_C. \end{aligned}$$

Then $B\cup C\in G(x,m,t)$, and hence there is a Chebyshev t-greedy operator so that

$$\begin{aligned} x-{\mathfrak {CG}}^t_m(x)=\mathbf{1}_{{\varepsilon }A}+\sum _{n\in B\cup C} a_n{\mathbf {e}}_n, \end{aligned}$$

for some scalars $a_n\in \mathbb {K}$. Clearly,

$$\begin{aligned} \Vert x-{\mathfrak {CG}}^t_m(x)\Vert \le {\mathbf {L}}_m^{{\mathrm{ch}}, t}\sigma _m(x)\,\le \,{\mathbf {L}}_m^{{\mathrm{ch}}, t}\,\Vert t\mathbf{1}_{\eta B}\Vert , \end{aligned}$$

using $z=\mathbf{1}_{{\varepsilon }A}+t\mathbf{1}_C$ an m-term approximant. On the other hand, let $N=\max A$. Since $\min B\cup C> cN$, then (3.9) yields

$$\begin{aligned} V_{N,cN}(x-{\mathfrak {CG}}^t_m(x))=\mathbf{1}_{{\varepsilon }A}. \end{aligned}$$

Therefore, (3.10) implies that

$$\begin{aligned} \Vert x-{\mathfrak {CG}}^t_m(x)\Vert \ge \frac{\Vert V_{N,cN}(x-{\mathfrak {CG}}^t_m(x))\Vert }{\Vert V_{N,cN}\Vert }\ge \frac{c-1}{(c+1)\beta }\,\Vert \mathbf{1}_{{\varepsilon }A}\Vert . \end{aligned}$$

We have therefore proved (3.11).

We next show that when $|A|=|B|\le m$ satisfy $A>cB$ then

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\ge \frac{1}{t\beta ^2}\,\frac{c-1}{c+1}\,\frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert },\quad \forall \;|{\varepsilon }|=|\eta |=1. \end{aligned}$$

(3.12)

This together with (3.11) is enough to establish Theorem 1.3. We shall actually show a slightly stronger result:

Lemma 3.4

Let $|A|=|B|\le m$ and let $y\in \mathbb {X}$ be such that $|y|_\infty :=\sup _n|{\mathbf {e}}^*_n(y)|\le 1$ and . Then

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\ge \frac{1}{t\beta ^2}\,\frac{c-1}{c+1}\,\frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}+y\Vert },\quad \forall \;|{\varepsilon }|=|\eta |=1. \end{aligned}$$

(3.13)

Observe that the case $y=0$ in (3.13) yields (3.12). We now show (3.13). Pick a large integer ${\lambda }>1$ and a set $C>{\lambda }A$ such that $|B\cup C|=m$. Let

$$\begin{aligned} x=\mathbf{1}_{{\varepsilon }A}+ty+t\mathbf{1}_{\eta B}+t\mathbf{1}_C. \end{aligned}$$

As before, $B\cup C\in G(x,m,t)$, and hence for some Chebyshev t-greedy operator we have

$$\begin{aligned} x-{\mathfrak {CG}}^t_m(x)=\mathbf{1}_{{\varepsilon }A}+ty+\sum _{n\in B\cup C} a_n{\mathbf {e}}_n, \end{aligned}$$

for suitable scalars $a_n\in \mathbb {K}$. Choosing $\mathbf{1}_{{\varepsilon }A}+t\mathbf{1}_C$ as m-term approximant of x we see that

$$\begin{aligned} \Vert x-{\mathfrak {CG}}^t_m(x)\Vert \le {\mathbf {L}}_m^{{\mathrm{ch}}, t}\sigma _m(x)\,\le \,{\mathbf {L}}_m^{{\mathrm{ch}}, t}\,t\,\Vert \mathbf{1}_{\eta B}+y\Vert . \end{aligned}$$

On the other hand, calling and $L=\max A$ we have

$$\begin{aligned} (I-V_{N,cN})\circ V_{L,{\lambda }L}\big (x-{\mathfrak {CG}}^t_m(x)\big )=\mathbf{1}_{{\varepsilon }A} \end{aligned}$$

Thus,

$$\begin{aligned} \Vert x-{\mathfrak {CG}}^t_m(x)\Vert \ge \frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert I-V_{N,cN}\Vert \Vert V_{L,{\lambda }L}\Vert }\ge \frac{c-1}{(c+1)\beta }\,\frac{{\lambda }-1}{({\lambda }+1)\beta }\,\Vert \mathbf{1}_{{\varepsilon }A}\Vert . \end{aligned}$$

Therefore we obtain

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\ge \frac{1}{t\beta ^2}\,\frac{c-1}{c+1}\,\frac{{\lambda }-1}{{\lambda }+1}\,\frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}+y\Vert } \end{aligned}$$

which letting ${\lambda }\rightarrow \infty$ yields (3.13). This completes the proof of Lemma 3.4, and hence of Theorem 1.3.

Remark 3.5

When $\mathscr {B}$ is a Schauder basis, a similar proof gives the following lower bound, which is also obtained in [19, Theorem 2.2]

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\ge \,\frac{1}{(K_b+1)t}\,\sup \left\{ \frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }{\,\,\,:\,\,\,}|A|=|B|= m, \;A>B \text{ or } B>A ,\;|{\varepsilon }|=|\eta |=1\right\} . \end{aligned}$$

The statement for Cesàro bases, however, will be needed for the applications in §4.3.

3.4 Lower bounds for general M-bases

Observe that

$$\begin{aligned} \vartheta _{m,c}=\sup _{|A|\le m}\vartheta _c(A),\quad \text{ where }\quad \vartheta _c(A)=\sup _{\begin{array}{c} B{\,\,\,:\,\,\,}|B|=|A| \\ B>cA \\ {\varepsilon },\eta \in \Upsilon \end{array}} \max \left\{ \frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert },\frac{\Vert \mathbf{1}_{\eta B}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }\right\} . \end{aligned}$$

We consider a new parameter

$$\begin{aligned} \vartheta _m=\sup _{|A|\le m}\;\inf _{c\ge 1}\;\vartheta _c(A). \end{aligned}$$

(3.14)

We remark that, unlike $\vartheta _{m,c}$, the parameter $\vartheta _m$ depends on $\{{\mathbf {e}}_n\}_{n=1}^\infty$ but not on the reorderings of the system. We shall give a lower bound for ${\mathbf {L}}_m^{{\mathrm{ch}}, t}$ in terms of $\vartheta _m$ in a less restrictive situation than the Cesàro basis assumption on $\{{\mathbf {e}}_n\}_{n=1}^\infty$.

Given $\rho \ge 1$, we say that $\{{\mathbf {e}}_n\}_{n=1}^\infty$ is $\rho$-admissible if the following holds: for each finite set $A\subset \mathbb {N}$, there exists $n_0=n_0(A)> \max A$such that, for all sets B with $\min B\ge n_0$and $|B|\le |A|$,

$$\begin{aligned} \left\| \sum _{n\in A}{\alpha }_n{\mathbf {e}}_n\right\| \le \rho \,\left\| \sum _{n\in A\cup B}{\alpha }_n{\mathbf {e}}_n\right\| ,\quad \forall \;{\alpha }_n\in \mathbb {K}. \end{aligned}$$

(3.15)

Observe that (3.15) implies that

$$\begin{aligned} \left\| \sum _{n\in B}{\alpha }_n{\mathbf {e}}_n\right\| \le (\rho +1)\,\left\| \sum _{n\in A\cup B}{\alpha }_n{\mathbf {e}}_n\right\| ,\quad \forall \;{\alpha }_n\in \mathbb {K}. \end{aligned}$$

(3.16)

This condition is clearly satisfied by all Schauder and Cesàro bases (with $\rho =K_b$ or $\rho >\beta$), but we shall see below that it also holds in more general situations.

