3.1 Towards new parameters in greedy approximation

Recall that a basis $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ of $\mathbb {X}$ is symmetric if it is equivalent to all its permutations, that is, there is a constant $C\ge 1$ such that

(3.1) $$ \begin{align} \frac{1}{C} \left\lVert \sum_{n=1}^{\infty} a_n \, \boldsymbol{x}_n\right\rVert \le \left\lVert \sum_{n=1}^{\infty} a_n \, \boldsymbol{x}_{\pi(n)}\right\rVert \le C \left\lVert \sum_{n=1}^{\infty} a_n \, \boldsymbol{x}_n\right\rVert \end{align} $$

for all $(a_n)_{n=1}^{\infty }\in c_{00}$ and all permutations $\pi $ on $\mathbb {N}$ . Democracy is the weakest symmetry condition that a basis can have, where we demand to a basis to verify equation (3.1) when all coefficients $a_{n}$ of $f\in \mathbb {X}$ are equal (without loss of generality) to 1.

Symmetric bases are in particular unconditional, which permits improving the previous inequality to have

(3.2) $$ \begin{align} \frac{1}{C} \left\lVert \sum_{n=1}^{\infty} a_n \, \boldsymbol{x}_n\right\rVert \le \left\lVert \sum_{n=1}^{\infty} \varepsilon_n\, a_{n} \, \boldsymbol{x}_{\pi(n)}\right\rVert \le C \left\lVert \sum_{n=1}^{\infty} a_n \, \boldsymbol{x}_n\right\rVert \end{align} $$

for all $(\varepsilon _n)_{n=1}^{\infty }\in \mathbb {E}^{\mathbb {N}}$ , where $\mathbb {E}$ denotes the subset of $\mathbb {F}$ consisting of all scalars of modulus $1$ , for a possibly larger constant C. If a basis $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ of $\mathbb {X}$ satisfies equation (3.2) when all coefficients $a_{n}$ are 1 it is called superdemocratic. To quantify the superdemocracy of $\mathcal {X}$ , we use the mth superdemocracy parameter,

where

so that $\mathcal {X}$ is superdemocratic if $\sup _m {\boldsymbol{\mu}^{\boldsymbol{s}}_m}<\infty $ .

Although the parameters $({\boldsymbol{\mu}^{\boldsymbol{s}}_m})_{m=1}^{\infty }$ are one step up in the scale of symmetry, in practice they do not provide asymptotically better estimates than $(\boldsymbol {\mu }_m)_{m=1}^{\infty }$ for $(\boldsymbol {L}_m)_{m=1}^{\infty }$ . Indeed, with a smaller constant $C_1$ and a larger constant $C_2$ , we have

$$\begin{align*}\frac{1}{C_2} \max\left\lbrace \boldsymbol{k}_m, \boldsymbol{\mu}^{\boldsymbol{s}}_m\right\rbrace \le \boldsymbol{L}_m \le C_1 \boldsymbol{k}_m \, \boldsymbol{\mu}^{\boldsymbol{s}}_m, \quad m\in\mathbb{N} \end{align*}$$

(cf. [Reference Berná, Blasco and Garrigós13, Proposition 1.1]).

Another condition related to symmetry which has been successfully implemented in the theory is the so-called symmetry for largest coefficients. Let

$$\begin{align*}\operatorname{\mathrm{supp}}(f)=\lbrace n\in\mathbb{N} \colon \boldsymbol{x}_n^*(f) \not=0\rbrace \end{align*}$$

denote the support of $f\in \mathbb {X}$ with respect to the basis $\mathcal {X}$ . We define

the supremum being taken over all finite subsets A, B of $\mathbb {N}$ with $ \left \lvert A\right \rvert =\left \lvert B\right \rvert \le m$ , all signs $\varepsilon \in \mathbb {E}^{A}$ and $\delta \in \mathbb {E}^{B}$ and all $f\in \mathbb {X}$ with $\left \lVert f\right \rVert _{\infty }\le 1$ and $\operatorname {\mathrm {supp}}(f)\cap (A\cup B)=\emptyset $ . A basis $\mathcal {X}$ is symmetric for largest coefficients (SLC for short) if $\sup _m \boldsymbol {\nu }_m<\infty $ . Imposing the extra assumption $A\cap B=\emptyset $ in the definition, we obtain the ‘disjoint’ counterpart of the SLC parameters, herein denoted by ${\boldsymbol{\nu}^{\boldsymbol{d}}_m}={\boldsymbol{\nu}^{\boldsymbol{d}}_m}[\mathcal {X},\mathbb {X}]$ .

