A remark on approximation with polynomials and greedy bases

TL;DR

This paper studies polynomial approximation errors with constant and modulus-constant coefficients in Banach spaces, characterizing their limits in relation to democracy properties of bases and extending previous results.

Contribution

It provides new characterizations of approximation error limits using democracy functions and extends prior results on polynomial approximation in Banach spaces.

Findings

Characterizes when approximation errors are equivalent to the norm.

Provides conditions for the limits of approximation errors to equal the norm.

Extends previous results on polynomial approximation with greedy bases.

Abstract

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

Equations

$$P_m(x)=\sum _{n=1}^m{e}_n^{\ast }(x)e_n,x\in \mathbb{X}$$

$$K_b:=\underset{m\in \mathbb{N}}{\sup }\Vert P_m\Vert <\infty .$$

$${\mathcal{G}}_m(x)=\sum _{j=1}^m{e}_{\rho (j)}^{\ast }(x)e_{\rho (j)}=\sum _{k\in {\Lambda }_m(x)}{e}_k^{\ast }(x)e_k,$$

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

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

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

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

$$1_A:=\sum _{n\in A}{e}_n\text{mbox}and{1}_{\epsilon A}:=\sum _{n\in A}{\epsilon }_n{e}_n.$$

$$h_l(m)=\underset{|A|=m,|\epsilon |=1}{\inf }\Vert 1_{\epsilon A}\Vert \text{mbox},h_r(m)=\underset{|A|=m,|\epsilon |=1}{\sup }\Vert 1_{\epsilon A}\Vert (m\in \mathbb{N}).$$

$$h_l^{\ast }(m)=\underset{|A|=m,|\epsilon |=1}{\inf }\Vert 1_{\epsilon A}\Vert \text{mbox},h_r^{\ast }(m)=\underset{|A|=m,|\epsilon |=1}{\sup }\Vert 1_{\epsilon A}\Vert (m\in \mathbb{N}).$$

$${\mathcal{D}}_m(x,\mathcal{B}{)}_{\mathbb{X}}$$

$${\mathcal{D}}_m^{\ast }(x,\mathcal{B}{)}_{\mathbb{X}}$$

$$\underset{m\to \infty }{\lim }{\mathcal{D}}_m(x)=\underset{m\to \infty }{\lim }{\mathcal{D}}_m^{\ast }(x)=\Vert x\Vert ,\text{mbox}foreveryx\in \mathbb{H}.$$

$$\underset{m\to +\infty }{\lim }{\mathcal{D}}_m(1_B)=\underset{m\to +\infty }{\lim }{\mathcal{D}}_m^{\ast }(1_B)=\Vert 1_B\Vert ,\text{mbox}foreveryfiniteB\subset \mathbb{N}.$$

$$c\Vert x\Vert \le \underset{m\to +\infty }{\mathrm{liminf}}{\mathcal{D}}_m^{\ast }(x)\le \underset{m\to +\infty }{\mathrm{limsup}}{\mathcal{D}}_m^{\ast }(x)\le \Vert x\Vert \text{mbox}foreveryx\in \mathbb{X},$$

$$\underset{m\to +\infty }{\lim }{\mathcal{D}}_m^{\ast }(x)=\Vert x\Vert \text{mbox}foreveryx\in \mathbb{X}.$$

$$\frac{1}{4K_b}\underset{n\in \mathbb{N}}{\sup }|e_n^{\ast }(x)|\le \underset{m\to \infty }{\mathrm{liminf}}{\mathcal{D}}_m^{\ast }(x).$$

$$\|x-\alpha\mathbf{1}_{\varepsilon A}\|\geq\sup_{n\in\mathbb{N}}\frac{|e_{n}^{\ast}(x-\alpha\mathbf{1}_{\eta A})|}{\|e_{n}^{\ast}\|}\geq\frac{\sup_{n\in\mathbb{N}}{|e_{n}^{\ast}(x-\alpha\mathbf{1}_{\varepsilon A})|}}{2K_{b}}\,\geq\frac{\sup_{n\in\mathbb{N}}\big{|}|e_{n}^{\ast}(x)|-|\alpha|\big{|}}{2K_{b}}\,\,.$$