Proposition 3.6

Let $\{{\mathbf {e}}_n,{\mathbf {e}}^*_n\}_{n=1}^\infty$ be an M-basis such that $\{{\mathbf {e}}_n\}_{n=1}^\infty$ is $\rho$-admissible. Then

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\ge \frac{\vartheta _m}{(\rho +1)t},\quad \forall \;m\in \mathbb {N},\quad t\in (0,1]. \end{aligned}$$

(3.17)

Proof

Fix $A\subset \mathbb {N}$ such that $|A|\le m$. Choose C disjoint with A such that $|A\cup C|=m$. Let $n_0=n_0(A\cup C)$ be as in the above definition, so that $n_0$ is larger than $\max A\cup C$. Pick any B with $\min B\ge n_0$ and $|B|=|A|$, and any ${\varepsilon },\eta \in \Upsilon$. Let $x=t\mathbf{1}_{{\varepsilon }A}+t \mathbf{1}_C+\mathbf{1}_{\eta B}$. Then $A\cup C\in G(x,m,t)$, and there is a Chebyshev t-greedy operator with ${\mathfrak {CG}}^t_m(x)$ supported in $A\cup C$. Thus,

$$\begin{aligned} \Vert x-{\mathfrak {CG}}^t_m(x)\Vert \le {\mathbf {L}}_m^{{\mathrm{ch}}, t}\,\sigma _m(x)\le {\mathbf {L}}_m^{{\mathrm{ch}}, t}\,\Vert x-(\mathbf{1}_{\eta B}+t\mathbf{1}_C)\Vert ={\mathbf {L}}_m^{{\mathrm{ch}}, t}\,t\,\Vert \mathbf{1}_{{\varepsilon }A}\Vert . \end{aligned}$$

On the other hand, using the property in (3.16) one obtains

$$\begin{aligned} \Vert x-{\mathfrak {CG}}^t_m(x)\Vert \ge \frac{\Vert \mathbf{1}_{\eta B}\Vert }{\rho +1}. \end{aligned}$$

Thus,

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\,\ge \,\frac{1}{(\rho +1)t}\,\frac{\Vert \mathbf{1}_{\eta B}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }. \end{aligned}$$

We now assume additionally that $\min B\ge n_0+m$, and pick $D\subset [n_0,n_0+m-1]$ such that $|B|+|D|=m$. Let $y=\mathbf{1}_{{\varepsilon }A}+t\mathbf{1}_{\eta B}+t\mathbf{1}_D$. Then $B\cup D\in G(y,m,t)$ and a similar reasoning gives

$$\begin{aligned} \frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\rho }\le \Vert y-{\mathfrak {CG}}^t_m(y)\Vert \le {\mathbf {L}}_m^{{\mathrm{ch}}, t}\,\sigma _m(y)\le \,{\mathbf {L}}_m^{{\mathrm{ch}}, t}\, t\,\Vert \mathbf{1}_{\eta B}\Vert . \end{aligned}$$

Thus,

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\,\ge \,\frac{1}{(\rho +1)t}\,\max \left\{ \frac{\Vert \mathbf{1}_{\eta B}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert },\frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }\right\} , \end{aligned}$$

and taking the supremum over all $|B|=|A|$ with $B\ge (n_0+m)A$ and all ${\varepsilon },\eta \in \Upsilon$, we see that

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\,\ge \,\frac{\vartheta _{n_0+m}(A)}{(\rho +1)t}\,\ge \frac{\inf _{c\ge 1} \vartheta _{c}(A)}{(\rho +1)t}. \end{aligned}$$

Finally, a supremum over all $|A|\le m$ leads to (3.17).

We now give some general conditions in $\{{\mathbf {e}}_n,{\mathbf {e}}^*_n\}_{n=1}^\infty$ and $\mathbb {X}$ under which $\rho$-admissibility holds. We recall a few standard definitions; see e.g. [13]. We use the notation $[{\mathbf {e}}_n]_{n\in A}={\overline{\text{ span }\,}}\{{\mathbf {e}}_n\}_{n\in A}$, for $A\subset \mathbb {N}$. A sequence $\{{\mathbf {e}}_n\}_{n=1}^\infty$ is weakly null if

$$\begin{aligned} \lim _{n\rightarrow \infty }x^*({\mathbf {e}}_n)=0,\quad \forall \;x^*\in \mathbb {X}^*. \end{aligned}$$

Given a subset $Y\subset \mathbb {X}^*$, we shall say that $\{{\mathbf {e}}_n\}_{n=1}^\infty$ is Y-null if

$$\begin{aligned} \lim _{n\rightarrow \infty }y({\mathbf {e}}_n)=0,\quad \forall \;y\in Y. \end{aligned}$$

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

$$\begin{aligned} \sup _{x^* \in Y, \Vert x^* \Vert \le 1} \vert x^*(x) \vert \,\ge \,\kappa \,\Vert x\Vert , \quad \forall \;x\in \mathbb {X}. \end{aligned}$$

We finally introduce a new abstract definition.

Definition 3.7

We say that a biorthogonal system $\{{\mathbf {e}}_n,{\mathbf {e}}^*_n\}_{n=1}^\infty \subset \mathbb {X}\times \mathbb {X}^*$ satisfies the property ${\mathscr {P}(\kappa )}$, for some $0<\kappa \le 1$, if the sequence $\{\Vert {\mathbf {e}}^*_n\Vert \,{\mathbf {e}}_n\}_{n=1}^\infty \subset \mathbb {X}$ is Y-null, for some subset $Y \subset \mathbb {X}^*$ which is $\kappa$-norming.

We remark that in every separable Banach space $\mathbb {X}$ there exists an M-basis $\{{\mathbf {e}}_n,{\mathbf {e}}^*_n\}_{n=1}^\infty$ with the property $\mathscr {P}(1)$; see e.g. [21, Theorem III.8.5].¹ Other examples are given in Remark 3.9 below.

Proposition 3.8

Let $\{{\mathbf {e}}_n,{\mathbf {e}}^*_n\}_{n=1}^\infty$ be a biorthogonal system in $\mathbb {X}\times \mathbb {X}^*$ with the property ${\mathscr {P}(\kappa )}$. Then $\{{\mathbf {e}}_n\}_{n=1}^\infty$ is $\rho$-admissible for every $\rho >1/\kappa$.

Proof

Let $Y\subset \mathbb {X}^*$ be the $\kappa$-norming set from Definition 3.7. Consider a finite set $A\subset \mathbb {N}$ with say $\vert A \vert = m$ and denote

$$\begin{aligned} E :=[{\mathbf {e}}_n]_{n\in A}. \end{aligned}$$

Given $\varepsilon >0$, one can find a finite set $S \subset Y \cap \{x^*\in \mathbb {X}^*{\,\,\,:\,\,\,}\Vert x^*\Vert =1\}$ so that

$$\begin{aligned} \max _{x^* \in S} \vert x^*(e) \vert \ge (1-\varepsilon ) \kappa \Vert e \Vert ,\quad \forall \;e\in E . \end{aligned}$$

(3.18)

Indeed, it suffices to verify the above inequality for e of norm 1. Pick an $\varepsilon \kappa /2$-net $(z_k)_{k=1}^N$ in the unit sphere of E. For any k find a norm one $z_k^* \in Y$ so that $|z_k^*(z_k)| > (1-\varepsilon /2) \kappa$. We claim that $S = \{z_k^* : 1 \le k \le N\}$ has the desired properties. To see this, pick a norm one $e \in E$, and find k with $\Vert e - z_k\Vert \le \varepsilon \kappa /2$. Then

$$\begin{aligned} \max _{x^* \in S} \vert x^*(e) \vert \ge |z_k^*(e)| \ge |z_k^*(z_k)| - \Vert e-z_k\Vert \ge (1-\varepsilon /2) \kappa - \varepsilon \kappa /2 = (1-\varepsilon ) \kappa . \end{aligned}$$

Next, since the sequence $\{\Vert {\mathbf {e}}^*_n\Vert \,{\mathbf {e}}_n\}$ is Y-null, for each $\delta >0$ we can find an integer $n_0> \max A$ so that

$$\begin{aligned} \max _{x^* \in S} \vert x^*({\mathbf {e}}_n) \vert \,\Vert {\mathbf {e}}^*_n\Vert \, \le \frac{\delta \kappa }{m},\quad \forall \; n\ge n_0 . \end{aligned}$$

Pick any B of cardinality m with $\min B\ge n_0$, and let

$$\begin{aligned} G := [{\mathbf {e}}_n]_{n\in B}. \end{aligned}$$

For $f = \sum _{n \in B} {\mathbf {e}}^*_n(f) {\mathbf {e}}_n \in G$, we have

$$\begin{aligned} \max _{x^* \in S} \vert x^*(f) \vert \le \max _{x^*\in S}\,\sum _{n\in B}|x^*({\mathbf {e}}_n)|\,\Vert {\mathbf {e}}^*_n\Vert \,\Vert f\Vert \,\le \, \delta \kappa \Vert f \Vert . \end{aligned}$$

(3.19)