Bases with $\sup _m \boldsymbol {\nu }_m=1$ were first considered in [Reference Albiac and Wojtaszczyk10], where they were called bases with Property (A). The parameters that quantify the symmetry for largest coefficients of a basis appear naturally when looking for estimates for the Lebesgue constants that are close to one (see [Reference Albiac and Wojtaszczyk10, Reference Dilworth, Odell, Schlumprecht and Zsák22, Reference Dilworth, Kutzarova, Odell, Schlumprecht and Zsák20, Reference Albiac, Ansorena and Wallis7]). In fact, if we put

$$\begin{align*}A_p=(2^p-1)^{1/p}, \quad 0<p\le 1, \end{align*}$$

and $\mathbb {X}$ is a p-Banach space,

(3.3) $$ \begin{align} \max\left\lbrace {\boldsymbol{k}^{\boldsymbol{c}}_m}, {\boldsymbol{\nu}^{\boldsymbol{d}}_m}\right\rbrace \le \boldsymbol{L}_m \le A_p^2 \boldsymbol{k}^{\boldsymbol{c}}_{2m} {\boldsymbol{\nu}^{\boldsymbol{d}}_m}, \quad m\in\mathbb{N}, \end{align} $$

(see [Reference Berná, Blasco and Garrigós13, Proposition 1.1] and [Reference Albiac, Ansorena, Berná and Wojtaszczyk4, Theorem 7.2]).

Thus, since ${\boldsymbol{\nu}^{\boldsymbol{d}}_m}\le \boldsymbol {\nu }_m \le ({\boldsymbol{\nu}^{\boldsymbol{d}}_m})^2$ , in the case when $\mathbb {X}$ is a Banach space, $\boldsymbol {C}_g=1$ if and only if ${\boldsymbol{k}^{\boldsymbol{c}}_m}=\boldsymbol {\nu }_m=1$ for all $m\in \mathbb {N}$ . Taking into account that for some constant C,

$$\begin{align*}\boldsymbol{\nu}_m\le C {\boldsymbol{\nu}^{\boldsymbol{d}}_m},\quad m\in \mathbb{N}, \end{align*}$$

(see Proposition 8.1 below), equation (3.3) yields

$$\begin{align*}\frac{1}{ C_1} \max\left\lbrace \boldsymbol{k}_m, \boldsymbol{\nu}_m \right\rbrace \le \boldsymbol{L}_m \le C_2 \boldsymbol{k}_m \, \boldsymbol{\nu}_m, \quad m\in\mathbb{N}, \end{align*}$$

for some constants $C_1$ and $C_2$ . Thus, again, the attempt to obtain better asymptotic estimates for $(\boldsymbol {L}_m)_{m=1}^{\infty }$ using parameters larger than $(\boldsymbol {\mu }_m)_{m=1}^{\infty }$ is futile.

The vestiges of symmetry found in some bases can also be measured qualitatively by means of the upper and lower democracy functions and using the concept of dominance between bases.

The upper superdemocracy function, also known as fundamental function, of a basis $\mathcal {X}$ of a quasi-Banach space $\mathbb {X}$ is defined as

(3.4)

while the lower superdemocracy function of $\mathcal {X}$ is

A basis $\mathcal {Y}=(\boldsymbol {y}_n)_{n=1}^{\infty }$ of a quasi-Banach space $\mathbb {Y}$ is said to dominate a basis $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ of a quasi-Banach space $\mathbb {X}$ if there is a bounded linear map $S\colon \mathbb {X}\to \mathbb {Y}$ such that $S(\boldsymbol {x}_n)=\boldsymbol {y}_n$ for all $n\in \mathbb {N}$ . If $\left \lVert S\right \rVert \le D$ , we will say that $\mathcal {Y} \ D$ -dominates $\mathcal {X}$ .

Roughly speaking, we are interested in bases that can be ‘squeezed’ (using domination) between two symmetric bases with equivalent fundamental functions.

Squeeze-symmetry guarantees in a certain sense the optimality of the compression algorithms with respect to the basis (see [Reference Donoho24]). For an approach to squeeze-symmetric bases from this angle, we refer the reader to [Reference Albiac and Ansorena1, Reference Wojtaszczyk55, Reference Cohen, DeVore, Petrushev and Xu17]. Squeeze-symmetry also serves in some situations as a tool to derive other properties of the bases like being quasi-greedy for instance (see [Reference Konyagin and Temlyakov34, Reference Dilworth, Kalton and Kutzarova18, Reference Berná, Blasco and Garrigós13, Reference Berná, Blasco, Garrigós, Hernández and Oikhberg14, Reference Albiac, Ansorena, Dilworth and Kutzarova6, Reference Wojtaszczyk56]).