$$|e_n^{\ast }(x)|\le \delta \text{mbox}foreveryn\ge n_0.$$

$$\Vert x-\alpha 1_{\epsilon A}\Vert \ge \frac{|e_j^{\ast }(x)-|\alpha ||}{2K_b}\ge \frac{||\alpha |-\delta |}{2K_b}.$$

$$\displaystyle\|x-\alpha\mathbf{1}_{\varepsilon A}\|\geq\frac{||\alpha|-\delta|+\sup_{n\in\mathbb{N}}\big{|}|e_{n}^{\ast}(x)|-|\alpha|\big{|}}{4K_{b}}\geq\sup_{n\in\mathbb{N}}{\frac{|e_{n}^{\ast}(x)|-\delta}{4K_{b}}}\,.$$

$${\mathcal{D}}_m^{\ast }(x)\ge \underset{n\in \mathbb{N}}{\sup }\frac{|e_n^{\ast }(x)|-\delta }{4K_b}.$$

$$h_l^{\ast }(k)\le K_b{h}_l^{\ast }(m),h_r^{\ast }(k)\le K_b{h}_r^{\ast }(m)\text{mbox}foreveryk\le m.$$

$$c\Vert x\Vert \le \underset{m\to +\infty }{\mathrm{liminf}}{\mathcal{D}}_m^{\ast }(x)\le \underset{m\to +\infty }{\mathrm{limsup}}{\mathcal{D}}_m^{\ast }(x)\le \Vert x\Vert \text{mbox}foreveryx\in \mathbb{X}.$$

$$\underset{m\to +\infty }{\lim }{\mathcal{D}}_m^{\ast }(x)=\Vert x\Vert .$$

$${\mathbf{x}}_1=e_1,{\mathbf{x}}_n=e_n-e_{n-1},n=2,3,...$$

$${\mathbf{s}}_n:=(\underset{n-1}{\underbrace{0,\dots ,0}},1,1,\dots ),n\in \mathbb{N}.$$

$$\mathbb{X}:={\overline{\mathrm{span}\{{\mathbf{x}}_n:n\in \mathbb{N}\}}}^{{\ell }^1}.$$

$$\underset{\begin{array}{c}A\subset \mathbb{N}\\ finite\end{array},|\eta |=1}{\sup }\underset{m\to +\infty }{\mathrm{liminf}}{\mathcal{D}}_m^{\ast }(1_{\eta A})$$

$$\underset{\begin{array}{c}A\subset \mathbb{N}\\ finite\end{array}}{\sup }\underset{m\to +\infty }{\mathrm{liminf}}{\mathcal{D}}_m(1_A)$$

$$\lambda <\underset{m\to +\infty }{\mathrm{liminf}}{\mathcal{D}}_m^{\ast }(1_{\eta A}).$$

Full text

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

Key words and phrases: thresholding greedy algorithm, greedy bases, almost 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). Also, the first author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for hospitality during the program Approximation, Sampling and Compression in Data Science where some work on this paper was undertaken. The second author is acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554).

1. Introduction

Let $(\mathbb{X},\|\cdot\|)$ be a real Banach space and let $\mathcal{B}=(e_{n})_{n=1}^{\infty}$ be a semi-normalized (Schauder) basis of $\mathbb{X}$ with biorthogonal functionals $(e_{n}^{*})_{n=1}^{\infty}$, that is:

1. (i)

There exist $a,b>0$ such that $a\leq\|e_{n}\|,\|e_{n}^{\ast}\|\leq b$ for every $n\in\mathbb{N}$, 2. (ii)

$e_{k}^{\ast}(e_{n})=\delta_{kn}$ for every $k,n\in\mathbb{N}$, 3. (iii)

The sequence of projections $P_{m}:\mathbb{X}\longrightarrow\mathbb{X}$ given by

[TABLE]

satisfy $\lim_{n}{\|P_{m}(x)-x\|}=0$ for every $x\in\mathbb{X}$. In this case, the basis constant of $\mathcal{B}$ is

[TABLE]

We say that $\mathcal{B}$ is monotone whether $K_{b}=1$.