We claim that

$$\begin{aligned} \Vert e+f \Vert \ge \frac{(1-{\varepsilon }-{\delta })\kappa }{1+{\delta }\kappa }\,\Vert e\Vert ,\, \, \, {\text {for any }} \, e \in E, \, f \in G . \end{aligned}$$

(3.20)

To show this, we fix ${\gamma }>0$ (to be chosen later), and assume first that $\Vert f\Vert \ge (1+{\gamma })\Vert e\Vert$. Then,

$$\begin{aligned} \Vert e+f\Vert \ge \Vert f\Vert -\Vert e\Vert \ge {\gamma }\Vert e\Vert . \end{aligned}$$

Next assume that $\Vert f\Vert <(1+{\gamma })\Vert e\Vert$, then using (3.18) and (3.19) we obtain that

$$\begin{aligned} \Vert e+f \Vert \ge \max _{x^* \in S} \vert x^* (e+f) \vert \ge (1-\varepsilon )\kappa \Vert e \Vert - \delta \kappa \Vert f \Vert > (1-\varepsilon - \delta (1+\gamma ))\kappa \Vert e \Vert . \end{aligned}$$

We now choose ${\gamma }$ so that ${\gamma }=(1-\varepsilon - \delta (1+\gamma ))\kappa$, that is,

$$\begin{aligned} {\gamma }=\frac{(1-\varepsilon - \delta )\kappa }{1+{\delta }\kappa }, \end{aligned}$$

which shows the claim in (3.20). Now, given $\rho >1/\kappa$, we may pick ${\delta }={\varepsilon }$ sufficiently small so that the above number ${\gamma }>1/\rho$. Then, (3.20) becomes

$$\begin{aligned} \Vert e+f \Vert \ge \frac{1}{\rho }\,\Vert e\Vert ,\, \, \, {\text {for any }} \, e \in [e_n]_{n\in A}, \, f \in [e_n]_{n\in B},\quad \end{aligned}$$

for all B with $\min B\ge n_0$ and $|B|=|A|=m$. Thus, $\{{\mathbf {e}}_n\}_{n=1}^\infty$ is $\rho$-admissible.

Remark 3.9

We give some more examples where property ${\mathscr {P}(\kappa )}$ holds.

• (1) If the sequence $\{\Vert {\mathbf {e}}^*_n\Vert \,{\mathbf {e}}_n\}_{n=1}^\infty$ is weakly null then $\mathscr {P}(1)$ holds (since $Y = \mathbb {X}^*$ is always 1-norming).

• (2) If $\{{\mathbf {e}}_n\}_{n=1}^\infty$ is a Schauder basis then ${\mathscr {P}(\kappa )}$ holds with $\kappa =1/K_b$; see [20, Theorems I.3.1 and I.12.2].

• (3) Let $\mathbb {X}=C(K)$, where K is a compact Hausdorff set, and let $\mu$ be a Radon probability measure in K with $\mathrm{supp}\mu =K$. Let $\{{\mathbf {e}}_n\}_{n=1}^\infty$ be a complete system in $\mathbb {X}$ which is orthonormal with respect to $\mu$ and uniformly bounded, that is,

  $$\begin{aligned} \int _K {\mathbf {e}}_n {\overline{{\mathbf {e}}_m}} \, d\mu = {\delta }_{n,m}{\quad \text{ and }\quad }\sup _n\Vert {\mathbf {e}}_n\Vert _\infty <\infty . \end{aligned}$$

  Then $\{{\mathbf {e}}_n\}_{n=1}^\infty$ has the property $\mathscr {P}(1)$ in $\mathbb {X}=C(K)$. Indeed, the sequence $\{{\mathbf {e}}_n\}_{n=1}^\infty$ is $L_1(\mu )$-null in $\mathbb {X}$, while $Y=L_1(\mu )$ is 1-norming in $\mathbb {X}$ (since the natural embedding of C(K) into $L_\infty (\mu )$ is isometric). Specific examples are the trigonometric system in C[0, 1] (in the real or complex case), as well as certain polygonal versions of the Walsh system [7, 17, 27], or any reorderings of them (which may cease to be Cesàro bases).

• (4) As a dual of the previous, if $\mathbb {X}=L^1(\mu )$ then every system $\{{\mathbf {e}}_n\}_{n=1}^\infty$ as in (3) is weakly null, and hence case (1) applies.

• (5) If $\{{\mathbf {e}}_n,{\mathbf {e}}^*_n\}_{n=1}^\infty$ is an M-basis such that

  $$\begin{aligned} {\varvec{\varphi }}(m) := \sup _{|A|\le m} \left\| \sum _{n\in A} {\mathbf {e}}_n \right\| \,=\,{\mathbf {o}}(m), \quad \mathrm{as}\quad m\rightarrow \infty , \end{aligned}$$

  then $\{{\mathbf {e}}_n\}_{n=1}^\infty$ is weakly null (and in particular, $\mathscr {P}(1)$ holds). Indeed, first note that also ${{{\tilde{\varvec{\varphi }}}}}(m) = \sup \lbrace \Vert \mathbf{1}_{\eta A} \Vert : \vert A \vert \le m,\;|\eta |=1 \rbrace ={\mathbf {o}}(m)$. Assume that the system is not weakly null. Then there exist a norm one $x^* \in \mathbb {X}^*$ and $\varepsilon _0 > 0$ so that the set $A=\{ n \in \mathbb {N} : \vert x^*({\mathbf {e}}_n) \vert \ge \varepsilon _0 \}$ is infinite. For every $m\ge 1$, pick a set $F \subset A$ with $|F| = m$ and let $\eta _n=\mathrm{sign}[ x^*({\mathbf {e}}_n)]$; then

  $$\begin{aligned} {{{\tilde{\varvec{\varphi }}}}}(m) \ge \Vert \mathbf{1}_{{\overline{\eta }}F} \Vert \ge \left| x^*\left( \sum _{n \in F} {\overline{\eta _n}}{\mathbf {e}}_n\right) \right| =\sum _{n\in F}|x^*({\mathbf {e}}_n) | \ge m\varepsilon _0 , \end{aligned}$$

  contradicting our assumption.

Finally, as a consequence of Propositions 3.6 and 3.8 one obtains

Theorem 3.10

Let $\{{\mathbf {e}}_n,{\mathbf {e}}^*_n\}_{n=1}^\infty$ be a seminormalized M-basis with the property ${\mathscr {P}(\kappa )}$. Then, if $\vartheta _m$ is as in (3.14), we have

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\ge \frac{\kappa \,\vartheta _m}{(\kappa +1)t},\quad \forall \;m\in \mathbb {N},\quad t\in (0,1]. \end{aligned}$$

(3.21)

4 Examples

The first two examples are variants of those in [5, §5.1] and [6, §8.1].

4.1 Example 4.1: the summing basis

Let $\mathbb {X}$ be the closure of the set of all finite sequences $\mathbf {a}=(a_n)_n\in c_{00}$ with the norm

$$\begin{aligned} \Vert \mathbf {a}\Vert = \sup _m\left| \sum _{n=1}^m a_n\right| . \end{aligned}$$

The canonical system $\mathscr {B}=\lbrace {\mathbf {e}}_n\rbrace _{n=1}^\infty$ is a Schauder basis in $\mathbb {X}$ with $K_b=1$ and $\Vert {\mathbf {e}}_n\Vert = 1$ for all n. Also, $\Vert \mathbf {e}_1^*\Vert = 1$, $\Vert {{\mathbf {e}}^*_n}\Vert = 2$ if $n\ge 2$, so $\mathfrak {K}=2$ in Theorem 1.1; see [5, §5.1]. We now show that, for this example of $(\mathbb {X},\mathscr {B})$, the bound of Theorem 1.1 is sharp. As in [5, §5.1], we consider the element:

$$\begin{aligned} x=\left( \underbrace{\frac{1}{2},\frac{1}{t},\frac{1}{2}},\ldots ,\underbrace{\frac{1}{2},\frac{1}{t},\frac{1}{2}}; \frac{1}{2}; \underbrace{-\,1,1},\ldots ,\underbrace{-\,1,1},0,\ldots ,\right) , \end{aligned}$$

where we have m blocks of $\left( \frac{1}{2},\frac{1}{t},\frac{1}{2}\right)$ and m blocks of $(-\,1,1)$. Picking $A=\lbrace n : x_n=-\,1\rbrace$ as a t-greedy set of x, we see that

$$\begin{aligned} \Vert x-{\mathfrak {CG}}_m^t(x)\Vert= & {} \min _{a_i, i =1,\ldots ,m}\left\| \left( \frac{1}{2},\frac{1}{t},\frac{1}{2},\ldots ,\frac{1}{2},\frac{1}{t},\frac{1}{2}; \frac{1}{2}; a_1, 1, a_2, 1,\ldots ,a_m,1,0,\ldots ,\right) \right\| \\\ge & {} \left\| \left( \frac{1}{2},\frac{1}{t},\frac{1}{2},\ldots ,\frac{1}{2},\frac{1}{t},\frac{1}{2}; \frac{1}{2}; 0,\ldots ,\right) \right\| = m+\frac{m}{t}+\frac{1}{2}. \end{aligned}$$

