# The Thresholding Greedy Algorithm versus Approximations with Sizes Bounded
by Certain Functions $f$

Hùng Việt Chu<EMAIL_ADDRESS>Department of Mathematics, University of
Illinois at Urbana-Champaign, Urbana, IL 61820, USA

###### Abstract.

Let $X$ be a Banach space and $(e_{n})_{n=1}^{\infty}$ be a basis. For a
function $f$ in a large collection $\mathcal{F}$ (closed under composition),
we define and characterize $f$-greedy and $f$-almost greedy bases. We study
relations among these bases as $f$ varies and show that while a basis is not
almost greedy, it can be $f$-greedy for some $f\in\mathcal{F}$. Furthermore,
we prove that for all non-identity function $f\in\mathcal{F}$, we have the
surprising equivalence

$f\mbox{-greedy}\ \Longleftrightarrow\ f\mbox{-almost greedy}.$

We give various examples of Banach spaces to illustrate our results.

###### Key words and phrases:

Thresholding Greedy Algorithm; almost greedy; bases

###### 2020 Mathematics Subject Classification:

41A65; 46B15

The author is thankful to Timur Oikhberg for helpful feedback on earlier
drafts of this paper and for showing the author that for certain functions
$f$, a quasi-greedy and $f$-disjoint superdemocratic basis is
$f$-unconditional.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary results
  3. 3 About $f$-(disjoint) democracy and Property (A, $f$)
  4. 4 About $f$-unconditionality
  5. 5 Characterizations of $f$-(almost) greedy bases
  6. 6 When $f$ varies - A modification of the Schreier space
  7. 7 Examples

##  1\. Introduction

###  1.1. Background and motivation

Let $X$ be an infinite-dimensional Banach space (with dual $X^{*}$) over the
field $\mathbb{F}=\\{\mathbb{R},\mathbb{C}\\}$. In this paper, we work with a
general basis, which is a countable collection of vectors
$(e_{n})_{n=1}^{\infty}\subset X$ such that: i)
$\overline{\operatorname{span}(e_{n})_{n=1}^{\infty}}=X$; ii) there exists a
collection $(e_{n}^{*})_{n=1}^{\infty}\subset X^{*}$, called the biorthogonal
functionals, that satisfy $e^{*}_{n}(e_{m})=\delta_{n,m}$; iii)
$\sup_{n}\|e_{n}\|\|e_{n}^{*}\|_{*}<\infty$ and semi-normalization:

$0\ <\ \inf_{n}\|e_{n}\|\ \leqslant\ \sup_{n}\|e_{n}\|\ <\ \infty.$

M-boundedness and semi-normalization imply the existence of
$\mathbf{c}_{1},\mathbf{c}_{2}>0$ such that

$0\ <\ \mathbf{c}_{1}\ :=\ \inf_{n}\\{\|e_{n}\|,\|e_{n}^{*}\|\\}\ \leqslant\
\sup_{n}\\{\|e_{n}\|,\|e_{n}^{*}\|\\}\ :=\ \mathbf{c}_{2}\ <\ \infty.$

One important goal of approximation theory is to efficiently approximate each
vector $x\in X$ by a finite linear combination of vectors in a given basis. In
1999, Konyagin and Temlyakov [15] introduced a natural, order-free method of
approximation, called the Thresholding Greedy Algorithm (TGA). The algorithm
chooses the largest coefficients (in modulus) of each vector $x$ in its
representation with respect to a basis $(e_{n})_{n=1}^{\infty}$. Formally, a
set $A\subset\mathbb{N}$ is a greedy set of a vector $x$ of order $m$ if

$|A|\ =\ m\mbox{ and }\min_{n\in A}|e_{n}^{*}(x)|\ \geqslant\ \max_{n\notin
A}|e_{n}^{*}(x)|.$

Let $G(x,m)$ denote the set of all greedy sets of $x$ of order $m$. For each
$x\in X$, the TGA thus produces a sequence of approximants
$(\sum_{n\in\Lambda_{m}}e_{n}^{*}(x)e_{n})_{m=1}^{\infty}$, where
$\Lambda_{m}\in G(x,m)$.

Konyagin and Temlyakov [15] defined greedy and quasi-greedy bases. For a
finite set $A\subset\mathbb{N}$, let $P_{A}(x)=\sum_{n\in A}e_{n}^{*}(x)e_{n}$
and $P_{A^{c}}(x)=x-P_{A}(x)$. Also, let $\mathbb{N}^{<\infty}$ denote the
collection of all finite subsets of $\mathbb{N}$.

######  Definition 1.1.

A basis is said to be greedy if there exists a constant $\mathbf{C}\geqslant
1$ such that

$\|x-P_{\Lambda}(x)\|\ \leqslant\ \mathbf{C}\sigma_{m}(x),\forall x\in
X,\forall m\in\mathbb{N},\forall\Lambda\in G(x,m),$

where $\sigma_{m}(x)=\inf_{\begin{subarray}{c}|A|\leqslant m\\\
(a_{n})\subset\mathbb{F}\end{subarray}}\|x-\sum_{n\in A}a_{n}e_{n}\|$.

######  Definition 1.2.

A basis $(e_{n})_{n=1}^{\infty}$ is said to be quasi-greedy111This definition
of the quasi-greedy property is equivalent to the convergence
$P_{\Lambda_{m}}(x)\ \longrightarrow\ x,\forall x\in X,\forall\Lambda_{m}\in
G(x,m).$ For details, see [3, Theorem 4.1]. if there exists a constant
$\mathbf{C}>0$ such that

$\|x-P_{\Lambda}(x)\|\ \leqslant\ \mathbf{C}\|x\|,\forall x\in X,\forall
m\in\mathbb{N},\forall\Lambda\in G(x,m).$ (1.1)

In this case, we let $\mathbf{C}_{\ell}$ denote the smallest constant
$\mathbf{C}$ for (1.1) to hold, and we say that $(e_{n})_{n=1}^{\infty}$ is
$\mathbf{C}_{\ell}$-suppression quasi-greedy. Also, let $\mathbf{C}_{q}$ be
the smallest constant such that

$\|P_{\Lambda}(x)\|\ \leqslant\ \mathbf{C}_{q}\|x\|,\forall x\in X,\forall
m\in\mathbb{N},\forall\Lambda\in G(x,m).$

We say that $(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{q}$-quasi-greedy.

A beautiful characterization of greedy bases involves unconditionality
(Definition 4.1) and democracy (Definition 3.1).

######  Theorem 1.3 (Konyagin and Temlyakov [15]).

A basis is greedy if and only if it is unconditional and democratic.

Observe that in the definition of greedy bases, we compare the TGA, which
involves projections onto greedy sets, against approximations with arbitrary
coefficients. Hence, it is also natural to compare the TGA against projections
only (without arbitrary coefficients). This thinking leads us to the so-called
almost greedy bases, introduced by Dilworth et al. [11].

######  Definition 1.4.

A basis is said to be almost greedy if there exists a constant
$\mathbf{C}\geqslant 1$ such that

$\|x-P_{\Lambda}(x)\|\ \leqslant\ \mathbf{C}\widetilde{\sigma}_{m}(x),\forall
x\in X,\forall m\in\mathbb{N},\forall\Lambda\in G(x,m),$

where $\widetilde{\sigma}_{m}(x)=\inf_{|A|\leqslant m}\|x-P_{A}(x)\|$.

It is worth noting that the requirement $``|A|\leqslant m"$ in the definition
of $\widetilde{\sigma}_{m}(x)$ can be replaced by $``|A|=m"$, and the basis is
still almost greedy with the same constant $\mathbf{C}$. While greedy bases
are almost greedy, the converse does not necessarily hold. A famous example of
an almost greedy basis that is not greedy is due to Konyagin and Temlyakov
[15] (also see [4, Example 10.2.9]). Dilworth et al. [11] gave the following
nice and surprising characterization of almost greedy bases.

######  Theorem 1.5.

[11, Theorem 3.3] Let $(e_{n})_{n=1}^{\infty}$ be a basis of a Banach space.
The following are equivalent

  1. i)

$(e_{n})_{n=1}^{\infty}$ is almost greedy.

  2. ii)

$(e_{n})_{n=1}^{\infty}$ is quasi-greedy and democratic.

  3. iii)

For any (respectively, every) $\lambda>1$, there is a constant
$\mathbf{C}_{\lambda}>0$ such that

$\|x-P_{\Lambda}(x)\|\ \leqslant\ \mathbf{C}_{\lambda}\sigma_{m}(x),\forall
x\in X,\forall m\in\mathbb{N},\forall\Lambda\in G(x,\lceil\lambda m\rceil).$

The equivalence between i) and ii) shares the same spirit as Theorem 1.3,
while the equivalence between i) and iii) shows that enlarging the greedy sums
in the definition of greedy bases brings us to the strictly larger realm of
almost greedy bases. In other words, for almost greedy bases, an approximation
using greedy sums of size $\lceil\lambda m\rceil$ gives us essentially the
smallest error term from using an arbitrary $m$-term approximation.

Besides greedy and almost greedy bases, two other notable greedy-type bases
are the so-called partially greedy (PG) (introduced by Dilworth et al. [11])
and reverse partially greedy (RPG) bases (introduced by Dilworth and Khurana
[12]). They are not the focus of this paper; however, we would like to make a
few remarks about these bases. Unlike (almost) greedy bases, the notion of
both PG and RPG bases depends on the ordering of the basis. For example, in
the definition of partially greedy bases, we compare the efficiency of the TGA
against partial sums, the more convenient method of approximation. Recently,
some authors expressed their concerns about the notion of partially greedy
bases as it is order-dependent, while the TGA is order-free [2, page 6]. While
that is a well-founded concern, one can also argue that partially greedy bases
are crucial because they give us information about how the TGA performs
against the most convenient method of approximation, namely partial
summations. In other words, there is no point in choosing the largest
coefficients for our approximation if we cannot outperform partial summations.

###  1.2. Motivation

Inspired by Theorem 1.5, the author of the present paper [7, 9] investigated
what happens if we enlarge the greedy sums (by a constant factor $\lambda>1$)
in the definition of almost greedy, PG, and RPG bases. Surprisingly, while
doing so gives new, weaker greedy-type bases in the case of PG and RPG bases
(see [7, Theorem 1.5] and [9, Theorem 1.4]), [7, Theorem 5.4] states that we
do not obtain a weaker greedy-type basis in the case of almost greedy bases.
To put it another way, if the sizes of greedy sums and of the projective
approximation terms in the definition of almost greedy bases are of the same
order, i.e., $\lceil\lambda m\rceil$ and $m$, respectively, then we obtain
only equivalences of being almost greedy. However, we suspect that such an
equivalence no longer holds if the sizes are of different order, say $m$ and
$\log m$, for example. This paper examines what happens when the sizes are of
different order.