Along the paper we will refer to every such $\mathcal{B}$ simply as a basis. Of course, as $m$ increases $P_{m}(x)$ offers a good approximation of $x$ by linear combinations of $m$-elements of the basis, but it is natural to ask whether a suitable (and systematic) rearrangement can provide better convergence rates. A natural proposal is the Thresholding Greedy Algorithm (TGA) introduced by S. V. Konyagin and V. N. Temlyakov ([10]): given $x\in\mathbb{X}$ we first consider the rearranging function $\rho:\mathbb{N}\longrightarrow\mathbb{N}$ satisfying 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 then

[TABLE]

where $\Lambda_{m}(x)=\{\rho(n):n\leq m\}$ is the greedy set of $x$ with cardinality $m$. Related to this, S. V. Konyagin and V. N. Temlyakov defined in [10] the concepts of greedy and quasi-greedy bases.

Definition 1.1**.**

We say that $\mathcal{B}$ is quasi-greedy if there exists a positive constant $C_{q}$ such that

[TABLE]

P. Wojtaszczyk proved in [12] that quasi-greediness is equivalent to the convergence of the algorithm, that is, $\mathcal{B}$ is quasi-greedy if and only if

[TABLE]

Definition 1.2**.**

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

[TABLE]

where

[TABLE]

Konyagin and Temlykov [10] showed that, although every greedy basis is quasigreedy, the converse does not holds (see also [1, Section 10.2]). They also characterize greedy bases as those which are unconditional and democratic. To define the last notion we have to introduce some notation. For each finite subset $A\subset\mathbb{N}$ and every scalar sequence $\varepsilon=(\varepsilon_{n})$ with $|\varepsilon_{n}|=1$ for each $n\in\mathbb{N}$ (from now on we will write $|\varepsilon|=1$, for simplicity) let us denote

[TABLE]

As usual, $|A|$ stands for the cardinal of $A$. We then define the democracy functions as

[TABLE]

and the superdemocracy functions as

[TABLE]

Definition 1.3**.**

We say that $\mathcal{B}$ is democratic (resp. superdemocratic) if there exists $C>0$ such that $h_{r}(m)\leq C\,h_{l}(m)$ (  resp. $h_{r}^{*}(m)\leq C\,h_{l}^{*}(m)$  ) for every $m\in\mathbb{N}$.

Another characterization of greedy bases was more recently provided by Ó. Blasco and the first author by means of the best $m$-th error in the approximation using polynomials of constant $($resp. modulus-constant$)$ coefficients:

[TABLE]

Theorem 1.4**.**

[2, Corollary 1.8]** Let $\mathcal{B}$ be a basis of a Banach space $\mathbb{X}$. The following assertions are equivalent:

1. (i)

$\mathcal{B}$* is greedy;* 2. (ii)

There is $C>0$ such that $\|x-\mathcal{G}_{m}(x)\|\leq C\,\mathcal{D}_{m}(x)$ for every $x\in\mathbb{X}$ and $m\in\mathbb{N}$. 3. (iii)

There is $C>0$ such that $\|x-\mathcal{G}_{m}(x)\|\leq C\,\mathcal{D}_{m}^{\ast}(x)$ for every $x\in\mathbb{X}$ and $m\in\mathbb{N}$.