On the other hand,

$$\begin{aligned} \sigma _m(x)\le & {} \Big \Vert x-\frac{t+1}{t}(0,1,0,\ldots ,0,1,0; 0,\ldots )\Big \Vert \\= & {} \Big \Vert \Big (\frac{1}{2},-\,1,\frac{1}{2},\ldots ,\frac{1}{2},-\,1,\frac{1}{2}; \frac{1}{2};-\,1,1,\ldots ,-\,1,1,0\ldots ,\Big )\Big \Vert =\frac{1}{2}. \end{aligned}$$

Hence, ${\mathbf {L}}_m^{{\mathrm{ch}}, t}\ge 1+2(1+\frac{1}{t})m$ and we conclude that ${\mathbf {L}}_m^{{\mathrm{ch}}, t}= 1+2(1+\frac{1}{t})m$ by Theorem 1.1. As a consequence, observe that in this case ${\mathfrak {CG}}^t_m(x)=0$.

Remark 4.1

The above example strengthens [19, Theorem 2.4], where the authors are only able to show that $1+4m\le {\mathbf {L}}_m^{\mathrm{ch}}\le 1+6m$.

4.2 Example 4.2: the difference basis

Let $\lbrace {\mathbf {e}}_n\rbrace _{n=1}^\infty$ be the canonical basis in $\ell ^1(\mathbb {N})$ and define the elements

$$\begin{aligned} y_1 = {\mathbf {e}}_1,\; y_n = {\mathbf {e}}_n-{\mathbf {e}}_{n-1},\; n=2,3,\ldots \end{aligned}$$

The new system $\mathscr {B}= \lbrace y_n\rbrace _{n=1}^\infty$ is called the difference basis of $\ell ^1$. We recall some basic properties used in [6, §8.1]. If $(b_n)_n\in c_{00}$ then

$$\begin{aligned} \left\| \sum _{n=1}^\infty b_n y_n\right\| = \sum _{n=1}^\infty \vert b_n-b_{n+1}\vert . \end{aligned}$$

Also, $\mathscr {B}$ is a monotone basis with $\Vert y_1\Vert = 1$, $\Vert y_n\Vert = 2$ if $n\ge 2$, and $\Vert y_n^*\Vert = 1$ for all $n\ge 1$ (in fact, the dual system corresponds to the summing basis). So, $\mathfrak {K}=2$ and Theorem 1.1 gives ${\mathbf {L}}_m^{{\mathrm{ch}}, t}\le 1+2(1+\frac{1}{t})m$ for all $t\in (0,1]$. To show the equality we consider the vector $x=\sum _nb_n y_n$ with coefficients $(b_n)$ given by

$$\begin{aligned} \left( 1,\underbrace{1,1,-\,\tfrac{1}{t},1},\ldots ,\underbrace{1,1,-\,\tfrac{1}{t},1},0,\ldots ,\right) , \end{aligned}$$

where the block $\Big (1,1,\frac{-\,1}{t},1\Big )$ is repeated m times. If we take $\Gamma =\lbrace 2,6,\ldots ,4m-2\rbrace$ as a t-greedy set for x of cardinality m, then

$$\begin{aligned} \Vert x-{\mathfrak {CG}}_m^t(x)\Vert= & {} \inf _{(a_j)_{j=1}^m}\left\| x-\sum _{j=1}^m a_jy_{4j-2}\right\| \\= & {} \inf _{(a_j)_{j=1}^m}\left\| \left( 1,1-a_1,1,\frac{-1}{t},1,\ldots ,1-a_m,1,\frac{-1}{t},1,0,\ldots ,\right) \right\| \\= & {} \inf _{(a_j)_{j=1}^m}2\sum _{j=1}^m \vert a_j\vert + 2m\left( 1+\frac{1}{t}\right) +1=2m\left( 1+\frac{1}{t}\right) +1. \end{aligned}$$

Hence, in this case we also have ${\mathfrak {CG}}^t_m(x)=0$. On the other hand

$$\begin{aligned} \sigma _m(x)\le \big \Vert x+\big (1+\tfrac{1}{t}\big )\sum _{j=1}^m y_{4j}\big \Vert = \Vert (1,1,1,1,1,\ldots ,1,1,1,1,0,\ldots )\Vert =1. \end{aligned}$$

This shows that ${\mathbf {L}}_m^{{\mathrm{ch}}, t}= 1+2(1+\frac{1}{t})m$.

4.3 Example 4.3: the trigonometric system in $L^p(\mathbb {T})$

Consider $\mathscr {B}=\{e^{i nx}\}_{n\in \mathbb {Z}}$ in $L^p(\mathbb {T})$ for $1\le p<\infty$, and in $C(\mathbb {T})$ if $p=\infty$. In [22], Temlyakov showed that

$$\begin{aligned} c_pm^{|\frac{1}{p}-\frac{1}{2}|}\le {\mathbf {L}}_m\le 1+3m^{\left| \frac{1}{p}-\frac{1}{2}\right| }, \end{aligned}$$

for some $c_p>0$ and all $1\le p\le \infty$. Adapting his argument, Shao and Ye have recently established, in [19, Theorem 2.1], that for $1<p\le \infty$ it also holds

$$\begin{aligned} {\mathbf {L}}_m^{\mathrm{ch}}\approx m^{|\frac{1}{p}-\frac{1}{2}|}. \end{aligned}$$

(4.1)

The case $p=1$ is left as an open question, and only the estimate $\frac{\sqrt{m}}{\ln (m)}\lesssim {\mathbf {L}}_m^{\mathrm{ch}}\lesssim \sqrt{m}$ is given; see [19, (2.24)]. Moreover, the proof of the case $p=\infty$ seems to contain some gaps and may not be complete.

Here, we shall give a short proof ensuring the validity of (4.1) in the full range $1\le p\le \infty$, with a reasoning similar to [5, §5.4]. More precisely, we shall prove the following.

Proposition 4.2

Let $1\le p\le \infty$. Then there exists $c_p>0$ such that

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\,\ge \, c_p\,t^{-1}\, m^{|\frac{1}{p}-\frac{1}{2}|},\quad \forall \;m\in \mathbb {N},\quad t\in (0,1]. \end{aligned}$$

(4.2)

We remark that in the cases $p=1$ and $p=\infty$ the trigonometric system is not a Schauder basis, but it is a Cesàro basis.² So we may use the lower bounds in Theorem 1.3, namely

$$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\ge \,{c'_p}\;t^{-1}\,\sup _{\begin{array}{c} |A|=|B|\le m \\ A>2B\;\;\text {or}\;\;B>2A \end{array}}\,\sup _{|{\varepsilon }|=|\eta |=1}\;\frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }. \end{aligned}$$

(4.3)

• Case $1<p\le 2$. Assume that $m=2\ell +1$ or $2\ell +2$ (that is, $\ell =\lfloor \frac{m-1}{2}\rfloor$). We choose $B=\{-\,\ell ,\ldots ,\ell \}$, so that $\mathbf{1}_B=D_\ell$ is the $\ell$th Dirichlet kernel, and hence

  $$\begin{aligned} \Vert \mathbf{1}_B\Vert _p=\Vert D_\ell \Vert _{L^p(\mathbb {T})}\approx m^{1-\frac{1}{p}}. \end{aligned}$$

  Next we take a lacunary set $A=\lbrace 2^j : j_0\le j\le j_0+2\ell \rbrace$, so that

  $$\begin{aligned} \Vert \mathbf{1}_A\Vert _p\approx \sqrt{m}, \end{aligned}$$

  (4.4)

  and where $j_0$ is chosen such that $2^{j_0}\ge m$, and hence $A>2B$. Then, (4.3) implies

  $$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\ge \,{c_p}\;t^{-1}\frac{m^{1/2}}{m^{1-\frac{1}{p}}}\,=\,c_p\,t^{-1}\, m^{\left| \frac{1}{p}-\frac{1}{2}\right| }. \end{aligned}$$

• Case $2\le p<\infty$. The same proof works in this case, just reversing the roles of A and B.