Our main results not only give new greedy-type bases and an unexpected
surprising equivalence among these bases but also provide a partial solution
to the above-mentioned concern of some researchers regarding the order-
dependent notion of PG and RPG bases. In particular, while some PG and PRG
greedy bases are not almost greedy, they may satisfy a certain order-free
condition, which involves resizing projective approximations in the definition
of (almost) greedy bases.

###  1.3. Main results

Throughout this paper, let $f:\mathbb{R}_{\geqslant
0}\rightarrow\mathbb{R}_{\geqslant 0}$ satisfy

  1. (1)

$f(0)=0$, $f(1)\leqslant 1$, and $f(x)>0$ for all $x>0$,

  2. (2)

$f$ is continuous, increasing, and concave on $[0,\infty)$, and

  3. (3)

$f$ is differentiable on $(0,\infty)$.

Let $\mathcal{F}$ denote the set of all such functions. Note that
$\mathcal{F}$ contains functions like $f(x)=cx^{\gamma}$ for
$c,\gamma\in(0,1]$. Also, $\mathcal{F}$ is closed under composition.

######  Definition 1.6.

A basis $(e_{n})_{n=1}^{\infty}$ is said to be $f$-almost greedy if there
exists a constant $\mathbf{C}>0$ such that

$\|x-P_{\Lambda}(x)\|\ \leqslant\
\mathbf{C}\widetilde{\sigma}_{f(m)}(x),\forall x\in X,\forall
m\in\mathbb{N},\forall\Lambda\in G(x,m).$ (1.2)

The least $\mathbf{C}$ is denoted by $\mathbf{C}_{a,f}$.

######  Definition 1.7.

A basis $(e_{n})_{n=1}^{\infty}$ is said to be $f$-greedy if there exists a
constant $\mathbf{C}>0$ such that

$\|x-P_{\Lambda}(x)\|\ \leqslant\ \mathbf{C}\sigma_{f(m)}(x),\forall x\in
X,\forall m\in\mathbb{N},\forall\Lambda\in G(x,m).$ (1.3)

The least such $\mathbf{C}$ is denoted by $\mathbf{C}_{g,f}$.

At the first glance, these definitions only make sense for functions $f$
satisfying $f(m)<m+1$ for all $m\in\mathbb{N}$. Otherwise, even the canonical
basis of $c_{0}$ does not satisfy (1.2). For example, suppose that $f(3)=4$
and consider the vector $x=(1,1,1,1,0,\ldots)$. Let $\Lambda=\\{1,2,3\\}$ be a
greedy set of $x$. While $\|x-P_{\Lambda}(x)\|=1$,
$\widetilde{\sigma}_{f(3)}(x)=0$. Therefore, there is no $\mathbf{C}$ for
(1.2) to hold. However, as we prove later, if $f\in\mathcal{F}$, then
$f(m)\leqslant m$ for all $m\in\mathbb{N}$.

The main goal of this paper is to establish the following unexpected result.

######  Theorem 1.8.

Let $f$ be a non-identity function in $\mathcal{F}$. Then a basis is
$f$-almost greedy if and only if it is $f$-greedy.

Theorem 1.8 says that if $f$ grows slower than linear functions, it does not
matter whether we compare the TGA against projections or linear combinations
of basis vectors. We shall prove Theorem 1.8 by characterizing $f$-greedy and
$f$-almost greedy bases separately, then show that the two characterizations
are equivalent when $f$ is non-identity. Below we list several interesting
results we shall encounter along the way. See Definition 3.5 for Property (A,
$f$) and Definition 4.4 for $f$-unconditionality, a new concept that lies
strictly between quasi-greediness and unconditionality.

######  Theorem 1.9.

Let $(e_{n})_{n=1}^{\infty}$ be a basis of a Banach space $X$. The following
hold.

  1. i)

If $(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{a,f}$-$f$-almost greedy, then
$(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{a,f}$-suppression quasi-greedy and
has $\mathbf{C}_{a,f}$-Property (A, $f$).

  2. ii)

Conversely, if $(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{\ell}$-suppression
quasi-greedy and has $\Delta_{A,f}$-Property (A, $f$), then
$(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{\ell}\Delta_{A,f}$-$f$-almost greedy.

######  Theorem 1.10.

Let $(e_{n})_{n=1}^{\infty}$ be a basis of a Banach space. The following hold.

  1. i)

If $(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{g,f}$-$f$-greedy, then it is
$\mathbf{C}_{g,f}$-suppression $f$-unconditional and has
$\mathbf{C}_{g,f}$-Property (A, $f$).

  2. ii)

If $(e_{n})_{n=1}^{\infty}$ is $\mathbf{K}_{su}$-suppression $f$-unconditional
and has $\Delta_{A,f}$-Property (A, $f$), then it is
$\Delta_{A,f}(\mathbf{K}^{2}_{su}+2\mathbf{K}_{su})$-$f$-greedy.

While the proof of Theorem 1.9 follows standard argument from the literature,
the proof of Theorem 1.10 item ii) requires a nontrivial deviation that relies
on properties of $f\in\mathcal{F}$ and a new concept, called $f$-greedy sets.
The next corollary involves (super)democracy (see Definition 3.2).

######  Corollary 1.11.

Let $(e_{n})_{n=1}^{\infty}$ be a basis of a Banach space $X$ and
$f\in\mathcal{F}$ be non-identity. The following are equivalent

  1. i)

$(e_{n})_{n=1}^{\infty}$ is $f$-greedy.

  2. ii)

$(e_{n})_{n=1}^{\infty}$ is $f$-almost greedy.

  3. iii)

$(e_{n})_{n=1}^{\infty}$ is quasi-greedy and has Property (A, $f$).

  4. iv)

$(e_{n})_{n=1}^{\infty}$ is quasi-greedy and $f$-(disjoint) superdemocratic.

  5. v)

$(e_{n})_{n=1}^{\infty}$ is quasi-greedy and $f$-(disjoint) democratic.

Furthermore, we can replace the quasi-greedy requirement in each of the above
statements by $f$-unconditionality.

Let us discuss an application of Corollary 1.11 to the space
$\ell_{p}\oplus\ell_{q}$ ($1\leqslant p<q<\infty$). Èdel’šteǐn and Wojtaszczyk
[13] showed that any unconditional basis $\mathcal{B}$ of the direct sum
$\ell_{p}\oplus\ell_{q}$ consists of two subsequences, one of which, denoted
by $\mathcal{B}_{1}$, is a basis for $\ell_{p}$, while the other, denoted by
$\mathcal{B}_{2}$, is a basis for $\ell_{q}$. By the gliding hump argument, we
can find subsequences of $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ that are
equivalent to the canonical bases of $\ell_{p}$ and $\ell_{q}$, respectively.
Therefore, none of the unconditional bases of $\ell_{p}\oplus\ell_{q}$ is
democratic, so $\ell_{p}\oplus\ell_{q}$ has no greedy basis. However,
$\ell_{p}\oplus\ell_{q}$ has an $x^{p/q}$-greedy basis. For example, let
$(e_{n})_{n=1}^{\infty}$ and $(f_{n})_{n=1}^{\infty}$ be the canonical bases
of $\ell_{p}$ and $\ell_{q}$, respectively. Define the basis
$\mathcal{X}=(x_{n})_{n=1}^{\infty}$, where $x_{2k-1}=(e_{k},0)$ and
$x_{2k}=(0,f_{k})$. It is easy to check that $\mathcal{X}$ is
$x^{p/q}$-democratic. By Corollary 1.11, $\mathcal{X}$ is $x^{p/q}$-greedy.

Another application of Corollary 1.11 is to the multivariate Haar system
$\mathcal{H}^{d}_{p}$ ($d>1$) in $L_{p}([0,1]^{d})$ for $1<p<\infty$, which
shall be discussed in Section 7. Roughly speaking, though the system is not
greedy, it is close to being greedy in the sense that $\mathcal{H}^{d}_{p}$ is
$x^{1-\varepsilon}$-greedy for any $\varepsilon>0$.

The paper is structured as follows:

  * •

Section 2 discusses some properties of functions in $\mathcal{F}$ and of
quasi-greedy bases. These results will be used throughout the paper.

  * •

Section 3 studies $f$-(disjoint) (super)democratic bases and Property (A,
$f$). These are the first ingredient in characterizing $f$-(almost) greedy
bases.

  * •

Section 4 introduces and characterizes the new notion, $f$-unconditionality,
which is the second ingredient in characterizing $f$-(almost) greedy bases.

  * •

Section 5 characterizes $f$-(almost) greedy bases and show the unexpected
equivalence between $f$-greedy and $f$-almost greedy bases when $f$ is non-
identity.

  * •

Section 6 investigates the relationship among $f$-(almost) greedy bases when
$f$ varies.

  * •

Section 7 gives various examples to complement our results.

##  2\. Preliminary results

Let us prove some important properties of functions in $\mathcal{F}$.

######  Proposition 2.1.

Let $f\in\mathcal{F}$. The following hold

  1. i)

If $0<a\leqslant b$, then $\frac{f(a)}{a}\geqslant\frac{f(b)}{b}$.

  2. ii)

For $x\geqslant 1$, $f(x)\leqslant x$.

  3. iii)

Let $m,n\in\mathbb{N}$ and $k$ be a nonnegative integer such that $n\geqslant
k$. If $n\leqslant f(m)$, then $n-k\leqslant f(m-k)$.

###### Proof.

i) Consider $g:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{\geqslant 0}$ defined by
$g(x)=f(x)/x$. It suffices to prove that

$g^{\prime}(x)\ =\ (f^{\prime}(x)x-f(x))/x^{2}\ \leqslant\ 0,\forall x>0.$
(2.1)

Recall that if $f$ is concave and differentiable on an interval $(u,v)$, then

$f(s)\ \leqslant\ f(t)+f^{\prime}(t)(s-t),\forall s,t\in(u,v).$

Therefore, for $0<\varepsilon<x$, we have

$f(\varepsilon)\ \leqslant\ f(x)+f^{\prime}(x)(\varepsilon-x).$

Letting $\varepsilon\rightarrow 0$, by continuity, we obtain

$0\ \leqslant\ f(x)-xf^{\prime}(x),\forall x>0,$

which implies (2.1).

ii) Let $x\geqslant 1$. By item i),
$\frac{f(x)}{x}\leqslant\frac{f(1)}{1}\leqslant 1$. Hence, $f(x)\leqslant x$.

iii) Since $f(m)\leqslant m$ by item ii), we get $k\leqslant n\leqslant m$. If
$n=k$, there is nothing to prove. If $n>k$, by item i), we have

$\frac{f(m-k)}{f(m)}\ \geqslant\ \frac{m-k}{m}\ \geqslant\ \frac{n-k}{n}.$

Hence,

$f(m-k)\ \geqslant\ f(m)\frac{n-k}{n}\ \geqslant\ n-k.$

This completes our proof. ∎

######  Proposition 2.2.

If $f\in\mathcal{F}$, then either $f$ is the identity or there exists
$\lambda>1$ such that $f(x)\ \leqslant\ \frac{x}{\lambda}$ for all
sufficiently large $x$.