The striking feature of this theorem compared to (1) is that, while $\lim_{m}\sigma_{m}(x)=0$ for every $x\in\mathbb{X}$, the terms $\mathcal{D}_{m}^{\ast}(x)$ and $\mathcal{D}_{m}(x)$ do not necessarily converge to zero if $x\neq 0$. Indeed, we have the following examples:

• $\rhd$

[2, Theorem 3.2],[3, Theorem 1.4] If $\mathbb{X}=\mathbb{H}$ is a (separable) Hilbert space and $\mathcal{B}$ is an orthonormal basis, then

[TABLE]

• $\rhd$

[2, Proposition 3.4] If $\mathbb{X}=\ell^{p}$ ($1<p<\infty$) and $\mathcal{B}$ is the canonical basis, then

[TABLE]

In the present paper, we aim to delve into this aspect. Let us briefly explain the structure of the paper. In Section 2 we show that $\mathcal{D}_{m}^{\ast}(x)$ and $\mathcal{D}_{m}(x)$ do not converge to zero as $m\rightarrow+\infty$ for any $x\neq 0$. In Section 3 we prove the main result of the paper (Theorem 3.2), namely a characterization of those bases $\mathcal{B}$ for which there is a positive constant $c>0$ such that

[TABLE]

in terms of the democracy and superdemocracy functions. We also provide a quite general condition ensuring that

[TABLE]

In Section 4 we deal with the notion of almost-greedy bases. We study how this property can be also characterized in terms of polynomials of constant or modulus-constant coefficients, extending a recent result of S. J. Dilworth and D. Khurana in [6].

Let us point out [1] as our basic reference for notation and fundamental results on greedy basis.

2. The limit of errors $\mathcal{D}_{m}^{*}(x)$ and $\mathcal{D}_{m}(x)$ is nonzero

Since $\mathcal{D}_{m}^{*}(x)\leq\mathcal{D}_{m}(x)\leq\|x\|$ for every $m\in\mathbb{N}$ and every $x\in\mathbb{X}$, it is only necessary to study lower bounds of $\mathcal{D}_{m}^{*}(x)$.

Proposition 2.1**.**

Let $\mathcal{B}=(e_{n})_{n=1}^{\infty}$ be a basis of a Banach space $\mathbb{X}$. Then, for every $x\in\mathbb{X}$

[TABLE]

Proof.

Let $x\in\mathbb{X}$. Note that for every finite set $A\subset\mathbb{N}$, $\alpha\in\mathbb{R}$ and $|\varepsilon|=1$ it holds that

[TABLE]

Let us also fix $\delta>0$ and $n_{0}\in\mathbb{N}$ with the property that

[TABLE]

If $A$ satisfies $|A|>n_{0}$, then there is $j\in A$ with $j>n_{0}$, and so

[TABLE]

In particular, combining both lower estimations we get that for $|A|>n_{0}$

[TABLE]

Therefore, for $m>n_{0}$

[TABLE]

∎

3. Main result: equivalence with the norm

The issue of when $\liminf_{m}{\mathcal{D}^{\ast}_{m}(x)}$ (resp. $\liminf_{m}{\mathcal{D}_{m}(x)}$) is equivalent to $\|x\|$ is going to be determined by the behaviour of the superdemocracy functions (resp. democracy functions), see Section 1 for the definitions. Along the present section we are going to focus on proving the results for superdemocracy case, namely for $h_{l}^{\ast}(m)$, $h_{r}^{\ast}(m)$ and the error $\mathcal{D}^{\ast}_{m}(x)$. The arguments for the case $h_{l}(m)$, $h_{r}$ and the error $\mathcal{D}_{m}(x)$ are completely analogous. First of all, we recall a trivial estimates of the superdemocray functions for any basis:

[TABLE]

These relations together with the trivial inequality $h_{l}^{\ast}(m)\leq h_{r}^{\ast}(m)$ $(m\in\mathbb{N})$ yield that there are three possible cases:

• $\rhd$

$h_{l}^{\ast}(m)$ and $h_{r}^{\ast}(m)$ are bounded.

• $\rhd$

$h_{l}^{\ast}(m)$ is bounded and $h_{r}^{\ast}(m)\rightarrow+\infty$ as $m\rightarrow+\infty$.