Although it was not originally given this name, squeeze-symmetry was introduced in [Reference Albiac, Ansorena, Berná and Wojtaszczyk4] to highlight a feature that had been implicit in greedy approximation with respect to bases since the early stages of the theory. The techniques developed in [Reference Albiac, Ansorena, Berná and Wojtaszczyk4, Section 9] show that squeeze-symmetric bases are closely related to embeddings involving Lorentz sequence spaces. Before providing a precise formulation of this connection, we recall that if a basis $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ of $\mathbb {X}$ is $1$ -symmetric, that is, equation (3.2) holds with $C_1=C_2=1$ , then $\boldsymbol {\varphi _u}[\mathcal {X},\mathbb {X}]=\boldsymbol {\varphi _l}[\mathcal {X},\mathbb {X}]$ . Note also that every symmetric basis is $1$ -symmetric under a suitable renorming of the space; thus, there is no real restriction in assuming that all symmetric bases are $1$ -symmetric.

Recall that a weight $\boldsymbol {\sigma }=(s_m)_{m=1}^{\infty }$ is the primitive weight of a weight $\boldsymbol {w}=(w_n)_{n=1}^{\infty }$ , in which case we say that $\boldsymbol {w}$ is the discrete derivative of $\boldsymbol {\sigma }$ , if $s_m=\sum _{n=1}^m w_n$ for all $m\in \mathbb {N}$ .

Theorem 3.2 (see [Reference Albiac, Ansorena, Berná and Wojtaszczyk4, Equation (9.4), Lemma 9.3 and Theorem 9.12])

Let $\mathbb {X}$ be a quasi-Banach space with a basis $\mathcal {X}$ . Let $\boldsymbol {w}$ be the discrete derivative of the fundamental function of $\mathcal {X}$ . Then, $\mathcal {X}$ is squeeze-symmetric if and only if the coefficient transform defines a bounded linear operator from $\mathbb {X}$ into $d_{1,\infty }(\boldsymbol {w})$ . Quantitatively, if we set $\Gamma =\inf D_1 D_2$ , where the infimum is taken over all $1$ -symmetric bases $\mathcal {X}_1$ and $\mathcal {X}_2$ of quasi-Banach spaces $\mathbb {X}_1$ and $\mathbb {X}_2$ such that $\mathcal {X}_1 \ D_1$ -dominates $\mathcal {X}$ , $\mathcal {X} \ D_2$ -dominates $\mathcal {X}_2$ , and $\boldsymbol {\varphi _u}[\mathcal {X}_1,\mathbb {X}_1]\le \boldsymbol {\varphi _u}[\mathcal {X}_2,\mathbb {X}_2]$ , then there are constants $C_1$ and $C_2$ depending only on the modulus of concavity of $\mathbb {X}$ such that

$$\begin{align*}\frac{1}{C_2} \Gamma \le \left\lVert \mathcal{F}\right\rVert_{\mathbb{X}\to d_{1,\infty}(\boldsymbol{w})} \le C_2 \Gamma. \end{align*}$$

Theorem 3.2 serves as motivation to define a new kind of Lebesgue constants associated with an arbitrary basis $\mathcal {X}$ , which will eventually be key in solving Temlyakov’s problem.

For each $f\in \mathbb {X}$ , let $(\boldsymbol {a}_m(f))_{m=1}^{\infty }$ be the nonincreasing rearrangement of $\left \lvert \mathcal {F}(f)\right \rvert $ . Note that, if we put

(3.5) $$ \begin{align} \boldsymbol{\psi}_m= \boldsymbol{\psi}_m[\mathcal{X},\mathbb{X}]=\sup_{f\in\mathbb{X}\setminus\lbrace0\rbrace} \frac{\boldsymbol{a}_m(f)}{\left\lVert f\right\rVert}, \quad m\in\mathbb{N}, \end{align} $$

then the norm of the coefficient transform as an operator from $\mathbb {X}$ into $d_{1,\infty }(\boldsymbol {w})$ is $\sup _m \boldsymbol {\psi }_m \boldsymbol {\varphi _u}(m)$ . So, for $m\in \mathbb {N}$ , we define the mth squeeze-symmetry parameter of $\mathcal {X}$ ,

$$\begin{align*}\boldsymbol{\lambda}_m=\boldsymbol{\lambda}_m[\mathcal{X},\mathbb{X}]= \boldsymbol{\psi}_m[\mathcal{X},\mathbb{X}] \boldsymbol{\varphi_u}[\mathcal{X},\mathbb{X}](m). \end{align*}$$