###### Proof.

By Proposition 2.1 items i) and ii), we know that
$\lim_{x\rightarrow\infty}\frac{f(x)}{x}=:s\leqslant 1$. If $s<1$, then for
sufficiently large $x$, $\frac{f(x)}{x}<\frac{s+1}{2}<1$, which gives
$f(x)<\frac{x}{2/(s+1)}$ and we are done. Suppose that $s=1$. Since
$\frac{f(x)}{x}$ is decreasing for $x>0$ and
$\lim_{x\rightarrow\infty}\frac{f(x)}{x}=1$, it follows that $f(x)\geqslant x$
for $x>0$. On the other hand, Proposition 2.1 item ii) states that
$f(x)\leqslant x$ for $x\geqslant 1$. Therefore, we get $f(x)=x$ for all
$x\geqslant 1$. It remains to show that $f(x)\leqslant x$ for all $x\in[0,1)$.
Suppose, for a contradiction, that there exists a $p\in(0,1)$ such that
$f(p)=p+\varepsilon$ for some $\varepsilon>0$. Choose $q=2-p$. We have

$f(q)+f(p)\ =\ (2-p)+(p+\varepsilon)\ =\ 2+\varepsilon\ >\ 2f(1)\
\Longrightarrow\ \frac{1}{2}f(q)+\frac{1}{2}f(p)\ >\
f\left(\frac{q}{2}+\frac{p}{2}\right),$

which contradicts concavity. Therefore, if $s=1$, then $f$ is the identity. ∎

Next, we mention two well-known properties of a quasi-greedy basis. For
$A\in\mathbb{N}^{<\infty}$, let $1_{A}:=\sum_{n\in A}e_{n}$.

######  Definition 2.3.

A basis is said to have the UL property if there exist
$\mathbf{C}_{1},\mathbf{C}_{2}>0$ such that

$\frac{1}{\mathbf{C}_{1}}\min|a_{n}|\|1_{A}\|\ \leqslant\ \left\|\sum_{n\in
A}a_{n}e_{n}\right\|\ \leqslant\ \mathbf{C}_{2}\max|a_{n}|\|1_{A}\|,$

for all $A\in\mathbb{N}^{<\infty}$ and scalars $(a_{n})_{n\in
A}\subset\mathbb{F}$.

######  Theorem 2.4.

[11, Lemma 2.1] If $(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{\ell}$-suppression
quasi-greedy, then $(e_{n})_{n=1}^{\infty}$ has the UL property. In
particular,

$\frac{1}{2\mathbf{C}_{\ell}}\min|a_{n}|\|1_{A}\|\ \leqslant\
\left\|\sum_{n\in A}a_{n}e_{n}\right\|\ \leqslant\
2\mathbf{C}_{\ell}\max|a_{n}|\|1_{A}\|,$

for all $A\in\mathbb{N}^{<\infty}$ and scalars $(a_{n})_{n\in
A}\subset\mathbb{F}$.

Another property of a quasi-greedy basis is the uniform boundedness of the
truncation operators. Let $\alpha>0$ and define $T_{\alpha}:X\rightarrow X$ by

$T_{\alpha}(x)\ =\ \sum_{n\in
A_{x}}\alpha\operatorname{sgn}(e_{n}^{*}(x))e_{n}+P_{A_{x}^{c}}(x),$

where $A_{x}=\\{n:|e_{n}^{*}(x)|>\alpha\\}$.

######  Theorem 2.5.

[5, Lemma 2.5] If $(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{\ell}$-suppression
quasi-greedy, then $\|T_{\alpha}\|\leqslant\mathbf{C}_{\ell}$ for all
$\alpha>0$.

The final result in this section follows from a well-known Lebesgue-type
inequality.

######  Lemma 2.6.

Let $(e_{n})_{n=1}^{\infty}$ be a basis of a Banach space $X$. For each
$N\in\mathbb{N}$, we have

$\|x-P_{\Lambda}(x)\|\ \leqslant\
(1+3\mathbf{c}_{2}^{2}N)\sigma_{m}(x),\forall x\in X,\forall m\leqslant
N,\forall\Lambda\in G(x,m).$

###### Proof.

The statement comes from [5, Theorem 1.8]. ∎

As an application of Lemma 2.6, we have

######  Proposition 2.7.

Let $f\in\mathcal{F}$ such that $f$ is not the identity. If
$(e_{n})_{n=1}^{\infty}$ is almost greedy, then $(e_{n})_{n=1}^{\infty}$ is
$f$-greedy.

###### Proof.

Since $f$ is not the identity, by Proposition 2.2, there exists $\lambda>1$
such that $f(x)\leqslant\frac{x}{\lambda}$ for all sufficiently large $x$. As
$(e_{n})_{n=1}^{\infty}$ is almost greedy, Theorem 1.5 implies that
$(e_{n})_{n=1}^{\infty}$ is $x/\lambda$-greedy. For sufficiently large $m$,
since $f(m)\leqslant\frac{m}{\lambda}$, we have
$\sigma_{m/\lambda}(x)\leqslant\sigma_{f(m)}(x)$ for all $x\in X$, which gives
(1.3). For small $m$, use Lemma 2.6. ∎

######  Remark 2.8.

The converse of Proposition 2.7 is not true, as evidenced by unconditional
bases of $\ell_{p}\oplus\ell_{q}$ ($1\leqslant p<q<\infty$) and the
multivariate Haar system in $L_{p}([0,1]^{d})$ ($p,d>1$). See the discussion
after Corollary 1.11.

##  3\. About $f$-(disjoint) democracy and Property (A, $f$)

We recall the democratic property of bases was due to Konyagin and Temlyakov
[15].

######  Definition 3.1.

A basis is said to be democratic if there exists a constant
$\mathbf{C}\geqslant 1$ such that

$\|1_{A}\|\ \leqslant\ \mathbf{C}\|1_{B}\|,\forall
A,B\in\mathbb{N}^{<\infty},|A|\leqslant|B|.$

A sign $\varepsilon$ is a sequence of scalars
$(\varepsilon_{n})_{n=1}^{\infty}\subset\mathbb{F}$ with modulus $1$. If
$A\in\mathbb{N}^{<\infty}$, we let $1_{\varepsilon A}:=\sum_{n\in
A}\varepsilon_{n}e_{n}$. For $x\in X$ and $A,B\in\mathbb{N}^{<\infty}$, we
write $x\sqcup A\sqcup B$ to mean that $\operatorname{supp}(x),A$, and $B$ are
pairwise disjoint. Similarly, write $\sqcup_{i}A_{i}$ to mean that the sets
$A_{i}$ are pairwise disjoint. Finally,
$\|x\|_{\infty}:=\max_{n\in\mathbb{N}}|e_{n}^{*}(x)|$.

######  Definition 3.2.

A basis $(e_{n})_{n=1}^{\infty}$ is said to be $f$-disjoint superdemocratic if
there exists a constant $\mathbf{C}>0$ such that

$\|1_{\varepsilon A}\|\ \leqslant\ \mathbf{C}\|1_{\delta B}\|,$ (3.1)

for all disjoint $A,B\in\mathbb{N}^{<\infty}$ with $|A|\leqslant f(|B|)$ and
for all signs $\varepsilon,\delta$. The least $\mathbf{C}$ that verifies (3.1)
is denoted by $\Delta_{sd,f}$. If $\varepsilon\equiv\delta\equiv 1$, we say
that $(e_{n})_{n=1}^{\infty}$ is $f$-disjoint democratic, and let
$\Delta_{d,f}$ denote the corresponding constant.

######  Remark 3.3.

For the definition of $f$-(super)democratic, we drop the requirement that $A$
and $B$ are disjoint in Definition 3.2.

######  Proposition 3.4.

Let $(e_{n})_{n=1}^{\infty}$ be a quasi-greedy basis. Then
$(e_{n})_{n=1}^{\infty}$ is $f$-disjoint democratic if and only if it is
$f$-disjoint superdemocratic. Similarly, a quasi-greedy basis is
$f$-democratic if and only if it is $f$-superdemocratic.

###### Proof.

Let $(e_{n})_{n=1}^{\infty}$ be a $\mathbf{C}_{\ell}$-suppression quasi-greedy
basis. By definition, if $(e_{n})_{n=1}^{\infty}$ is $f$-disjoint
superdemocratic, then it is $f$-disjoint democratic. Conversely, assume that
$(e_{n})_{n=1}^{\infty}$ is $\Delta_{d,f}$-$f$-disjoint democratic. Let
$A,B\in\mathbb{N}^{<\infty}$ be disjoint and $|A|\leqslant f(|B|)$. Pick signs
$\varepsilon,\delta$. By the UL property, we have

$\frac{1}{2\mathbf{C}_{\ell}}\|1_{\varepsilon A}\|\ \leqslant\ \|1_{A}\|\
\leqslant\ \Delta_{d,f}\|1_{B}\|\ \leqslant\
2\Delta_{d,f}\mathbf{C}_{\ell}\|1_{\delta B}\|.$

Therefore, $\|1_{\varepsilon A}\|\leqslant
4\Delta_{d,f}\mathbf{C}^{2}_{\ell}\|1_{\delta B}\|$. ∎

Finally, we introduce Property (A, $f$), which is closely related to the
democratic property and will be used in due course.

######  Definition 3.5.

A basis $(e_{n})_{n=1}^{\infty}$ is said to have Property (A, $f$) if there
exists a constant $\mathbf{C}\geqslant 1$ such that

$\|x+1_{\varepsilon A}\|\ \leqslant\ \mathbf{C}\|x+1_{\delta B}\|,$ (3.2)

for all $x\in X$ with $\|x\|_{\infty}\leqslant 1$, for all
$A,B\subset\mathbb{N}^{<\infty}$ with $|A|\leqslant f(|B|)$ and $x\sqcup
A\sqcup B$, and for all signs $\varepsilon,\delta$. The least such
$\mathbf{C}$ is denoted by $\Delta_{A,f}$.

######  Proposition 3.6.

A basis $(e_{n})_{n=1}^{\infty}$ has $\Delta_{A,f}$-Property (A, $f$) if and
only if

$\|x\|\ \leqslant\ \Delta_{A,f}\|x-P_{A}(x)+1_{\varepsilon B}\|,$ (3.3)

for all $x\in X$ with $\|x\|_{\infty}\leqslant 1$, for all
$A,B\subset\mathbb{N}^{<\infty}$ such that $|A|\leqslant f(|B|)$ and
$(x-P_{A}(x))\sqcup A\sqcup B$, and for all signs $\varepsilon$.