• $\rhd$

$h_{l}^{\ast}(m),h_{r}^{\ast}(m)\rightarrow+\infty$ as $m\rightarrow+\infty$.

Definition 3.1**.**

The functions $h_{l}^{\ast}(m)$ and $h_{r}^{\ast}(m)$ (resp. $h_{l}(m)$ and $h_{r}(m)$) are said to be comparable if they are both bounded or divergent to infinity.

The main result of the section is the following theorem.

Theorem 3.2**.**

Let $\mathcal{B}$ be a basis of a Banach space $\mathbb{X}$. The following assertions are equivalent:

1. (i)

There is a positive constant $c>0$ such that

[TABLE] 2. (ii)

$h_{l}^{*}(m)$* and $h_{r}^{*}(m)$ are comparable.*

Moreover, if $\mathcal{B}$ is monotone and $h_{l}^{*}(m)\rightarrow+\infty$ as $m\rightarrow+\infty$, then

[TABLE]

$($The theorem also holds if we replace $\mathcal{D}_{m}^{\ast}(x)$, $h_{l}^{*}(m)$, $h_{r}^{*}(m)$ respectively by $\mathcal{D}_{m}(x)$, $h_{l}(m)$, $h_{r}(m)$.$)$

Before going into the proof let us make a few observations:

• $\rhd$

From Theorem 3.2 we recover (2) and (3). Indeed, if $\mathbb{H}$ is a (separable) Hilbert space and $\mathcal{B}$ is an orthonormal basis of $\mathbb{H}$ then $h_{l}(m)=h_{l}^{*}(m)=m^{1/2}$. On the other hand, for $\mathbb{X}=\ell_{p}$ with $1\leq p<\infty$ and $\mathcal{B}$ is the canonical basis, it holds that $h_{l}(m)=h_{l}^{*}(m)=m^{1/p}$.

• $\rhd$

For $\mathbb{X}=L_{p}[0,1]$ we have that the Haar basis $\mathcal{B}$ is monotone (see [7, Theorem 5.18]) and satisfies $h_{l}^{*}(m)=h_{l}(m)\approx m^{1-1/p}$ for $1\leq p<\infty$. Hence, it satisfies that $\lim_{m}{\mathcal{D}^{*}_{m}(x)}=\lim_{m}{\mathcal{D}_{m}(x)}=\|x\|$ for every $x\in\mathbb{X}$.

• $\rhd$

If $\mathcal{B}$ is superdemocratic (resp. democratic), then it satisfies Theorem 3.2.(ii) (resp. Theorem 3.2.(ii) for $h_{r}(m)$ and $h_{l}(m)$). However, there are easy examples showing that converse is not true. For instance, the canonical basis of $\ell^{2}\oplus_{1}\ell^{4}$ satisfies that $h_{l}(m)=h_{l}^{*}(m)\approx m^{1/4}$ and $h_{r}(m)=h_{r}^{*}(m)\approx m^{1/2}$.

• $\rhd$

Example of basis not satisfying Theorem 3.2.(ii): Let us consider $\mathbb{X}=\ell_{1}$ and let $\mathcal{B}=(\mathbf{x}_{n})_{n=1}^{\infty}$ be the difference basis, which in terms of the canonical basis $(e_{n})_{n=1}^{\infty}$ is given by

[TABLE]

By [4, Lemma 8.1], it holds that $h_{l}^{*}(m)=h_{l}(m)=1$ and $h_{r}^{*}(m)=h_{r}(m)=2m$.