• Case $p=\infty$. We replace the lacunary set by a Rudin-Shapiro polynomial of the form

  $$\begin{aligned} R(x)=e^{iNx}\,\sum _{n=0}^{2^L-1}{\varepsilon }_n e^{inx},\quad \text{ with } \;{\varepsilon }_n\in \{\pm \,1\}, \end{aligned}$$

  where L is such that $2^L\le m<2^{L+1}$; see e.g. [14, p. 33]. Then, $R=\mathbf{1}_{{\varepsilon }B}$ with $B=N+\{0,1,\ldots ,2^L-1\}$ and

  $$\begin{aligned} \Vert \mathbf{1}_{{\varepsilon }B}\Vert _\infty =\Vert R\Vert _{L^\infty (\mathbb {T})}\approx \sqrt{m}. \end{aligned}$$

  If we pick $N\ge 2\cdot 2^L$, then $B>2A$ with $A=\{\pm \,1,\ldots ,\pm \, (2^L-1)\}$. Finally,

  $$\begin{aligned} \Vert \mathbf{1}_A\Vert _\infty =\Vert D_{2^L-1}-1\Vert _{L^\infty (\mathbb {T})} \approx \, m. \end{aligned}$$

  So, (4.3) implies the desired bound.

• Case $p=1$. We use the lower bound in Lemma 3.4, namely

  $$\begin{aligned} {\mathbf {L}}_m^{{\mathrm{ch}}, t}\,\ge \, c_1'\;t^{-1}\;\frac{\Vert \mathbf{1}_A\Vert }{\Vert \mathbf{1}_B+y\Vert }, \end{aligned}$$

  (4.5)

  for all $|A|=|B|\le m$ and all y such that and $\sup _n|{\mathbf {e}}^*_n(y)|\le 1$. As before, let $m=2\ell +1$ or $2\ell +2$, and choose the same sets A and B as in the case $1<p\le 2$. Next choose y so that the vector

  $$\begin{aligned} V_\ell =\mathbf{1}_B+y \end{aligned}$$

  is a de la Vallée-Poussin kernel as in [14, p. 15]. Then, the Fourier coeffients ${\mathbf {e}}^*_n(y)$ have modulus $\le 1$ and are supported in $\{n{\,\,\,:\,\,\,}\ell <|n|\le 2\ell +1\}$, so the condition holds if $2^{j_0}\ge 2m+1$. Finally,

  $$\begin{aligned} \Vert \mathbf{1}_B+y\Vert _1=\Vert V_\ell \Vert _{L^1(\mathbb {T})}\le 3, \end{aligned}$$

  so the bound ${\mathbf {L}}_m^{{\mathrm{ch}}, t}\gtrsim t^{-1}\sqrt{m}$ follows from (4.5).

Remark 4.3

Using the trivial upper bound ${\mathbf {L}}_m^{{\mathrm{ch}}, t}\le {\mathbf {L}}_m^t\lesssim t^{-1} m^{|\frac{1}{p}-\frac{1}{2}|}$, we conclude that ${\mathbf {L}}_m^{{\mathrm{ch}}, t}\approx t^{-1}m^{\vert \frac{1}{p}-\frac{1}{2}\vert }$ for all $1\le p\le \infty$.

5 Comparison between ${\tilde{\mu }}_m$ and ${\tilde{\mu }}^d_m$

In this section we compare the democracy constants ${\tilde{\mu }}_m$ and ${\tilde{\mu }}^d_m$ defined in §1 above. Let us first note that

$$\begin{aligned} {\tilde{\mu }}^d_m\le {\tilde{\mu }}_m\le ({\tilde{\mu }}^d_m)^2 \end{aligned}$$

(5.1)

and

$$\begin{aligned} {\tilde{\mu }}_m^d\le {\tilde{\mu }}_m\le (1+2\kappa ){\gamma }_m{\tilde{\mu }}_m^d, \end{aligned}$$

(5.2)

where $\kappa =1$ or 2 depending if $\mathbb {K}=\mathbb {R}$ or $\mathbb {C}$. Indeed, the left inequality in (5.1) is immediate by definition, and the right one follows from

$$\begin{aligned} \frac{\Vert \mathbf{1}_{\eta B}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }= \frac{\Vert \mathbf{1}_{\eta B}\Vert }{\Vert \mathbf{1}_{C}\Vert }\,\frac{\Vert \mathbf{1}_{C}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }\le ({\tilde{\mu }}^d_m)^2, \end{aligned}$$

for any $|A|=|B|\le m$ and any C disjoint with $A\cup B$ with $|C|=|A|=|B|$. Concerning the right inequality in (5.2), we use that if $|A|=|B|\le m$ then

$$\begin{aligned} \frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }\le \frac{\Vert \mathbf{1}_{{\varepsilon }(A{\setminus } B)}\Vert +\Vert \mathbf{1}_{{\varepsilon }(A\cap B)}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }\le {\gamma }_m \frac{\Vert \mathbf{1}_{{\varepsilon }(A{\setminus } B)}\Vert }{\Vert \mathbf{1}_{\eta (B{\setminus } A)}\Vert } +\frac{\Vert \mathbf{1}_{{\varepsilon }(A\cap B)}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }\le {\gamma }_m\,{\tilde{\mu }}_m^d+2\kappa {\gamma }_m, \end{aligned}$$

using in the last step [5, Lemma 3.3]. From (5.2) we see that ${\tilde{\mu }}_m\approx {\tilde{\mu }}_m^d$ when $\mathscr {B}$ is quasi-greedy for constant coefficients.

In the next subsection we shall show that ${\tilde{\mu }}_m\approx {\tilde{\mu }}^d_m$ for all Schauder bases, a result which seems new in the literature.

5.1 Equivalence for Schauder bases

We begin with a simple observation.

Lemma 5.1

$$\begin{aligned} {\tilde{\mu }}^d_m=\sup \left\{ \frac{\Vert \mathbf{1}_{\eta B}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\,\,\,:\,\,\,}|B|\le |A|\le m,\;\;A\cap B=\emptyset ,\;\;|{\varepsilon }|=|\eta |=1\right\} . \end{aligned}$$

(5.3)

Proof

Let $|{\varepsilon }|=|\eta |=1$ and $|B| \le |A| \le m$ with $A\cap B=\emptyset$. We must show that $\Vert \mathbf{1}_{\eta B}\Vert /\Vert \mathbf{1}_{{\varepsilon }A}\Vert \le {\tilde{\mu }}^d_m$. Pick any set C disjoint with $A\cup B$ such that $|B|+|C| = |A|$. We now use the elementary inequality

$$\begin{aligned} \Vert x\Vert =\Big \Vert \frac{x+y}{2}+\frac{x-y}{2}\Big \Vert \le \max \{\Vert x+y\Vert ,\Vert x-y\Vert \}, \end{aligned}$$

(5.4)

with $x=\mathbf{1}_{\eta B}$ and $y=\mathbf{1}_C$. Let $\eta '\in \Upsilon$ be such that $\eta '|_B = \eta |_B$ and $\eta '|_C=\pm 1$, according to the sign that reaches the maximum in (5.4). Then $\Vert \mathbf{1}_{\eta B}\Vert \le \Vert \mathbf{1}_{\eta '(B\cup C)}\Vert \le {\tilde{\mu }}^d_m\Vert \mathbf{1}_{{\varepsilon }A}\Vert$, and the result follows.