###### Proof.

Assume that $(e_{n})_{n=1}^{\infty}$ has $\Delta_{A,f}$-Property (A, $f$). Let
$x,A,B,\varepsilon$ be chosen as in (3.3). We have

$\displaystyle\|x\|\ =\ \left\|x-P_{A}(x)+\sum_{n\in
A}e_{n}^{*}(x)e_{n}\right\|$ $\displaystyle\ \leqslant\
\sup_{\delta}\left\|x-P_{A}(x)+1_{\delta A}\right\|$ $\displaystyle\
\leqslant\ \Delta_{A,f}\left\|x-P_{A}(x)+1_{\varepsilon B}\right\|.$

Conversely, let $x,A,B,\varepsilon,\delta$ be chosen as in Definition 3.5. Let
$y=x+1_{\varepsilon A}$. We have

$\displaystyle\|x+1_{\varepsilon A}\|\ =\ \|y\|\ \leqslant\
\Delta_{A,f}\|y-P_{A}(y)+1_{\delta B}\|\ =\ \Delta_{A,f}\|x+1_{\delta B}\|.$

This completes our proof. ∎

######  Proposition 3.7.

Let $(e_{n})_{n=1}^{\infty}$ be a quasi-greedy basis and $f\in\mathcal{F}$.
Then the following are equivalent

  1. i)

$(e_{n})_{n=1}^{\infty}$ is $f$-disjoint democratic.

  2. ii)

$(e_{n})_{n=1}^{\infty}$ is $f$-democratic.

  3. iii)

$(e_{n})_{n=1}^{\infty}$ is $f$-disjoint superdemocratic.

  4. iv)

$(e_{n})_{n=1}^{\infty}$ is $f$-superdemocratic.

  5. v)

$(e_{n})_{n=1}^{\infty}$ has Property (A, $f$).

###### Proof.

In light of Proposition 3.4, it suffices to show that for quasi-greedy bases,

  * a)

an $f$-disjoint democratic basis is $f$-democratic.

  * b)

Property (A, $f$) is equivalent to $f$-disjoint superdemocracy.

a) Let $A,B\in\mathbb{N}^{<\infty}$ and $|A|\leqslant f(|B|)$. By Proposition
2.1 item iii), $|A\backslash(A\cap B)|\leqslant f(|B\backslash(A\cap B)|)$.
Hence, $f$-disjoint democracy and quasi-greediness imply that

$\|1_{A\backslash(A\cap B)}\|\ \leqslant\ \Delta_{d,f}\|1_{B\backslash(A\cap
B)}\|\ \leqslant\ \Delta_{d,f}\mathbf{C}_{q}\|1_{B}\|\mbox{ and }\|1_{A\cap
B}\|\ \leqslant\ \mathbf{C}_{q}\|1_{B}\|.$

We, therefore, obtain

$\|1_{A}\|\ \leqslant\ \|1_{A\backslash(A\cap B)}\|+\|1_{A\cap B}\|\
\leqslant\ \mathbf{C}_{q}(1+\Delta_{d,f})\|1_{B}\|.$

b) Clearly, Property (A, $f$) implies $f$-disjoint superdemocracy. We prove
the converse. Assume that $(e_{n})_{n=1}^{\infty}$ is
$\mathbf{C}_{\ell}$-suppression quasi-greedy and $\Delta_{sd,f}$-$f$-disjoint
superdemocratic. We have

$\displaystyle\|x+1_{\varepsilon A}\|\ \leqslant\ \|x\|+\|1_{\varepsilon A}\|$
$\displaystyle\ \leqslant\ \mathbf{C}_{\ell}\|x+1_{\delta
B}\|+\Delta_{sd,f}\|1_{\delta B}\|$ $\displaystyle\ \leqslant\
\mathbf{C}_{\ell}\|x+1_{\delta
B}\|+\Delta_{sd,f}(\mathbf{C}_{\ell}+1)\|x+1_{\delta B}\|$ $\displaystyle\ =\
(\mathbf{C}_{\ell}+\Delta_{sd,f}(\mathbf{C}_{\ell}+1))\|x+1_{\delta B}\|.$

Hence, $(e_{n})_{n=1}^{\infty}$ has Property (A, $f$). ∎

##  4\. About $f$-unconditionality

Unconditionality is a well-known and strong requirement on bases. In this
section, we introduce a weaker concept, called $f$-unconditionality, that
shall be shown to lie strictly between quasi-greediness and unconditionality.
We study this new property and use it to later characterize $f$-greedy bases.
First of all, we have the classical definition of unconditionality.

######  Definition 4.1.

A basis is said to be unconditional if there exists a constant
$\mathbf{C}\geqslant 1$ such that

$\|x-P_{A}(x)\|\ \leqslant\ \mathbf{C}\|x\|,\forall x\in X,\forall
A\subset\mathbb{N}.$

In this case, the least constant $\mathbf{C}$ is denoted by $\mathbf{K}_{su}$,
called the suppression unconditional constant.

######  Definition 4.2.

Fix $f\in\mathcal{F}$. A set $A\in\mathbb{N}^{<\infty}$ is an $f$-greedy set
of $x$ if $A$ can be written as the disjoint union of $E\sqcup F$, where
$|E|\leqslant f(|A|)$, and

$|e_{n}^{*}(x)|\ \geqslant\ \|x-P_{A}(x)\|_{\infty},\forall n\in F.$

We call $(E,F)$ an $f$-greedy decomposition of $A$. Let $G(x,f)$ denote the
collection of all $f$-greedy sets of $x$.

######  Proposition 4.3.

Fix $x\in X$ and $f\in\mathcal{F}$. A set $A\in\mathbb{N}^{<\infty}$ is an
$f$-greedy set of $x$ if and only if there exists $B\subset A$ such that $B$
is a greedy set of $x$ of size at least $|A|-f(|A|)$.

###### Proof.

Let $x\in X$ and $f\in\mathcal{F}$. Assume that $A\in G(x,f)$. Let $(E,F)$ be
an $f$-greedy decomposition of $A$. Then $|E|\leqslant f(|A|)$. If
$F=\emptyset$, then $A=E$ and $|A|=|E|\leqslant f(|A|)$; that is,
$|A|-f(|A|)=0$ and there is nothing to prove. If $F\neq\emptyset$, let
$\alpha=\min_{n\in F}|e_{n}^{*}(x)|$. Define $E^{\prime}=\\{n\in
E:|e_{n}^{*}(x)|\geqslant\alpha\\}$. It is easy to check that $E^{\prime}\cup
F$ is a greedy set of $x$. Furthermore,

$|E^{\prime}\cup F|\ =\ |E^{\prime}|+|F|\ =\ |A|-|E\backslash E^{\prime}|\
\geqslant\ |A|-f(|A|).$

Conversely, pick $A\in\mathbb{N}^{<\infty}$ such that there exists $B\subset
A$ and $B$ is a greedy set of $x$ of size at least $|A|-f(|A|)$. We claim that
$(A\backslash B,B)$ is an $f$-greedy decomposition of $A$. Indeed,

$\min_{n\in B}|e_{n}^{*}(x)|\ \geqslant\ \|x-P_{B}(x)\|_{\infty}\ \geqslant\
\|x-P_{A}(x)\|_{\infty}$

and

$|A\backslash B|\ =\ |A|-|B|\ \leqslant\ |A|-(|A|-f(|A|))\ =\ f(|A|).$

Therefore, $A$ is an $f$-greedy set of $x$. ∎

We now introduce $f$-unconditionality.

######  Definition 4.4.

A basis $(e_{n})_{n=1}^{\infty}$ is said to be $f$-unconditional if there
exists a constant $\mathbf{C}\geqslant 1$ such that

$\|x-P_{A}(x)\|\ \leqslant\ \mathbf{C}\|x\|,\forall x\in X,\forall A\in
G(x,f).$

The smallest such $\mathbf{C}$ is denoted by $\mathbf{K}_{su}$, and we say
that $(e_{n})_{n=1}^{\infty}$ is $\mathbf{K}_{su}$-suppression
$f$-unconditional. In this case, let $\mathbf{K}_{u}$ denote the smallest
constant such that

$\|P_{A}(x)\|\ \leqslant\ \mathbf{K}_{u}\|x\|,\forall x\in X,\forall A\in
G(x,f),$

and say that $(e_{n})_{n=1}^{\infty}$ is $\mathbf{K}_{u}$-$f$-unconditional.

######  Remark 4.5.

We can obtain Theorem 1.3 from Theorem 1.10 and Proposition 3.7 by choosing
$f$ to be identity function. Our notion of $f$-unconditionality lies between
quasi-greediness and unconditionality. By definition, we have the implications

$\mbox{unconditional}\Longrightarrow
f\mbox{-unconditional}\Longrightarrow\mbox{quasi-greedy}.$

To show that our notion of $f$-unconditionality does not overlap with either
of these notions, we show that there exists a conditional basis that is
$x/\lambda$-unconditional for all $\lambda>1$ (Example 7.2), and there exists
a quasi-greedy basis that is not $f$-unconditional for any unbounded
$f\in\mathcal{F}$ (Example 7.4).

Finally, observe that when $f$ is the identity function, $f$-unconditionality
is the same as unconditionality. When $f$ is bounded, it is not hard to show
that $f$-unconditionality is the same as quasi-greedy.

######  Proposition 4.6.

If a basis $(e_{n})_{n=1}^{\infty}$ is $\mathbf{K}_{su}$-suppression
$f$-unconditional, then it is $\mathbf{K}_{su}$-suppression quasi-greedy. If a
basis is $\mathbf{K}_{u}$-$f$-unconditional, then it is
$\mathbf{K}_{u}$-quasi-greedy.

###### Proof.

Simply note that for all $m\in\mathbb{N}$, $G(x,m)\subset G(x,f)$ by
Proposition 4.3. ∎

We characterize $f$-unconditionality as follows.

######  Proposition 4.7.

Let $f\in\mathcal{F}$. A basis $(e_{n})_{n=1}^{\infty}$ is $f$-unconditional
if and only if it is quasi-greedy and there exists a constant $\mathbf{C}>0$
such that

$\|P_{A}(x)\|\ \leqslant\ \mathbf{C}\|x\|,$ (4.1)

for all $x\in\mathbb{X}$ and for all $A\in\mathbb{N}^{<\infty}$ such that
$|A|\leqslant f(|A|+|B|)$ for some greedy set $B$ of $x$ that is disjoint from
$A$.

###### Proof.

Assume that $(e_{n})_{n=1}^{\infty}$ is $\mathbf{K}_{u}$-$f$-unconditional. By
Proposition 4.6, $(e_{n})_{n=1}^{\infty}$ is $\mathbf{K}_{u}$-quasi-greedy. To
see (4.1), we fix $x\in\mathbb{X}$, a greedy set $B$ of $x$, and
$A\subset\mathbb{N}^{<\infty}$ such that $A\sqcup B$ and $|A|\leqslant
f(|A|+|B|)$. We have

$\|P_{A}(x)\|\ \leqslant\ \|P_{A\cup B}(x)\|+\|P_{B}(x)\|\ \leqslant\
\|P_{A\cup B}(x)\|+\mathbf{K}_{u}\|x\|.$

Since $A\cup B\in G(x,f)$, we get

$\|P_{A\cup B}(x)\|\ \leqslant\ \mathbf{K}_{u}\|x\|.$

We conclude that $\|P_{A}(x)\|\leqslant 2\mathbf{K}_{u}\|x\|$.