• $\rhd$

Example of basis satisfying $\lim_{m}{\mathcal{D}_{m}(x)}=\|x\|$ for every $x\in\mathbb{X}$, but $\liminf_{m}{\mathcal{D}^{\ast}(x)}$ is not even equivalent to $\|x\|$: Let $\mathbb{X}=\mathbf{c}$ be the space of convergent sequences and let $\mathcal{B}=(\mathbf{s}_{n})_{n=1}^{\infty}$ be the summing basis, defined as

[TABLE]

By [4, Lemma 8.1] we know that $h_{l}^{*}(m)\approx 1$ and $h_{r}^{*}(m)\approx m$, so Theorem 3.2.(ii) does not hold. On the other hand, $\mathcal{B}$ is monotone and $h_{l}(m)\approx h_{r}(m)\approx m$ by the same reference. Thus, $\lim_{m}{\mathcal{D}_{m}(x)}=\|x\|$ for every $x\in\mathbb{X}$.

• $\rhd$

Condition Theorem 3.2.(ii) is not preserved for dual bases: If $(e_{n})_{n=1}^{\infty}$ is the canonical basis of $\ell_{1}$, let us consider the sequence $\mathbf{x}_{n}=e_{n}-(e_{2n+1}+e_{2n+2})/2$, $n\in\mathbb{N}$ and the space

[TABLE]

This is known as the Lindenstrauss space [8] and the sequence $\mathcal{B}=(\mathbf{x}_{n})_{n=1}^{\infty}$ is actually a monotone basis for $\mathbb{X}$ (see [11, pg 457]). In [4, Section 8.2] it is shown that $h_{l}^{*}(m)\approx m$. On the other hand, in the same reference it is proved that the dual space $\mathbb{X}^{\ast}$ with the corresponding dual basis $\mathcal{B}^{\ast}$ satisfies $h_{l}^{*}(m)\approx 1$ and $h_{r}^{*}(m)\approx\ln(m)$.

3.1. Proof of the main result

Proposition 3.3**.**

Let $\mathcal{B}$ be a basis of a Banach space $\mathbb{X}$. Then,

[TABLE]

Proof.

We explain the argument for (5), as the proof of (6) is completely analogous with the obvious replacements. Let us fix a finite set $A\subset\mathbb{N}$ and $\eta\in\{\pm 1\}^{A}$, and let us take $\lambda\in\mathbb{R}$ satisfying

[TABLE]

We can then find $m_{0},n_{0}\in\mathbb{N}$ with the following properties:

• $\rhd$

$\lambda\leq\|\mathbf{1}_{\eta A}-\alpha\mathbf{1}_{\varepsilon B}\|$ for every $\alpha\in\mathbb{R}$, $|\varepsilon|=1$ and $B\subset\mathbb{N}$ with $|B|\geq m_{0}$ ,

• $\rhd$

$A\subset\{1,\ldots,n_{0}\}$ .

Let $C\subset\mathbb{N}$ be a finite set with $|C|\geq m_{0}+n_{0}$. Then,

[TABLE]

where $C^{\prime}:=C\setminus\{1,\ldots,n_{0}\}$. Notice that $|C^{\prime}|\geq m_{0}$, so in particular

[TABLE]

Thus, we have the relation

[TABLE]

Taking supremums on $\lambda$ according to (7) we conclude that

[TABLE]

∎

Theorem 3.4**.**

Let $\mathcal{B}$ be a basis of a Banach space $\mathbb{X}$. Assume that there is a constant $C>0$ satisfying

[TABLE]

Then, for every $x\in\mathbb{X}$

[TABLE]

Proof.

Let us fix $x\in\mathbb{X}$. We just have to show that the left hand-side of (8) holds. For, let $0<\delta<1$ and $m_{0},n_{0}\in\mathbb{N}$ such that

[TABLE]

Given $\alpha\in\mathbb{R}$, $A\subset\mathbb{N}$ with $|A|\geq m_{0}+n_{0}$ and $\varepsilon\in\{\pm 1\}^{A}$, we are going to establish two lower bounds for $\|x-\alpha\mathbf{1}_{\varepsilon A}\|$.