Theorem 5.2

If $K_b$ is the basis constant and $\varkappa =\sup _{n}\Vert {{\mathbf {e}}^*_n}\Vert \Vert {\mathbf {e}}_n\Vert$, then

$$\begin{aligned} {\tilde{\mu }}_m\le 2(K_b+1){\tilde{\mu }}^d_m +\varkappa \,K_b. \end{aligned}$$

(5.5)

Proof

Let $|A|=|B|\le m$, and $|{\varepsilon }|=|\eta |=1$. Then

$$\begin{aligned} \frac{\Vert \mathbf{1}_{\eta B}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert } \le \frac{\Vert \mathbf{1}_{\eta (B{\setminus } A)}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }+\frac{\Vert \mathbf{1}_{\eta (B\cap A)}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert } = I+II. \end{aligned}$$

Lemma 5.1 implies $I\le {\tilde{\mu }}_m^d$. We now bound II. Pick an integer $n_0$ such that $A_1=\{n\in A{\,\,\,:\,\,\,}n\le n_0\}$ and $A_2=A{\setminus } A_1$ satisfy

$$\begin{aligned} |A_1|=|A_2| \quad \text{(if } |A| \text{ is } \text{ even), } \text{ or }\quad |A_1|=\frac{|A|-1}{2}=|A_2|-1\quad \text{(if } |A| \text{ is } \text{ odd) }. \end{aligned}$$

Then

$$\begin{aligned} II\le & {} \frac{\Vert \mathbf{1}_{\eta (B\cap A_1)}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }+\frac{\Vert \mathbf{1}_{\eta (B\cap A_2)}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }\\\le & {} (K_b+1)\frac{\Vert \mathbf{1}_{\eta (B\cap A_1)}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A_2}\Vert }+K_b\frac{\Vert \mathbf{1}_{\eta (B\cap A_2)}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A_1}\Vert }\;=\;II_1+II_2, \end{aligned}$$

using in the second line the basis constant bound for the denominator. Since $|B\cap A_1|\le |A_1|\le |A_2|$, we see that

$$\begin{aligned} II_1\le (K_b+1){\tilde{\mu }}^d_m. \end{aligned}$$

On the other hand, picking any number $n_1\in B\cap A_2$, and using $\Vert {\mathbf {e}}^*_{n_1}\Vert \Vert \mathbf{1}_{{\varepsilon }A}\Vert \ge |{\mathbf {e}}^*_{n_1}(\mathbf{1}_{{\varepsilon }A})|=1$, we see that

$$\begin{aligned} II_2\le K_b\frac{\Vert \mathbf{1}_{\eta (B\cap A_2{\setminus }\{n_1\})}\Vert }{\Vert \mathbf{1}_{{\varepsilon }A_1}\Vert }+ K_b\Vert {\mathbf {e}}_{n_1}\Vert \Vert {\mathbf {e}}^*_{n_1}\Vert \le K_b{\tilde{\mu }}^d_m+\varkappa K_b, \end{aligned}$$

the last bound due to $|B\cap A_2{\setminus }\{n_1\}|\le |A_2|-1\le |A_1|$ and Lemma 5.1. Putting together the previous bounds easily leads to (5.5).

Remark 5.3

A similar argument shows the equivalence of the standard (unsigned) democracy parameters

$$\begin{aligned} \mu _m=\sup _{|A|=|B|\le m}\frac{\Vert \mathbf{1}_{B}\Vert }{\Vert \mathbf{1}_{A}\Vert }{\quad \text{ and }\quad }\mu ^d_m=\sup _{\begin{array}{c} |A|=|B|\le m \\ A\cap B=\emptyset \end{array}}\frac{\Vert \mathbf{1}_{B}\Vert }{\Vert \mathbf{1}_{A}\Vert }. \end{aligned}$$

(5.6)

Indeed, in this case, the analog of (5.3) takes the weaker form

$$\begin{aligned} \mu ^d_m\le \sup _{\begin{array}{c} |B|\le |A|\le m \\ A\cap B=\emptyset \end{array}}\frac{\Vert \mathbf{1}_{ B}\Vert }{\Vert \mathbf{1}_{ A}\Vert }\le K_b\mu ^d_m. \end{aligned}$$

(5.7)

Then, (5.7) and the same proof we gave for Theorem 5.2 (with $\eta ={\varepsilon }\equiv 1$) leads to

$$\begin{aligned} \mu _m\le 2(K_b+1)K_b\,\mu ^d_m +\varkappa \,K_b. \end{aligned}$$

(5.8)

5.2 An example where ${\tilde{\mu }}_m$ grows faster than ${\tilde{\mu }}^d_m$

The following example also seems to be new in the literature. As in (5.6), we denote by $\mu _m$, $\mu ^d_m$ the democracy parameters corresponding to constant signs.

Theorem 5.4

There exists a Banach space $\mathbb {X}$ with an M-basis $\mathscr {B}$ such that

$$\begin{aligned} \limsup _{m\rightarrow \infty }\frac{{\tilde{\mu }}_m}{[{\tilde{\mu }}^d_m]^{2-{\varepsilon }}}=\limsup _{m\rightarrow \infty }\frac{\mu _m}{[\mu ^d_m]^{2-{\varepsilon }}}=\infty ,\quad \forall \;{\varepsilon }>0. \end{aligned}$$

Proof

Let $N_0=1$, and define recursively $N_k=2^{2^{N_{k-1}}}$, and $N_k'=N_1+\cdots +N_{k-1}$. Consider the blocks of integers

$$\begin{aligned} S_k=\big \{N'_k+1,\ldots , N'_k+N_k \big \}, \end{aligned}$$

and denote the tail blocks by $T_k=\cup _{j\ge k+1} S_{j}$. Finally, let

$$\begin{aligned} \mathfrak {N}_k=\left\{ (\sigma _j)_{j\in S_k}{\,\,\,:\,\,\,}\sigma _j\in \{\pm \,1\}{\quad \text{ and }\quad }\sum _{j\in S_k}\sigma _j=0\right\} . \end{aligned}$$

We define a real Banach space $\mathbb {X}$ as the closure of $c_{00}$ with the norm

$$\begin{aligned} \Vert x\Vert \;=\;\max \left\{ \;\Vert x\Vert _\infty , \;\sup _{k\ge 1} {\alpha }_k\,\sup _{\sigma \in \mathfrak {N}_k}\big |\langle \mathbf{1}_{\sigma S_k},x\rangle \big |,\; \sup _{k\ge 1}\beta _k\sup _{\begin{array}{c} S\subset T_k \\ |S|=N_k \end{array}}\sum _{j\in S}|x_j|\;\right\} , \end{aligned}$$

where the weights ${\alpha }_k$ and $\beta _k$ are chosen as follows:

$$\begin{aligned} {\alpha }_k=2^{-N_{k-1}}=\frac{1}{\log _2 N_k}{\quad \text{ and }\quad }\beta _k=\frac{1}{\sqrt{N_k}}. \end{aligned}$$

Observe that

$$\begin{aligned} N'_k=N_1+\cdots +N_{k-1}\le 2 N_{k-1}=2\log _2\log _2 N_k {\quad \text{ and }\quad }\frac{{\alpha }_k}{\beta _k}=\frac{\sqrt{N_k}}{\log _2 N_k}. \end{aligned}$$

Claim 1

$\quad \displaystyle {\tilde{\mu }}_{N_k}\ge \mu _{N_k}\ge \frac{N_k/2}{(\log _2 N_k)\sqrt{\log _2\log _2 N_k}}$, for all $k\ge 1$.

Proof

Pick any $A\subset S_k\cup S_{k+1}$ such that $|A|=N_k$ and $|A\cap S_k|=|A\cap S_{k+1}|=N_k/2$. Then

$$\begin{aligned} \Vert \mathbf{1}_A\Vert \ge {\alpha }_k\,N_k/2=\frac{N_k/2}{\log _2 N_k}. \end{aligned}$$

Next, pick $B=S_k$, so that $|B|=|A|=N_k$ and

$$\begin{aligned} \Vert \mathbf{1}_B\Vert =\max \Big \{\;1,\;{\alpha }_k\cdot 0,\;\sup _{n\le k-1}\beta _n N_n\;\Big \}=\beta _{k-1}N_{k-1}=\sqrt{N_{k-1}}= \sqrt{\log _2\log _2 N_k}. \end{aligned}$$

Then $\mu _{N_k}\ge \Vert \mathbf{1}_A\Vert /\Vert \mathbf{1}_B\Vert \ge \frac{N_k/2}{(\log _2 N_k)\sqrt{\log _2\log _2 N_k}}$.

Claim 2

$\quad \displaystyle \mu ^d_{N_k}\le {\tilde{\mu }}^d_{N_k}\le \sqrt{N_k}$, for all $k\ge 2$.