Conversely, assume that $(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{q}$-quasi-
greedy and satisfies (4.1). Let $x\in X$ and $A\in G(x,f)$ with an $f$-greedy
decomposition $(E,F)$. If $F=\emptyset$, then $A=E$ and $|E|\leqslant f(|E|)$.
By (4.1), we obtain

$\|P_{A}(x)\|\ \leqslant\ \mathbf{C}\|x\|.$

If $F\neq\emptyset$, let $\alpha=\min_{n\in F}|e_{n}^{*}(x)|$ and
$E^{\prime}=\\{n\in E:|e^{*}_{n}(x)|<\alpha\\}$. Then
$F^{\prime}=F\cup(E\backslash E^{\prime})$ is a greedy set of $x$. We get

$\|P_{A}(x)\|\ \leqslant\ \|P_{E^{\prime}}(x)\|+\|P_{F^{\prime}}(x)\|\
\leqslant\ \|P_{E^{\prime}}(x)\|+\mathbf{C}_{q}\|x\|.$

Furthermore,

$|E^{\prime}|\ \leqslant\ |E|\ \leqslant\ f(|A|)\ =\
f(|E^{\prime}|+|F^{\prime}|).$

Hence, (4.1) implies that $\|P_{E^{\prime}}(x)\|\leqslant\mathbf{C}\|x\|$.
Therefore,

$\|P_{A}(x)\|\ \leqslant\ (\mathbf{C}_{q}+\mathbf{C})\|x\|.$

This completes our proof. ∎

The above characterization inspires our introduction of Property ($f$).

######  Definition 4.8.

A basis $(e_{n})_{n=1}^{\infty}$ is said to have Property ($f$) if there
exists a constant $\mathbf{C}>0$ such that

$\|P_{A}(x)\|\ \leqslant\ \mathbf{C}\left\|x+1_{\varepsilon B}\right\|,$

for all $x\in\mathbb{X}$ with $\|x\|_{\infty}\leqslant 1$, for sets
$A,B\in\mathbb{N}^{<\infty}$ with $A\subset\operatorname{supp}(x)$, $x\sqcup
B$, and $|A|\leqslant f(|A|+|B|)$, and for all signs $\varepsilon$.

When $f$ is the identity function, Property ($f$) is the same as
unconditionality.

######  Proposition 4.9.

A basis is $f$-unconditional if and only if it is quasi-greedy and has
Property ($f$).

###### Proof.

Due to Propositions 4.6 and 4.7, it suffices to prove that for a quasi-greedy
basis, Property ($f$) is equivalent to (4.1). Assume (4.1). Let
$x,A,B,\varepsilon$ as in Definition 4.8. Set $y=x+1_{\varepsilon B}$. Then
$B$ is a greedy set of $y$ and $|A|\leqslant f(|A|+|B|)$. By (4.1),

$\|P_{A}(x)\|\ =\ \|P_{A}(y)\|\ \leqslant\ \mathbf{C}\|y\|\ =\
\mathbf{C}\|x+1_{\varepsilon B}\|.$

Conversely, assume that $(e_{n})_{n=1}^{\infty}$ has $\Delta$-Property ($f$)
and is $\mathbf{C}_{\ell}$-suppression quasi-greedy. Choose $x,A,B$ as in
(4.1). If $B=\emptyset$, then $|A|\leqslant f(|A|)$. By Property ($f$),
$\|P_{A}(x)\|\leqslant\Delta\|x\|$. If $B\neq\emptyset$, let
$\alpha=\min_{n\in B}|e_{n}^{*}(x)|$ and
$\varepsilon=(\operatorname{sgn}(e_{n}^{*}(x))$. By Theorem 2.5, we have

$\|P_{B^{c}}(x)+\alpha 1_{\varepsilon B}\|\ \leqslant\
\mathbf{C}_{\ell}\|x\|.$

Let $A^{\prime}=A\cap\operatorname{supp}(P_{B^{c}}(x))$, which satisfies
$|A^{\prime}|\leqslant f(|A^{\prime}|+|B|)$ due to Proposition 2.1 item iii).
By Property ($f$),

$\|P_{A}(x)\|\ =\ \|P_{A^{\prime}}(P_{B^{c}}(x))\|\ \leqslant\
\Delta\|P_{B^{c}}(x)+\alpha 1_{\varepsilon B}\|.$

Therefore,

$\|P_{A}(x)\|\ \leqslant\ \mathbf{C}_{\ell}\Delta\|x\|,$

as desired. ∎

######  Remark 4.10.

We provide several examples to show that our characterization is not
redundant. In particular, there exists a quasi-greedy basis that does not have
Property ($f$) for any unbounded $f$ (Example 7.4). On the other hand, there
exists a non-quasi-greedy basis that has Property ($x/\lambda$) for any
$\lambda>1$ (Example 7.3). This is the best we can do since Property ($x$) is
the same as unconditionality.

##  5\. Characterizations of $f$-(almost) greedy bases

In this section, we prove Theorems 1.9, 1.10 and Corollary 1.11, all of which
give equivalences of $f$-(almost) greedy bases.

###### Proof of Theorem 1.9.

i) Assume that $(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{a,f}$-$f$-almost
greedy. Since by definition, $\widetilde{\sigma}_{m}(x)\leqslant\|x\|$ for all
$x\in X$ and $m\in\mathbb{N}$, (1.2) implies that $(e_{n})_{n=1}^{\infty}$ is
$\mathbf{C}_{a,f}$-suppression quasi-greedy. Let $x,A,B,\varepsilon,\delta$ be
chosen as in Definition 3.5. Form $y=x+1_{\varepsilon A}+1_{\delta B}$. We
have

$\|x+1_{\varepsilon A}\|\ =\ \|y-P_{B}(y)\|\ \leqslant\
\mathbf{C}_{a,f}\widetilde{\sigma}_{f(|B|)}(y)\ \leqslant\
\mathbf{C}_{a,f}\|x+1_{\delta B}\|.$

ii) Assume that $(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{\ell}$-suppression
quasi-greedy and has $\Delta_{A,f}$-Property (A, $f$). Let $x\in X$,
$m\in\mathbb{N}$, $\Lambda\in G(x,m)$, and a set $B\subset\mathbb{N}$ with
$|B|\leqslant f(|\Lambda|)$, which implies that $|B\backslash\Lambda|\leqslant
f(|\Lambda\backslash B|)$ by Proposition 2.1 item iii). Let
$\varepsilon=(\operatorname{sgn}(e_{n}^{*}(x))$ and
$\alpha=\min_{n\in\Lambda}|e_{n}^{*}(x)|$. By Proposition 3.6 and Theorem 2.5,
we have

$\displaystyle\|x-P_{\Lambda}(x)\|$ $\displaystyle\ \leqslant\
\Delta_{A,f}\|x-P_{\Lambda}(x)-P_{B\backslash\Lambda}(x)+\alpha
1_{\varepsilon\Lambda\backslash B}\|$ $\displaystyle\ \leqslant\
\Delta_{A,f}\|T_{\alpha}(x-P_{\Lambda}(x)-P_{B\backslash\Lambda}(x)+P_{\Lambda\backslash
B}(x))\|$ $\displaystyle\ \leqslant\
\Delta_{A,f}\mathbf{C}_{\ell}\|x-P_{B}(x)\|.$

Since $B$ is arbitrary, we know that $(e_{n})_{n=1}^{\infty}$ is
$\Delta_{A,f}\mathbf{C}_{\ell}$-$f$-almost greedy. ∎

###### Proof of Theorem 1.10.

i) Assume that $(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{g,f}$-$f$-greedy.
First, we show that it is $\mathbf{C}_{g,f}$-suppression $f$-unconditional.
Let $x\in X$, $A\in G(x,f)$, and $(E,F)$ be an $f$-greedy decomposition of
$A$. Choose $\alpha>0$ sufficiently large such that $y:=x+\alpha 1_{E}$ has
$E$ as a greedy set. It follows that $A$ is a greedy set of $y$. We have

$\|x-P_{A}(x)\|\ =\ \|y-P_{A}(y)\|\ \leqslant\
\mathbf{C}_{g,f}\sigma_{f(|A|)}(y)\ \leqslant\ \mathbf{C}_{g,f}\|y-\alpha
1_{E}\|\ =\ \mathbf{C}_{g,f}\|x\|.$

Therefore, $(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{g,f}$-suppression
$f$-unconditional. To see that $(e_{n})_{n=1}^{\infty}$ has
$\mathbf{C}_{g,f}$-Property (A, $f$), we use a similar argument as in the
proof of Theorem 1.9 item i).

ii) Assume that $(e_{n})_{n=1}^{\infty}$ is $\mathbf{K}_{su}$-suppression
$f$-unconditional and has $\Delta_{A,f}$-Property (A, $f$). Let $x\in X$,
$\Lambda\in G(x,m)$, $A\in\mathbb{N}^{<\infty}$ with $|A|\leqslant f(m)$, and
$(a_{n})_{n\in A}$ be arbitrary scalars. Set
$\varepsilon=(\operatorname{sgn}(e_{n}^{*}(x)))$ and
$\alpha=\min_{n\in\Lambda}|e_{n}^{*}(x)|$. Note that $|A\backslash\Lambda|\
\leqslant\ f(|\Lambda\backslash A|)$. By Proposition 3.6, we have

$\displaystyle\|x-P_{\Lambda}(x)\|$ $\displaystyle\ \leqslant\
\Delta_{A,f}\|x-P_{\Lambda}(x)-P_{A\backslash\Lambda}(x)+\alpha
1_{\varepsilon\Lambda\backslash A}\|$ $\displaystyle\ =\
\Delta_{A,f}\|x-P_{\Lambda\cup A}(x)+\alpha 1_{\varepsilon\Lambda\backslash
A}\|.$ (5.1)

On the other hand, write

$y\ :=\ x-\sum_{n\in A}a_{n}e_{n}\ =\ x-P_{\Lambda\cup A}(x)+\sum_{n\in
A}(e_{n}^{*}(x)-a_{n})e_{n}+P_{\Lambda\backslash A}(x).$

Observe that $\Lambda\cup A\in G(y,f)$ because $f(|\Lambda\cup A|)\geqslant
f(m)\geqslant|A|$ and

$|e_{n}^{*}(y)|\ \geqslant\ \alpha\ \geqslant\ \|x-P_{\Lambda\cup
A}(x)\|_{\infty}\ =\ \|y-P_{\Lambda\cup A}(y)\|_{\infty},\forall
n\in\Lambda\backslash A.$