• $\rhd$

Since $|A\cap(n_{0},+\infty)|\geq m_{0}$ we can find $n\geq n_{0}$ such that $|A\cap(n,+\infty)|=m_{0}$. Thus, applying the operator $\operatorname{Id}-P_{n}$ to $x-\alpha\mathbf{1}_{\varepsilon A}$ we have that

[TABLE]

• $\rhd$

As $|A|\geq n_{0}$ we can find $n\geq n_{0}$ with $|A\cap[1,n]|=n_{0}$, so that

[TABLE]

Note that the lower estimations (9) and (11) are respectively increasing and decreasing linear functions $f(t)$ and $g(t)$ on $t=|\alpha|$. Moreover these functions have a unique point of intersection $t_{0}>0$ which can be easily checked to satisfy

[TABLE]

Thus

[TABLE]

Taking the infimum of $\|x-\alpha\mathbf{1}_{\varepsilon A}\|$ on $\alpha\in\mathbb{R}$ and $A$ satisfying the conditions above, we deduce that

[TABLE]

Finally, making $\delta\rightarrow 0^{+}$ we get the desired conclusion. ∎

Proof of Theorem 3.2.

To check (i) $\Rightarrow$ (ii), note that using Proposition 3.3 we then deduce that

[TABLE]

It is clear from this inequality that $h_{l}^{\ast}(m)$ and $h_{r}^{\ast}(m)$ are then comparable. To see the converse (ii) $\Rightarrow$ (i), note first that if $h_{l}^{\ast}(m)$ and $h_{r}^{\ast}(m)$ are comparable, then there exists $C>0$ such that

[TABLE]

and so Theorem 3.4 applies. The second statement of the theorem follows also from Theorem 3.4 since $\mathcal{B}$ being monotone means that $K_{b}=1$, and condition $\lim_{m}h_{l}^{\ast}(m)=+\infty$ means that (13) holds for every $C>0$. ∎

4. Almost-greediness and polynomials with constant coefficients

Definition 4.1**.**

Let $\mathcal{B}=(e_{n})_{n=1}^{\infty}$ be a basis of a Banach space $\mathbb{X}$. We say that $\mathcal{B}$ is almost-greedy if there exists a constant $C>0$ such that

[TABLE]

where

[TABLE]

This notion was introduced by S. J. Dilworth, N. J. Kalton, D. Kutzarova and V. N. Temlyakov in [5], together with two characterizations. First, that a basis is almost-greedy if and only if it is quasi-greedy and democratic. The second characterization is given in the next theorem.

Theorem 4.2** ([5, Theorem 3.3]).**

Let $\mathcal{B}$ be a basis of a Banach space $\mathbb{X}$. Then, $\mathcal{B}$ is almost-greedy if and only if for some (resp. every) $\lambda>1$, there exists a positive constant $C_{\lambda}$ such that

[TABLE]

Indeed, $C_{\lambda}\approx\frac{1}{\lambda-1}$.

As in the case of greedy basis, we can replace the error $\sigma_{m}(x)$ by the $m$-th error of approximation by polynomials with constant (resp. modulus-constant) coefficients.

Theorem 4.3**.**

Let $\mathcal{B}$ be a basis of a Banach space $\mathbb{X}$ and let $\lambda>1$. The following assertions are equivalent:

1. (i)

$\mathcal{B}$* is almost-greedy.* 2. (ii)

There is $C>0$ such that $\|x-\mathcal{G}_{[\lambda m]}(x)\|\leq C_{\lambda}\,\mathcal{D}_{m}(x)$ for every $x\in\mathbb{X}$ and every $m\in\mathbb{N}$. 3. (iii)

There is $C>0$ such that $\|x-\mathcal{G}_{[\lambda m]}(x)\|\leq C_{\lambda}\,\mathcal{D}_{m}^{*}(x)$ for every $x\in\mathbb{X}$ and every $m\in\mathbb{N}$.

Proof.