We can also approach the definition of the squeeze-symmetry parameters from a functional angle. If for $A\subseteq \mathbb {N}$ finite, we denote by $\mathcal {F}_A$ the projection of the coefficient transform onto A,

$$\begin{align*}\mathcal{F}_A\colon\mathbb{X} \to \mathbb{F}^{\mathbb{N}}, \quad f\mapsto \mathcal{F}(f) \chi_A, \end{align*}$$

then

$$\begin{align*}\boldsymbol{\lambda}_m[\mathcal{X},\mathbb{X}] = \sup \lbrace \left\lVert \mathcal{F}_A\right\rVert_{\mathbb{X}\to d_{1,\infty}(\boldsymbol{w})} \colon \left\lvert A\right\rvert\le m\rbrace. \end{align*}$$

Using that $(\boldsymbol {a}_m(f))_{m=1}^{\infty }$ is nonincreasing for all $f\in \mathbb {X}$ yields the properties of $(\boldsymbol {\lambda }_m)_{m=1}^{\infty }$ gathered in the following lemma for further reference.

Although the parameters $\boldsymbol {\lambda }_m$ could appear to be nonintuitive at first glance, it should be understood that they capture a very natural feature of a basis, namely, the inverse of $\boldsymbol {\lambda }_m$ is the optimal constant which bounds below the norm of vectors whose coefficients are greater than $1/\boldsymbol {\varphi _u}(m)$ on a set of cardinality m. In other words, $\boldsymbol {\lambda }_m$ is the inverse of the distance from the origin to such set of vectors.

3.2 Optimal estimates in terms of the squeeze-symmetry parameters

We get started with a lemma which connects some important constants that we will need by using the p-convexitity techniques developed in [Reference Albiac, Ansorena, Berná and Wojtaszczyk4, Section 2]. The symbol ${\boldsymbol {\Upsilon }}$ stands for $2$ if $\mathbb {F}=\mathbb {R}$ , and for $4$ if $\mathbb {F}=\mathbb {C}$ .

Lemma 3.4. Let $0<p\le 1$ , and let $\mathcal {X}$ be a basis of a p-Banach space $\mathbb {X}$ . Given $A\subseteq \mathbb {N}$ finite, $f\in \mathbb {X}$ and $\delta =(\delta _n)_{n\in A}\in \mathbb {E}^A$ , we put

Set $C_p={\boldsymbol {\Upsilon }}^{1/p}$ if $f=0$ and $C_p=(1+{\boldsymbol {\Upsilon }})^{1/p}$ otherwise. Then:

$$ \begin{align*} K[A,f]&\le \min\lbrace C_p\, M[\delta,A,f], A_p\, L[A,f]\rbrace \text{ and }\\ M[\delta,A,f]&\le A_p\, N[\delta,A,f]. \end{align*} $$

Proof. In the case when $\mathbb {F}=\mathbb {C}$ , set $\gamma _j=i^j$ for $j=1$ , $2$ , $3$ , $4$ . In the case when $\mathbb {F}=\mathbb {R}$ , set $\gamma _j=(-1)^j$ for $j=1$ , $2$ . Given $(a_n)_{n\in A}\in \mathbb {F}^A$ with $\left \lvert a_n\right \rvert \le 1$ , there are $(a_{j,n})_{n\in A}\in \mathbb {F}^A$ , $j\in \mathbb {N}$ , $1\le j\le {\boldsymbol {\Upsilon }}$ , in $[0,1]$ such that $a_n=\sum _{j=1}^4 \gamma _j a_{j,n}$ . The identity

$$\begin{align*}g:= f+ \sum_{n\in A} a_n\,\boldsymbol{x}_n=f+ \sum_{j=1}^{\boldsymbol{\Upsilon}} \gamma_j \left( f+ \sum_{n\in A} a_{j,n}\,\boldsymbol{x}_n\right) \end{align*}$$

gives $\left \lVert g \right \rVert \le C_p M[\delta ,A,f]$ . The other inequalities follow readily from [Reference Albiac, Ansorena, Berná and Wojtaszczyk4, Corollary 2.3].

With all the previous ingredients, we are now in a position to provide a satisfactory answer to Temlyakov’s question by means of the squeeze-symmetry parameters.