Therefore,

$\|x-P_{\Lambda\cup A}(x)\|\ =\ \|y-P_{\Lambda\cup A}(y)\|\ \leqslant\
\mathbf{K}_{su}\|y\|.$ (5.2)

On the other hand, by Proposition 4.6 and Theorem 2.5, we get

$\displaystyle\|y\|$ $\displaystyle\ \geqslant\
\frac{1}{\mathbf{K}_{su}}\left\|x-P_{\Lambda\cup A}(x)+\sum_{n\in
A}T_{\alpha}(e_{n}^{*}(x)-a_{n})e_{n}+\alpha 1_{\varepsilon\Lambda\backslash
A}\right\|$ $\displaystyle\ \geqslant\
\frac{1}{\mathbf{K}_{su}(\mathbf{K}_{su}+1)}\left\|\alpha
1_{\varepsilon\Lambda\backslash A}\right\|,$ (5.3)

where the last inequality is due to $\Lambda\backslash A$ being a greedy set
of

$x-P_{\Lambda\cup A}(x)+\sum_{n\in
A}T_{\alpha}(e_{n}^{*}(x)-a_{n})e_{n}+\alpha 1_{\varepsilon\Lambda\backslash
A}.$

From (5.2) and (5), we obtain

$\|x-P_{A}(x)+\alpha 1_{\varepsilon\Lambda\backslash A}\|\ \leqslant\
(\mathbf{K}^{2}_{su}+2\mathbf{K}_{su})\|y\|,$

which, combined with (5), gives

$\|x-P_{\Lambda}(x)\|\ \leqslant\
\Delta_{A,f}(\mathbf{K}^{2}_{su}+2\mathbf{K}_{su})\left\|x-\sum_{n\in
A}a_{n}e_{n}\right\|,$

as desired. ∎

We are now ready to prove the equivalence between $f$-almost greedy and
$f$-greedy bases for each non-identity $f\in\mathcal{F}$.

###### Proof of Theorem 1.8.

We need only to show that an $f$-almost greedy basis is $f$-greedy. Assume
that $(e_{n})_{n=1}^{\infty}$ is $f$-almost greedy; that is,
$(e_{n})_{n=1}^{\infty}$ is $\mathbf{C}_{q}$-quasi-greedy and
$\Delta$-$f$-superdemocratic according to Theorem 1.9 and Proposition 3.7.

We know that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}=\mathbf{c}$ exists by
Proposition 2.1. Since $f$ is non-identity, the proof of Proposition 2.1 shows
that $\mathbf{c}<1$. Suppose first that $0<\mathbf{c}<1$. Since $f(x)/x$ is
decreasing, we know that $\mathbf{c}x\leqslant f(x)$ for all $x>0$. By Theorem
1.5, $(e_{n})_{n=1}^{\infty}$ is almost greedy. We use Proposition 2.7 to
conclude that the basis is $f$-greedy.

For the rest of the proof, we assume that
$\lim_{x\rightarrow\infty}\frac{f(x)}{x}=0$. By Theorem 1.10 and Proposition
4.9, it suffices to verify that $(e_{n})_{n=1}^{\infty}$ has Property ($f$).
Choose $x,A,B,\varepsilon$ as in Definition 4.8. We want to show that there
exists a constant $\mathbf{C}$ (independent of $x,A,B,\varepsilon$) such that

$\|P_{A}(x)\|\ \leqslant\ \mathbf{C}\|x+1_{\varepsilon B}\|.$

Since $f$ is concave, we have

$|A|\ \leqslant\ f(|A|+|B|)\ \leqslant\ f(|A|)+f(|B|).$

Choose $N\geqslant 6$ be such that $f(x)\leqslant x/3$ for all $x\geqslant N$.
If $|A|\leqslant N$, then

$\|P_{A}(x)\|\ \leqslant\ \|P_{A}\|\|x\|\ \leqslant\ \mathbf{c}_{2}^{2}\|x\|\
\leqslant\ \mathbf{c}_{2}^{2}(\mathbf{C}_{q}+1)\|x+1_{\varepsilon B}\|.$

If $|A|>N$, then

$f(|B|)\ \geqslant\ |A|-f(|A|)\ \geqslant\ \frac{2|A|}{3}.$

Partition $A$ into two sets $A_{1},A_{2}$ such that

$|A_{i}|\ \leqslant\ \frac{|A|}{2}+1\ \leqslant\ \frac{2|A|}{3}\ \leqslant\
f(|B|).$

By $\Delta$-$f$-superdemocracy, we get

$\sup_{\delta}\|1_{\delta A}\|\ \leqslant\
\sup_{\delta^{\prime}}\|1_{\delta^{\prime}A_{1}}\|+\sup_{\delta^{\prime\prime}}\|1_{\delta^{\prime\prime}A_{2}}\|\
\leqslant\ 2\Delta\|1_{\varepsilon B}\|.$

Therefore, by convexity and the quasi-greedy property,

$\|P_{A}(x)\|\ \leqslant\ \sup_{\delta}\|1_{\delta A}\|\ \leqslant\
2\Delta\|1_{\varepsilon B}\|\ \leqslant\
2\Delta\mathbf{C}_{q}\|x+1_{\varepsilon B}\|.$

We conclude that $(e_{n})_{n=1}^{\infty}$ has Property ($f$), which completes
our proof. ∎

###### Proof of Corollary 1.11.

The equivalence between i) and ii) is due to Theorem 1.8. The equivalence
between ii) and iii) is due to Theorem 1.9. According to Proposition 3.4, iii)
$\Longleftrightarrow$ iv). Next, Proposition 3.7 gives the equivalence among
iii), iv), and v). Finally, the second statement follows from the fact that
$f$-unconditionality implies quasi-greediess and Theorem 1.10. ∎

We shall end this section with an easy corollary that shall be used to prove
Theorem 6.3.

######  Corollary 5.1.

Let $g\in\mathcal{F}$ and $g$ is bounded. Then a basis is $g$-almost greedy if
and only if it is quasi-greedy.

###### Proof.

By Corollary 1.11, it suffices to show that if $g$ is bounded, then every
basis $(e_{n})_{n=1}^{\infty}$ is $g$-democratic. Indeed, let
$A,B\in\mathbb{N}^{<\infty}$ with $|A|\leqslant g(|B|)$. If $|B|=0$, then
$|A|=0$, and there is nothing to prove. Assume that $|B|\geqslant 1$ and let
$\sup_{m}g(m)=s$. We have

$\|1_{A}\|\ \leqslant\ |A|\mathbf{c}_{2}\ \leqslant\ s\mathbf{c}_{2}\
\leqslant\ s\mathbf{c}_{2}\sup_{n}\|e_{n}^{*}\|\|1_{B}\|\ \leqslant\
s\mathbf{c}^{2}_{2}\|1_{B}\|.$

Hence, $(e_{n})_{n=1}^{\infty}$ is $g$-democratic, as desired. ∎

##  6\. When $f$ varies - A modification of the Schreier space

The main goal of this section is to study relations among $f$-(almost) greedy
bases when $f$ varies.

######  Proposition 6.1.

Let $f,g\in\mathcal{F}$ be such that there exists a constant $\mathbf{c}>0$
satisfying $f(m)\geqslant\mathbf{c}g(m)$ for all $m\in\mathbb{N}$. If
$(e_{n})_{n=1}^{\infty}$ is $f$-almost greedy, then it is $g$-almost greedy.

###### Proof.

Let $f,g\in\mathcal{F}$ such that there exists $\mathbf{c}>0$ satisfying
$f(m)\geqslant\mathbf{c}g(m)$ for all $m\in\mathbb{N}$. Due to Corollary 1.11,
it suffices to prove that a $\Delta$-$f$-democratic basis is $g$-democratic.
Let $A,B\in\mathbb{N}^{<\infty}$ with $|A|\leqslant g(|B|)$. We have

$|A|\ \leqslant\ g(|B|)\ \leqslant\ \frac{1}{\mathbf{c}}f(|B|).$

If $\frac{1}{\mathbf{c}}\leqslant 1$, then by $f$-democracy, we obtain
$\|1_{A}\|\leqslant\Delta\|1_{B}\|$. Suppose that $\frac{1}{\mathbf{c}}>1$. We
consider two cases.

Case 1: $f(|B|)\leqslant 2$. Then $|A|\leqslant\frac{2}{\mathbf{c}}$. We have

$\|1_{A}\|\ \leqslant\ |A|\mathbf{c}_{2}\ \leqslant\
\frac{2}{\mathbf{c}}\mathbf{c}_{2}\ \leqslant\
\frac{2}{\mathbf{c}}\mathbf{c}_{2}\sup_{n}\|e_{n}^{*}\|\|1_{B}\|\ \leqslant\
\frac{2\mathbf{c}_{2}^{2}}{\mathbf{c}}\|1_{B}\|.$

Case 2: $f(|B|)>2$. Partition $A$ into $s:=\lceil\frac{2}{\mathbf{c}}\rceil$
disjoint subsets $A_{1},\ldots,A_{s}$ such that for all $1\leqslant i\leqslant
s$,

$|A_{i}|\ \leqslant\ \frac{|A|}{s}+1\ \leqslant\ \frac{\mathbf{c}|A|}{2}+1\
\leqslant\ \frac{f(|B|)}{2}+1\ <\ f(|B|).$

By $f$-democracy, we get

$\|1_{A_{i}}\|\ \leqslant\ \Delta\|1_{B}\|,\forall 1\leqslant i\leqslant s.$

Therefore,

$\|1_{A}\|\ \leqslant\ s\Delta\|1_{B}\|.$

In general, we have shown that

$\|1_{A}\|\ \leqslant\
\max\left\\{\frac{2\mathbf{c}_{2}^{2}}{\mathbf{c}},\left\lceil\frac{2}{\mathbf{c}}\right\rceil\Delta\right\\}\|1_{B}\|,$

and hence, $(e_{n})_{n=1}^{\infty}$ is $g$-democratic. ∎

######  Corollary 6.2.

Let $f,g\in\mathcal{F}$ be such that
$0<\inf_{m\in\mathbb{N}}\frac{f(m)}{g(m)}\leqslant\sup_{m\in\mathbb{N}}\frac{f(m)}{g(m)}<\infty$,
then a basis is $f$-almost greedy if and only if it is $g$-almost greedy.

###### Proof.

Use Proposition 6.1. ∎

We now turn our attention to a notable example (in [6]) of a PG basis that is
not almost greedy. We shall modify the basis there to prove the following
theorem

######  Theorem 6.3.

i) Let $f,g\in\mathcal{F}$ such that
$\sup_{m\in\mathbb{N}}\frac{f(m)}{g(m)}=\infty$. Then there exists a
$1$-unconditional basis that is $g$-greedy but is not $f$-almost greedy.

ii) There exists an $1$-unconditional basis that is not $f$-almost greedy for
all unbounded $f\in\mathcal{F}$.