Implication (i) $\Rightarrow$ (iii) $\Rightarrow$ (ii) are clear using Theorem 4.2 and the inequalities $\sigma_{m}(x)\leq\mathcal{D}_{m}^{*}(x)\leq\mathcal{D}_{m}(x)$. To show that (ii) $\Rightarrow$ (i) we follow the ideas from the proof of Theorem 4.2: using the hypothesis, we argue that $\mathcal{B}$ is democratic and quasi-greedy.

To see that it is democratic, let $m\in\mathbb{N}$ and $A,B\subset\mathbb{N}$ with $|A|=m$ and $|B|=[\lambda m]$. Let us consider a set $E\supset A,B$ with $|E|=m+[\lambda m]$, let $\delta>0$ and consider the element $x=\mathbf{1}_{A}+(1+\delta)\mathbf{1}_{E\setminus A}$. Then,

[TABLE]

As $\delta>0$ is arbitrary, taking supremum over $A$ and infimum over $B$ we deduce that

[TABLE]

where in the last inequality we have used the estimations mentioned at the beginning of Section 2.

Let show now that the basis $\mathcal{B}$ is quasi-greedy. For, take $m\in\mathbb{N}$ and $r\in\mathbb{N}\cup\{0\}$ such that $[\lambda r]\leq m<[\lambda(r+1)]$. Then,

[TABLE]

Note that $\mathcal{G}_{[\lambda r]}(x)-\mathcal{G}_{m}(x)$ contains at most $m-[\lambda r]<\lambda$ summands of the form $e_{n}^{\ast}(x)\,e_{n}$, so that

[TABLE]

On the other hand, using the hypothesis

[TABLE]

Thus, the basis is quasi-greedy. ∎

Recently, S. J. Dilworth and D. Khurana provided the following characterization of almost-greedy bases in the same spirit of Theorem 1.4. In order to present it we have to introduce some notation: if $A,B\subset\mathbb{N}$ are finite sets, we will write $A<B$ if $\max{A}<\min{B}$.

[TABLE]

where recall that $\Lambda_{m}(x)$ is the $m$-th greedy set associated to $x$ introduced in Section 1.

Theorem 4.4**.**

[6]** Let $\mathcal{B}$ be a basis of a Banach space $\mathbb{X}$. Then, $\mathcal{B}$ is almost-greedy if and only if there exists $C>0$ such that

[TABLE]

Inspiring on the previous theorem , we can prove the following result which is again strinking as $\mathcal{D}_{m}(x)\leq\mathcal{H}_{m}(x)$ and so $\liminf{\mathcal{H}_{m}(x)}\approx\|x\|$ when $h_{l}(m)$ and $h_{r}(m)$ are comparable by Theorem 3.2.

Corollary 4.5**.**

Let $\mathcal{B}$ be a basis of a Banach space $\mathbb{X}$. Then, $\mathcal{B}$ is almost-greedy if and only if there exists $C>0$ such that

[TABLE]

Proof.

If $\mathcal{B}$ is quasi-greedy then 14 holds by Theorem 4.4. To see the converse we use the aforementioned characterization of almost-greedy bases as those being quasi-greedy and democratic. The fact that $\mathcal{B}$ is quasi-greedy follows from the hypothesis and the trivial inequality $\mathcal{H}_{m}(x)\leq\|x\|$. Let us show that $\mathcal{B}$ is democratic. Let $A,B\subset\mathbb{N}$ be finite subsets of cardinality $m$, and take $E\subset\mathbb{N}$ also with $|E|=m$ and moreover $A<E$ and $B<E$. Fixed $\delta>0$ consider the elements $x=\mathbf{1}_{A}+(1+\delta)\mathbf{1}_{E}$ and $y=\mathbf{1}_{E}+(1+\delta)\mathbf{1}_{B}$. Then,

[TABLE]

Analogously,

[TABLE]

Since $\delta>0$ was arbitraty, we conclude that $h_{r}(m)\leq C^{2}\,h_{l}(m)$ for every $m\in\mathbb{N}$, and so the basis is democratic. ∎