Proof

Let A, B be any pair of disjoint sets with $|A|=|B|\le N_k$, and let $|{\varepsilon }|=|\eta |=1$. If $|A|=|B|\le \sqrt{N_k}$, then the trivial bounds $\Vert \mathbf{1}_{{\varepsilon }A}\Vert \le |A|$ and $\Vert \mathbf{1}_{\eta B}\Vert \ge 1$ give

$$\begin{aligned} \frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }\le \sqrt{N_k}. \end{aligned}$$

So, it remains to consider the cases $\sqrt{N_k}<|A|=|B|\le N_k$. We split A into three parts

$$\begin{aligned} A_0=A\cap S_k,\quad A_+=A\cap T_k,\quad A_-=A\cap [S_1\cup \ldots \cup S_{k-1}]. \end{aligned}$$

Then, we have the following upper bound

$$\begin{aligned} \Vert \mathbf{1}_{{\varepsilon }A}\Vert\le & {} \max \Big \{1,\; \sup _{n<k}{\alpha }_{n}|A_-|, \;{\alpha }_k|A_0|,\;\sup _{n>k}{\alpha }_n N_k,\; \sup _{n<k}\beta _nN_n,\;\sup _{n\ge k}\beta _n|A|\;\Big \}\\\le & {} \max \Big \{\; N'_k, \;{\alpha }_k|A_0|,\;\beta _k|A|\;\Big \}, \end{aligned}$$

due to the elementary inequalities

• $\sup _{n<k} {\alpha }_n|A_-|\le |A_-|\le N'_k$

• $\sup _{n>k}{\alpha }_n N_k={\alpha }_{k+1} N_k=N_k2^{-N_k}\le 1$

• $\sup _{n<k}\beta _nN_n=\sqrt{N_{k-1}}\le N_{k-1}\le N'_k$

• $\sup _{n\ge k}\beta _n|A|=\beta _k|A|$.

Moreover, since $\beta _k|A|\le \min \{\beta _kN_k=\sqrt{N_k},\;{\alpha }_k|A|\;\}$, we derive

$$\begin{aligned} \Vert \mathbf{1}_{{\varepsilon }A}\Vert \le \max \{\sqrt{N_k}, {\alpha }_k|A_0|\}{\quad \text{ and }\quad }\Vert \mathbf{1}_{{\varepsilon }A}\Vert \le \max \{N'_k,{\alpha }_k|A|\}. \end{aligned}$$

(5.9)

We now give a lower bound for $\Vert \mathbf{1}_{\eta B}\Vert$. The key estimate will rely on the following

Lemma 5.5

Let $B_0=B\cap S_k$ and $B^c_0=S_k{\setminus } B_0$. Then

$$\begin{aligned} \sup _{\sigma \in \mathfrak {N}_k}\big |\langle \mathbf{1}_{\sigma S_k},\mathbf{1}_{\eta B_0}\rangle \big |\,\ge \, \min \{|B_0|,|B_0^c|\}. \end{aligned}$$

(5.10)

Proof

If $|B_0| \le N_k/2$, then we may select any $\sigma \in \mathfrak {N}_k$ such that $\sigma |_{B_0} = \eta$ (which is possible since $\vert B_0^c \vert \ge \vert B_0 \vert$), which gives

$$\begin{aligned} |\langle \mathbf{1}_{\sigma S_k}, \mathbf{1}_{\eta B_0} \rangle | = |B_0|=\min \{|B_0|,|B_0^c|\}. \end{aligned}$$

Assume now that $\vert B_0 \vert > N_k/2$. Pick any $S \subset B_0$ with $\vert S \vert = \vert B_0^c \vert = N_k - \vert B_0 \vert$. Choose $\nu \in \lbrace -1,1\rbrace ^{B_0^c}$ so that $\sum _{i \in S} \eta _i + \sum _{i \in B_0^c} \nu _i = 0$. Choose $\tau \in \lbrace -1,1\rbrace ^{B_0 \backslash S}$ so that $\sum _{i \in B_0 \backslash S} \tau _i = 0$. Replacing $\tau$ by $-\tau$, if necessary, we may assume that $\sum _{i\in B_0{\setminus } S}\tau _i\eta _i\ge 0$. Finally, define $\sigma \in \mathfrak {N}_k$ by setting

$$\begin{aligned} \sigma |_S= \eta |_S, \quad \sigma |_{B^c_0}= \nu |_{B^c_0}, \quad \sigma |_{B_0 \backslash S}=\tau |_{B_0 \backslash S}. \end{aligned}$$

Then,

$$\begin{aligned} |\langle \mathbf{1}_{\sigma S_k}, \mathbf{1}_{\eta B_0} \rangle |\, =\, \sum _{i\in S} \eta _i^2 + \sum _{i \in B_0 \backslash S} \tau _i \eta _i \, \ge |S| = |B_0^c| =\min \{|B_0|,|B_0^c|\} \,. \end{aligned}$$

$\square$

From the lemma and the definition of the norm we see that

$$\begin{aligned} \Vert \mathbf{1}_{\eta B}\Vert \ge \max \Big \{\,1,\; {\alpha }_k\min \{|B_0|,|B^c_0|\},\;\beta _k|B_+|\;\Big \}. \end{aligned}$$

(5.11)

We shall finally combine the estimates in (5.9) and (5.11) to establish Claim 2. We distinguish two cases

Case 1: $\min \{|B_0|,|B^c_0|\}=|B^c_0|$. Then, since $A_0\subset B^c_0$, we see that

$$\begin{aligned} {\alpha }_k|A_0|\le {\alpha }_k|B^c_0|\le \Vert \mathbf{1}_{\eta B}\Vert , \end{aligned}$$

and therefore the first estimate in (5.9) gives

$$\begin{aligned} \frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }\le \frac{\max \{\sqrt{N_k},\Vert \mathbf{1}_{\eta B}\Vert \}}{\Vert \mathbf{1}_{\eta B}\Vert } \le \sqrt{N_k}. \end{aligned}$$

Case 2: $\min \{|B_0|,|B^c_0|\}=|B_0|$. Then, (5.11) reduces to

$$\begin{aligned} \Vert \mathbf{1}_{\eta B}\Vert \ge \max \big \{\,{\alpha }_k|B_0|,\;\beta _k|B_+|\;\big \}\ge \beta _k\frac{|B_0|+|B_+|}{2}= \beta _k\frac{|B|-|B_-|}{2}\ge \beta _k|B|/4, \end{aligned}$$

since $|B_-|\le N'_k\le \sqrt{N_k}/2\le |B|/2$, if $k\ge 2$. Also, the second bound in (5.9) reads

$$\begin{aligned} \Vert \mathbf{1}_{{\varepsilon }A}\Vert \le {\alpha }_k |A|, \end{aligned}$$

since $N'_k\le \sqrt{N_k}/\log _2 N_k={\alpha }_k\sqrt{N_k}\le {\alpha }_k|A|$, if $k\ge 2$. Thus

$$\begin{aligned} \frac{\Vert \mathbf{1}_{{\varepsilon }A}\Vert }{\Vert \mathbf{1}_{\eta B}\Vert }\le \frac{{\alpha }_k|A|}{\beta _k|B|/4}=\frac{4{\alpha }_k}{\beta _k}= \frac{4\sqrt{N_k}}{\log _2 N_k} \le \sqrt{N_k}. \end{aligned}$$

This establishes Claim 2.

From Claims 1 and 2 we now deduce that

$$\begin{aligned} \frac{\mu _{N_k}}{[{\tilde{\mu }}^d_{N_k}]^{2-{\varepsilon }}}\ge \frac{N_k^{{\varepsilon }/2}/2}{(\log _2 N_k)\sqrt{\log _2\log _2 N_k}}\rightarrow \infty , \end{aligned}$$

and therefore

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{\mu _{N}}{[\mu ^d_{N}]^{2-{\varepsilon }}}=\limsup _{N\rightarrow \infty }\frac{{\tilde{\mu }}_{N}}{[{\tilde{\mu }}^d_{N}]^{2-{\varepsilon }}}=\infty . \end{aligned}$$

$\square$

6 Norm convergence of ${\mathfrak {CG}}^t_m x$ and $\mathscr {G}_m^t x$

In this section we search for conditions on $\mathscr {B}=\{{\mathbf {e}}_n\}_{n=1}^\infty$ under which it holds

$$\begin{aligned} \Vert x-{\mathfrak {CG}}_m(x)\Vert \rightarrow 0,\quad \forall \;x\in \mathbb {X}. \end{aligned}$$

(6.1)

In [19, Theorem 1.1] this convergence is asserted for every basis in $\mathbb {X}$. Here we investigate whether (6.1) may be true for a general M-basis, as defined in §1.

The solution to this question requires the notion of strong M-basis; see [21, Def 8.4]. We say that $\mathscr {B}$ is a strong M-basis if additionally to the conditions (a)–(d) in §1 it also holds

$$\begin{aligned} \,{\overline{\text{ span }\,\{{\mathbf {e}}_n\}_{n\in A}}}\,=\,\big \{x\in \mathbb {X}{\,\,\,:\,\,\,}\mathrm{supp}x\subset A\big \},\quad \forall \;A\subset \mathbb {N}. \end{aligned}$$

(6.2)

Clearly, all Schauder or Cesàro bases (in some ordering) are strong M-bases; see e.g. [18] for further examples. However, there exist M-bases which are not strong M-bases, see e.g. [21, p. 244], or [11]³ for seminormalized examples in Hilbert spaces.