Let us generalize the construction in [6] to any function $f\in\mathcal{F}$.
Define

$\mathcal{A}=\left\\{A\in\mathbb{N}^{<\infty}:A\neq\emptyset\mbox{ and
}\frac{1}{f(1)}f(\min A)\geqslant|A|\right\\}$

(thus all singletons are in $\mathcal{A}$) and $X_{f}$ be the completion of
$c_{00}$ under the following norm: for $x=(x_{1},x_{2},\ldots)$,

$\|x\|\ =\ \sup_{A\in\mathcal{A}}\sum_{n\in A}|x_{n}|.$

Let $(e_{n})_{n=1}^{\infty}$ be the normalized canonical basis, which is
clearly $1$-unconditional.

######  Lemma 6.4.

For all $N\in\mathbb{N}$, it holds that

$\frac{1}{2}f(N)-1\ \leqslant\ \left\|\sum_{n=1}^{N}e_{n}\right\|\ \leqslant\
\frac{1}{f(1)}f(N).$

###### Proof.

Let $A\subset\\{1,2,\ldots,N\\}$ and $A\in\mathcal{A}$. Then $\min A\leqslant
N$. Because $f$ is increasing,

$\sum_{m\in A}\left|e_{m}^{*}\left(\sum_{n=1}^{N}e_{n}\right)\right|\ =\ |A|\
\leqslant\ \frac{1}{f(1)}f(\min A)\ \leqslant\ \frac{1}{f(1)}f(N).$

We conclude that $\left\|\sum_{n=1}^{N}e_{n}\right\|\ \leqslant\
\frac{1}{f(1)}f(N)$.

Next, we establish a lower bound. Choose a nonnegative integer $k\leqslant
N-1$ to be the smallest such that $A=\\{k+1,\ldots,k+(N-k)\\}\in\mathcal{A}$,
i.e., $f(k+1)\geqslant f(1)(N-k)$. If $k=0$, then $|A|=N\geqslant f(N)$.
Suppose that $k\geqslant 1$. Then $f(k)<f(1)(N-k+1)$. We consider two cases.

Case 1: if $k>N/2$, then

$|A|\ =\ N-k\ >\ \frac{f(k)}{f(1)}-1\ \geqslant\ \frac{f(N/2)}{f(1)}-1\
\geqslant\ \frac{1}{2f(1)}f(N)-1.$

Case 2: if $k\leqslant N/2$, then $N\geqslant 2$ because $k\geqslant 1$. By
Proposition 2.1 item ii),

$|A|\ =\ N-k\ \geqslant\ \frac{N}{2}\ \geqslant\ f\left(\frac{N}{2}\right)\
\geqslant\ \frac{1}{2}f(N).$

Therefore, $\left\|\sum_{n=1}^{N}e_{n}\right\|\ \geqslant\ \frac{1}{2}f(N)-1$.
∎

###### Proof of Theorem 6.3.

We prove item i). Let $f,g\in\mathcal{F}$ such that
$\sup_{m\in\mathbb{N}}\frac{f(m)}{g(m)}=\infty$. Clearly, $f$ is unbounded. We
first assume that $g$ is also unbounded. We claim that the canonical basis of
$X_{g}$ is $g$-greedy but is not $f$-almost greedy.

$(e_{n})_{n=1}^{\infty}$ is $g$-greedy: let $A,B\in\mathbb{N}^{<\infty}$ with
$|A|\leqslant g(|B|)$. By Lemma 6.4, we get

$\|1_{B}\|\ \geqslant\ \left\|\sum_{n=1}^{|B|}e_{n}\right\|\ \geqslant\
\frac{1}{2}g(|B|)-1\ \geqslant\ \frac{1}{2}|A|-1\ \geqslant\
\frac{1}{2}\|1_{A}\|-1.$

Hence,

$\frac{1}{2}\|1_{A}\|\ \leqslant\ \|1_{B}\|+1\ \leqslant\
\|1_{B}\|+\sup_{n}\|e_{n}^{*}\|\|1_{B}\|\ =\ (\mathbf{c}_{2}+1)\|1_{B}\|.$

Therefore, $(e_{n})_{n=1}^{\infty}$ is $g$-democratic. By unconditionality and
Corollary 1.11, $(e_{n})_{n=1}^{\infty}$ is $g$-greedy.

$(e_{n})_{n=1}^{\infty}$ is not $f$-almost greedy: let $m\in\mathbb{N}$,
$B=\\{1,2,\ldots,m\\}$ and $A\subset\mathbb{N}^{<\infty}$ such that
$|A|=\lfloor f(m)\rfloor$ and $\frac{1}{g(1)}g(\min A)\geqslant|A|$ (which is
possible since $\lim_{x\rightarrow\infty}g(x)=\infty$). By Lemma 6.4,

$\|1_{B}\|\ \leqslant\ \frac{1}{f(1)}g(m)\mbox{ and }\|1_{A}\|\ =\ \lfloor
f(m)\rfloor,$

which implies that

$\frac{\|1_{A}\|}{\|1_{B}\|}\ \geqslant\ f(1)\frac{\lfloor
f(m)\rfloor}{g(m)}.$

Since $\sup_{m\in\mathbb{N}}\frac{f(m)}{g(m)}=\infty$, we do not have
$f$-democracy. By Theorem 1.9 and Proposition 3.7, $(e_{n})_{n=1}^{\infty}$ is
not $f$-almost greedy.

In the case when $g$ is bounded, define $h=\sqrt{f}\in\mathcal{F}$ because
$\mathcal{F}$ is closed under compositions. Note that $h$ is unbounded. Using
the same argument as above, the canonical basis of $X_{h}$ is $h$-greedy but
is not $f$-almost greedy. By the proof of Corollary 5.1, the canonical basis
of $X_{h}$ is $g$-democratic, and according to Corollary 1.11, the basis is
$g$-greedy.

Next, we prove item ii). Consider the space $(\ell_{1}\oplus
c_{0})_{\ell_{1}}$ with the basis $\mathcal{X}=(x_{n})_{n=1}^{\infty}$ defined
as

$x_{n}\ =\ \begin{cases}e_{k}&\mbox{ if }n=2k-1,\\\ f_{k}&\mbox{ if
}n=2k.\end{cases}$

where $(e_{n})_{n=1}^{\infty}$ and $(f_{n})_{n=1}^{\infty}$ are the canonical
bases of $\ell_{1}$ and $c_{0}$, respectively. Clearly, $\mathcal{X}$ is
$1$-unconditional. Let $f\in\mathcal{F}$ be unbounded. Then $\mathcal{X}$ is
not $f$-disjoint democratic. Indeed, pick $m\in\mathbb{N}$,
$A=\\{1,3,\ldots,2\lfloor f(m)\rfloor-1\\}$, and $B=\\{2,4,\ldots,2m\\}$. It
follows that $\frac{\|1_{A}\|}{\|1_{B}\|}=\lfloor
f(m)\rfloor\rightarrow\infty$ as $m\rightarrow\infty$. Hence, $\mathcal{X}$ is
not $f$-almost greedy. ∎

######  Remark 6.5.

We consider the PG basis in [6, Proposition 6.10], which is the canonical
basis of $X_{f}$, when $f$ is the square-root function. According the proof of
Theorem 6.3, we know that the basis is $f$-greedy though it is not almost
greedy. Consequently, though some PG bases are not almost greedy, they can be
$f$-greedy for a well-chosen $f$. Since the notion of being $f$-greedy is
order-free, this provides a partial solution to the concern of some
researchers regarding the order-dependent nature of PG bases.

######  Corollary 6.6.

Let $f,g\in\mathcal{F}$.

  1. i)

If $\sup_{m\in\mathbb{N}}\frac{f(m)}{g(m)}<\infty$, then the canonical basis
of $X_{g}$ is $f$-greedy.

  2. ii)

If $\sup_{m\in\mathbb{N}}\frac{f(m)}{g(m)}=\infty$ and $g$ is unbounded, then
the canonical basis of $X_{g}$ is not $f$-almost greedy.

  3. iii)

If $g$ is bounded, then the canonical basis of $X_{g}$ is equivalent to the
canonical basis of $c_{0}$. In this case, the canonical basis of $X_{g}$ is
$f$-greedy for all $f\in\mathcal{F}$.

###### Proof.

i) Suppose that $\sup_{m\in\mathbb{N}}\frac{f(m)}{g(m)}=:\mathbf{c}<\infty$.
Then $f(m)\leqslant\mathbf{c}g(m),\forall m\in\mathbb{N}$. Since the canonical
basis $(e_{n})_{n=1}^{\infty}$ of $X_{g}$ is $g$-greedy according to the proof
of Theorem 6.3, Proposition 6.1 states that $(e_{n})_{n=1}^{\infty}$ is
$f$-democratic. By unconditionality, Theorem 1.10, and Proposition 3.7,
$(e_{n})_{n=1}^{\infty}$ is $f$-greedy.

ii) Use the same reasoning as in the proof of Theorem 6.3.

iii) Suppose that $\sup_{x\geqslant 0}g(x)=\mathbf{c}$. Let
$(e_{n})_{n=1}^{\infty}$ be the canonical basis of $X_{g}$. For all
$A\in\mathbb{N}^{<\infty}$ and sign $\varepsilon$, we have $\|1_{\varepsilon
A}\|\leqslant\frac{\mathbf{c}}{g(1)}$. Hence, $(e_{n})_{n=1}^{\infty}$ is
equivalent to the canonical basis of $c_{0}$. ∎

##  7\. Examples

###  7.1. The (normalized) multivariate Haar basis

Consider the multivariate Haar basis $\mathcal{H}^{d}_{p}$ of
$L^{p}([0,1]^{d})$ for $1<p<\infty$. A famous result by Temlyakov [16] states
that the univariate $\mathcal{H}^{1}_{p}$ is a greedy basis of $L^{p}[0,1]$.
However, the same conclusion does not hold for the general multivariate basis
$\mathcal{H}_{p}^{d}=\mathcal{H}_{p}\times\cdots\mathcal{H}_{p}$. While
$\mathcal{H}_{p}^{d}=\\{h_{n}\\}$ is an unconditional basis for
$L^{p}([0,1]^{d})$, the system is not democratic and thus, not greedy [17,
(1.3)]. However, one has the following result from [14] (see also [10,
Prosposition 7.6]).

######  Proposition 7.1.

For any $A\in\mathbb{N}^{<\infty}$ with $|A|=m\geqslant 2$, there exist
$\mathbf{C}_{1},\mathbf{C}_{2}>0$ such that if $p\in[2,\infty)$,

$\mathbf{C}_{1}m^{1/p}\ \leqslant\ \left\|\sum_{n\in A}h_{n}\right\|\
\leqslant\ \mathbf{C}_{2}(\ln m)^{h(p,d)}m^{1/p},$

while if $p\in(1,2)$,

$\mathbf{C}_{1}m^{1/p}(\ln m)^{-h(p,d)}\ \leqslant\ \left\|\sum_{n\in
A}h_{n}\right\|\ \leqslant\ \mathbf{C}_{2}m^{1/p},$

where $h(p,d):=(d-1)\left|\frac{1}{2}-\frac{1}{p}\right|$.

######  Theorem 7.2.

For any $\varepsilon>0$, let $f(x)=x^{1-\varepsilon}$. The $d$-multivariate
system $\mathcal{H}_{p}^{d}=\\{h_{n}\\}$ is $f$-greedy.

###### Proof.

By Corollary 1.11, it suffices to show that $\mathcal{H}_{p}^{d}$ is
$f$-democratic. Let $A,B\in\mathbb{N}^{<\infty}$ such that
$|A|\leqslant|B|^{1-\varepsilon}$. Also let $\mathbf{C}_{1},\mathbf{C}_{2}$ be
as in Proposition 7.1. Choose $N\geqslant 2$ to be sufficiently large such
that

$x^{\varepsilon/p}\ \geqslant\ \frac{\mathbf{C}_{2}}{\mathbf{C}_{1}}(\ln
x)^{h(p,d)},\forall x\geqslant N.$

Without loss of generality, assume $|B|^{1-\varepsilon}\geqslant|A|\geqslant
N$. (If $|A|\leqslant N$, we use semi-normalization of the basis). We consider
only the case $p\in[2,\infty)$ as the case $p\in(1,2)$ is similar. We have

$\displaystyle\left\|1_{A}\right\|\ \leqslant\
\mathbf{C}_{2}(\ln|A|)^{h(p,d)}|A|^{1/p}$ $\displaystyle\ \leqslant\
\mathbf{C}_{2}(\ln|B|)^{h(p,d)}|B|^{(1-\varepsilon)/p}$ $\displaystyle\
\leqslant\ (\mathbf{C}_{1}|B|^{\varepsilon/p})|B|^{(1-\varepsilon)/p}\ =\
\mathbf{C}_{1}|B|^{1/p}\ \leqslant\ \|1_{B}\|.$

Therefore, $\mathcal{H}_{p}^{d}$ is $f$-democratic, as desired. ∎

###  7.2. A conditional basis that is $x/\lambda$-unconditional for any
$\lambda>1$

We borrow the conditional, QG basis $\mathcal{B}$ by Konyagin and Temlyakov
[15], who considered the canonical basis $\mathcal{B}$ of the completion of
$c_{00}$ with respect to the norm

$\|(a_{n})_{n=1}^{\infty}\|\ :=\
\max\left\\{\left(\sum_{n}|a_{n}|^{2}\right)^{1/2},\sup_{N}\left|\sum_{n=1}^{N}\frac{a_{n}}{\sqrt{n}}\right|\right\\}.$

Fix $\lambda>1$ and let us show that the basis is $x/\lambda$-unconditional.
By Proposition 4.9, it suffices to show that $\mathcal{B}$ has Property
($x/\lambda$). Choose $x,A,B,\varepsilon$ as in Definition 4.8. Then
$|A|\leqslant|B|/(\lambda-1)$. Clearly, $\|x+1_{\varepsilon
B}\|\geqslant|B|^{1/2}$. We have

$\displaystyle\|P_{A}(x)\|$ $\displaystyle\ =\ \max\left\\{\left(\sum_{n\in
A}|e_{n}^{*}(x)|^{2}\right)^{1/2},\sup_{N}\left|\sum_{\begin{subarray}{c}1\leqslant
n\leqslant N\\\ n\in
A\end{subarray}}\frac{e_{n}^{*}(x)}{\sqrt{n}}\right|\right\\}$ $\displaystyle\
\leqslant\ \max\left\\{|A|^{1/2},\sum_{n=1}^{|A|}\frac{1}{\sqrt{n}}\right\\}\
\leqslant\ 2|A|^{1/2}$ $\displaystyle\ \leqslant\
2(\lambda-1)^{-1/2}|B|^{1/2}\ \leqslant\ 2(\lambda-1)^{-1/2}\|x+1_{\varepsilon
B}\|.$

Hence, $\mathcal{B}$ is $x/\lambda$-unconditional, as claimed.

###  7.3. A non-QG basis that has Property ($x/\lambda$) for any $\lambda>1$

We use the example in [8, Section 5.3]. Let us first construct the weight
sequence $\omega=(w_{n})_{n=1}^{\infty}$. For $n\geqslant 1$, let
$t_{n}=\frac{1}{\sqrt{n}}$, $L_{n}=e^{(\ln n)^{2}}$, and
$a_{n}=\frac{1}{\sqrt{n}\ln(n+1)}$. Define an increasing sequence
$(N_{n})_{n=0}^{\infty}$ recursively. Set $N_{0}=0$. Choose $N_{1}>10$ to be
the smallest such that

$b_{1}\ :=\ a_{1}t_{1}\left(\sum_{n=1}^{N_{1}}t_{n}\right)^{-1}\ <\
\frac{a_{1}}{L_{1}}.$

Once $N_{j}$ and $b_{j}$ are defined for $j\geqslant 1$, choose $N_{j+1}$ to
be the smallest such that

$b_{j+1}\ :=\
a_{j+1}t_{j+1}\left(\sum_{n=N_{j}+1}^{N_{j}+N_{j+1}}t_{n}\right)^{-1}\ <\
\min\left\\{\frac{a_{j+1}}{L_{j+1}},b_{j}\right\\}\mbox{ and
}\frac{N_{j+1}}{N_{j}}\ >\ 10.$ (7.1)

For $j\geqslant 1$, denote the finite sequence
$(t_{n})_{n=N_{j-1}+1}^{N_{j-1}+N_{j}}=(t_{N_{j-1}+1},\ldots,t_{N_{j-1}+N_{j}})$
by $B_{j}$.

We now define a weight $\omega:=(w_{n})_{n=1}^{\infty}$ on $\mathbb{N}$ as
follows:

$(w_{n})\ =\ (t_{1},B_{1},t_{2},B_{2},t_{3},B_{4},t_{4},\ldots).$

In words, the weight $(w_{n})$ is chosen such that the first one is $t_{1}$;
the next $N_{1}$ weights are taken from $B_{1}$ in the same order; the next
weight is $t_{2}$, then the next $N_{2}$ weights are taken from $B_{2}$ in the
same order, and so on.

Let $X$ be the completion of $c_{00}$ under the following norm

$\left\|\sum_{n}x_{n}e_{n}\right\|\ =\
\max\left\\{\sup_{N}\left|\sum_{n=N}^{\infty}w_{n}x_{n}\right|,\left(\sum_{n=1}^{\infty}|x_{n}|^{2}\right)^{1/2}\right\\}.$

For ease of notation, set

$\left\|\sum_{n}x_{n}e_{n}\right\|_{1}\>=\
\sup_{N}\left|\sum_{n=N}^{\infty}w_{n}x_{n}\right|\mbox{ and
}\left\|\sum_{n}x_{n}e_{n}\right\|_{2}\ =\
\left(\sum_{n=1}^{\infty}|x_{n}|^{2}\right)^{1/2}.$

Clearly, the standard unit vector basis $\mathcal{B}$ is Schauder and
normalized in $(X,\|\cdot\|)$. As shown in [8], the basis is not quasi-greedy.

We shall show that it has Property ($x/\lambda$) for any $\lambda>1$. Fix
$\lambda>1$ and let $x,A,B,\varepsilon$ be chosen as in Definition 4.8. Then
$|A|\leqslant(\lambda-1)^{-1}|B|$. We know that $\|x+1_{\varepsilon
B}\|\geqslant|B|^{1/2}$. On the other hand,

$\|P_{A}(x)\|\ \leqslant\ \max\left\\{\sum_{\begin{subarray}{c}n\geqslant 1\\\
n\in A\end{subarray}}w_{n},|A|^{1/2}\right\\}\ \leqslant\
\max\left\\{2\sum_{n=1}^{|A|}\frac{1}{\sqrt{n}},|A|^{1/2}\right\\}\ \leqslant\
4|A|^{1/2}.$

Hence,

$\|P_{A}(x)\|\ \leqslant\ 4|A|^{1/2}\ \leqslant\ 4(\lambda-1)^{-1/2}|B|^{1/2}\
\leqslant\ 4(\lambda-1)^{-1/2}\|x+1_{\varepsilon B}\|.$

We conclude that $\mathcal{B}$ has Property ($x/\lambda$).

###  7.4. A quasi-greedy basis that does not have Property $(f)$ for any
unbounded $f$

Let $(x_{n})_{n=1}^{\infty}$ be a Schauder, conditional, QG basis of a Banach
space $X$ such that $\|x\|_{X}\leqslant 1$ implies that
$|x_{n}^{*}(x)|\leqslant 1$ for all $n$. We know such a basis exists due to
the Konyagin and Temlyakov’s example mentioned in Example 7.2. Equip
$Z=X\oplus c_{0}$ with the norm

$\|(x,v)\|_{Z}=\max\\{\|x\|_{X},\|v\|_{c_{0}}\\},\forall x\in X,\forall v\in
c_{0}.$

Let $\mathcal{Z}=(z_{n})_{n=1}^{\infty}$ be the basis of $Z$ defined as

$z_{n}\ =\ \begin{cases}(x_{k},0)&\mbox{ if }n=2k-1,\\\ (0,e_{k})&\mbox{ if
}n=2k,\end{cases}$

where $(e_{n})_{n=1}^{\infty}$ is the canonical basis of $c_{0}$. Clearly,
$\mathcal{Z}$ is QG because the $c_{0}$-direct sum of two QG bases is again
QG. However, the basis $\mathcal{Z}$ is not $f$-unconditional for any
unbounded $f\in\mathcal{G}$. Indeed, pick $L>1$. Since
$(x_{n})_{n=1}^{\infty}$ is conditional, there exists a finitely supported
vector $x=\sum_{n=1}^{N}a_{n}x_{n}$ with $\|x\|_{X}=1$ (and thus,
$|a_{n}|\leqslant 1$) and a finite set $A\subset\operatorname{supp}(x)$ such
that $\|P_{A}(x)\|_{X}\ >\ L$. Choose $m$ such that $f(m)>|A|$. Consider the
following vector in $Z$

$z\ :=\ \sum_{j=1}^{N}a_{j}z_{2j-1}+\sum_{n=1}^{m}z_{2n}.$

Let $A^{\prime}=\\{2j-1:j\in A\\}$ and $B=\\{2,4,\ldots,2m\\}$. Observe that
$B$ is a greedy set of $z$ and

$|A^{\prime}\cup B|-f(|A^{\prime}\cup B|)\ \leqslant\ |A|+|B|-f(m)\ <\ |B|.$

However,

$\|P_{A^{\prime}\cup B}(z)\|_{Z}\ =\ \|P_{A}(x)\|_{X}\ >\ L\ =\ L\|z\|_{Z}.$

Since $L>1$ is arbitrary, our basis $\mathcal{Z}$ is not $f$-unconditional.

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