Lemma 6.1

If $\mathscr {B}$ is an M-basis which is not a strong M-basis, then there exists an $x_0\in \mathbb {X}$ such that, for all Chebyshev greedy operators ${\mathfrak {CG}}_m$,

$$\begin{aligned} \liminf _{m\rightarrow \infty }\Vert x_0-{\mathfrak {CG}}_m(x_0)\Vert >0. \end{aligned}$$

(6.3)

Proof

If $\mathscr {B}$ is not a strong M-basis there exists some set $A\subset \mathbb {N}$ (necessarily infinite) and some $x_0\in \mathbb {X}$ with $\mathrm{supp}x_0\subset A$ such that

$$\begin{aligned} {\delta }=\mathrm{dist}(x_0,[{\mathbf {e}}_n]_A)>0. \end{aligned}$$

Since $\mathrm{supp}{\mathfrak {CG}}_mx_0$ is always a subset of A, this implies (6.3).

Remark 6.2

The above reasoning also implies that $\liminf _{m}\Vert x_0-\mathscr {G}_mx_0\Vert >0$, for all greedy operators $\mathscr {G}_m$. In particular, if there exists a not strong M-basis with the quasi-greedy condition

$$\begin{aligned} C_q:=\sup _{\begin{array}{c} \mathscr {G}_m\in \mathbb {G}_m \\ m\in \mathbb {N} \end{array}}\Vert \mathscr {G}_m\Vert <\infty , \end{aligned}$$

(6.4)

it will not occur that $\mathscr {G}_mx$ converges to x for all $x\in \mathbb {X}$. This observation suggests that in the characterization of quasi-greedy biorthogonal systems given in [28, Theorem 1] one may need to assume that $\mathscr {B}$ is a strong M-basis, or else clarify if this property could be a consequence of (6.4).⁴

Here we show that under the strong M-basis assumption, the conclusions of [19, Theorem 1.1] (and also of “$3\Rightarrow 1$” in [28, Theorem 1]) hold.

Proposition 6.3

If $\mathscr {B}$ is a strong M-basis then, for all Chebyshev t-greedy operators ${\mathfrak {CG}}^t_m$,

$$\begin{aligned} \lim _{m\rightarrow \infty }\Vert x-{\mathfrak {CG}}^t_m(x)\Vert =0,\quad \forall \;x\in \mathbb {X}. \end{aligned}$$

(6.5)

If additionally $C_q<\infty$, then for all t-greedy operators $\mathscr {G}^t_m$,

$$\begin{aligned} \lim _{m\rightarrow \infty }\Vert x-\mathscr {G}^t_m(x)\Vert =0,\quad \forall \;x\in \mathbb {X}. \end{aligned}$$

(6.6)

Proof

Given $x\in \mathbb {X}$ and ${\varepsilon }>0$, by (6.2) there exists $z=\sum _{n\in B}b_n{\mathbf {e}}_n$ such that $\Vert x-z\Vert <{\varepsilon }$, for some finite set $B\subset \mathrm{supp}x$. Let ${\alpha }=\min _{n\in B}|{\mathbf {e}}^*_n(x)|$ and

$$\begin{aligned} {{\bar{\Lambda }}}_{{\alpha }}=\{n{\,\,\,:\,\,\,}|{\mathbf {e}}^*_n(x)|\ge {\alpha }\}. \end{aligned}$$

Since ${\alpha }>0$, this is a finite greedy set for x which contains B. Moreover, we claim that

$$\begin{aligned} {{\bar{\Lambda }}}_{{\alpha }}\subset \mathrm{supp}{\mathfrak {CG}}^t_m (x)=:A,\quad \forall \; m> |{{\bar{\Lambda }}}_{t{\alpha }}|. \end{aligned}$$

(6.7)

Indeed, if this was not the case there would exist $n_0\in {{\bar{\Lambda }}}_{{\alpha }}{\setminus } A$, and since A is a t-greedy set for x, then $\min _{n\in A}|{{\mathbf {e}}^*_n}(x)|\ge t|{\mathbf {e}}^*_{n_0}(x)|\ge t{\alpha }$. So, $A\subset {{\bar{\Lambda }}}_{t{\alpha }}$, which is a contradiction since $m=|A|>|{{\bar{\Lambda }}}_{t{\alpha }}|$. Therefore, (6.7) holds and hence

$$\begin{aligned} \Vert x-{\mathfrak {CG}}^t_m(x)\Vert \le \Vert x-\sum _{n\in B}b_n{\mathbf {e}}_n\Vert <{\varepsilon },\quad \forall \; m> |{{\bar{\Lambda }}}_{t{\alpha }}|. \end{aligned}$$

This establishes (6.5).

We now prove (6.6). As above, let $z=\sum _{n\in B}b_n{\mathbf {e}}_n$ with $B\subset \mathrm{supp}x$ and $\Vert x-z\Vert <{\varepsilon }$. Performing if necessary a small perturbation in the $b_n$’s, we may assume that $b_n\not ={{\mathbf {e}}^*_n}(x)$ for all $n\in B$. Let now

$$\begin{aligned} {\alpha }_1=\min _{n\in B}|{\mathbf {e}}^*_n(x)|,\quad {\alpha }_2=\min _{n\in B}|{\mathbf {e}}^*_n(x-z)|,{\quad \text{ and }\quad }{\alpha }=\min \{{\alpha }_1,{\alpha }_2\}>0. \end{aligned}$$

Consider the sets

$$\begin{aligned} {{\bar{\Lambda }}}_{t{\alpha }}=\{n{\,\,\,:\,\,\,}|{\mathbf {e}}^*_n(x)|\ge t{\alpha }\}=\{n{\,\,\,:\,\,\,}|{\mathbf {e}}^*_n(x-z)|\ge t{\alpha }\}, \end{aligned}$$

which for all $t\in (0,1]$ are greedy sets for both x and $x-z$, and contain B. We claim that,

$$\begin{aligned} \text{ if } m>|{{\bar{\Lambda }}}_{t{\alpha }}| \text{ and } A:=\mathrm{supp}\mathscr {G}^t_m(x), \quad {\mathrm{then }}\quad {{\bar{\Lambda }}}_{{\alpha }}\subset A {\quad \text{ and }\quad }A\in G(x-z,m,t). \end{aligned}$$

(6.8)

The assertion ${{\bar{\Lambda }}}_{{\alpha }}\subset A$ is proved exactly as in (6.7). Next, we must show that

$$\begin{aligned} \text{ if } n\in A \quad {\mathrm{then }}\quad |{\mathbf {e}}^*_n(x-z)|\;\ge \; t\,\max _{k\notin A}|{\mathbf {e}}^*_k(x-z)|\,=\; t\,\max _{k\notin A}|{\mathbf {e}}^*_k(x)|. \end{aligned}$$

This is clear if $n\in A{{\setminus }} B$ since ${{\mathbf {e}}^*_n}(x-z)={{\mathbf {e}}^*_n}(x)$, and $A\in G(x,m,t)$. On the other hand, if $n\in B$, then $|{{\mathbf {e}}^*_n}(x-z)|\ge {\alpha }_2\ge {\alpha }\ge \max _{k\in A^c}|{\mathbf {e}}^*_k(x)|$, the last inequality due to ${{\bar{\Lambda }}}_{\alpha }\subset A$. Thus (6.8) holds true, and therefore

$$\begin{aligned} \mathscr {G}^t_m(x)-z=\sum _{n\in A}{{\mathbf {e}}^*_n}(x-z){\mathbf {e}}_n\,=\,\bar{\mathscr {G}}^t_m(x-z), \end{aligned}$$

for some $\bar{\mathscr {G}}^t_m\in \mathbb {G}^t_m$. Thus,

$$\begin{aligned} \Vert \mathscr {G}^t_m(x)-x\Vert \,=\, \Vert (I-\bar{\mathscr {G}}^t_m)(x-z)\Vert \,\le \, (1+\Vert \bar{\mathscr {G}}^t_m\Vert )\,{\varepsilon }, \end{aligned}$$

and the result follows from $\sup _{m}\Vert \bar{\mathscr {G}}^t_m\Vert \le (1+4C_q/t)C_q$, by [10, Lemma 2.1].