Theorem 3.5. Let $\mathcal {X}$ be a basis of a quasi-Banach space $\mathbb {X}$ . There are constants $C_1$ and $C_2$ (depending only on the modulus of concavity of $\mathbb {X}$ ) such that

$$\begin{align*}\frac{1}{C_1} \boldsymbol{L}_m[\mathcal{X},\mathbb{X}] \le \max\lbrace\boldsymbol{\lambda}_m[\mathcal{X},\mathbb{X}], \boldsymbol{k}_m[\mathcal{X},\mathbb{X}]\rbrace\le C_2 \, \boldsymbol{L}_m[\mathcal{X},\mathbb{X}], \quad m\in\mathbb{N}. \end{align*}$$

Proof. Let us first note that by equation (1.1), we can replace $\boldsymbol {k}_m$ with ${\boldsymbol{k}^{\boldsymbol{c}}_m}$ . The inequality

(3.6) $$ \begin{align} {\boldsymbol{k}^{\boldsymbol{c}}_m} \le \boldsymbol{L}_m, \quad m\in\mathbb{N}, \end{align} $$

can be deduced from the fact that, given $f\in \mathbb {X}$ and $A\subseteq \mathbb {N}$ finite, there is $h\in \operatorname {\mathrm {span}}(\boldsymbol {x}_n \colon n\in A)$ such that A is a greedy set of $f+h$ (see [Reference Garrigós, Hernández and Oikhberg29]). Thus, to complete the proof it suffices to show that

(3.7) $$ \begin{align} \boldsymbol{L}_m\le C_1 \max\lbrace\boldsymbol{\lambda}_m,{\boldsymbol{k}^{\boldsymbol{c}}_m}\rbrace \end{align} $$

and

(3.8) $$ \begin{align} \boldsymbol{\lambda}_m\le C_2 \max\lbrace{\boldsymbol{k}^{\boldsymbol{c}}_m},\boldsymbol{L}_m\rbrace \end{align} $$

for all $m\in \mathbb {N}$ and some constants $C_1$ and $C_2$ .

To show equation (3.7), assume that $\mathbb {X}$ is a p-Banach space, $0<p\le 1$ . Let A be a greedy set of $f\in \mathbb {X}$ with $\left \lvert A\right \rvert =m<\infty $ , and pick $z=\sum _{n\in B}a_n\, \boldsymbol {x}_n$ with $\left \lvert B\right \rvert =\left \lvert A\right \rvert $ . Notice that

$$ \begin{align*} \max_{n\in B\setminus A}\left\lvert\boldsymbol{x}_n^*(f)\right\rvert\le\min_{n\in A\setminus B}\left\lvert\boldsymbol{x}_n^*(f)\right\rvert =\min_{n\in A\setminus B}\left\lvert\boldsymbol{x}_n^*(f-z)\right\rvert. \end{align*} $$

Set $k=\left \lvert B\setminus A\right \rvert =\left \lvert A\setminus B\right \rvert $ . On one hand, by Lemma 3.3 (ii) and Lemma 3.4,

$$\begin{align*}\left\lVert S_{B\setminus A}(f)\right\rVert \le A_p \boldsymbol{\lambda}_m \left\lVert f-z\right\rVert. \end{align*}$$

On the other hand, since $\left \lvert A\cup B\right \rvert =m+k$ ,

$$\begin{align*}\left\lVert (f-z) -S_{A\cup B)}(f-z)\right\rVert\le {\boldsymbol{k}^{\boldsymbol{c}}_{m+k}} \left\lVert f -z\right\rVert\le {\boldsymbol{k}^{\boldsymbol{c}}_{2m}}\left\lVert f -z\right\rVert. \end{align*}$$

Since

$$\begin{align*}f- S_A(f) = (f-z)-S_{A\cup B}(f-z) + S_{B\setminus A}(f), \end{align*}$$

combining both inequalities gives

$$ \begin{align*} \left\lVert f- S_A(f)\right\rVert^p &\le \left(({\boldsymbol{k}^{\boldsymbol{c}}_{2m}})^p + (A_p \boldsymbol{\lambda}_m)^p\right)\left\lVert f\right\rVert^p\\ &\le \left( 1+2 (\boldsymbol{k}_m)^p + (A_p \boldsymbol{\lambda}_m)^p\right)\left\lVert f\right\rVert^p. \end{align*} $$

Let us now prove inequality (3.8). In order to apply Lemma 3.3 (iii), we pick $A\subseteq \mathbb {N}$ , $f\in \mathbb {X}$ and B greedy set of f with $\left \lvert A\right \rvert =\left \lvert B\right \rvert \le m$ . Set $t=\min _{n\in B} \left \lvert \boldsymbol {x}_n^*(f)\right \rvert $ , and pick $(a_n)_{n\in A\cup B}$ in $[0,t]$ . Let us put

$$\begin{align*}y=\sum_{n\in B} a_n \, \varepsilon_n(f)\, \boldsymbol{x}_n, \quad z=\sum_{n\in A\setminus B} a_n \, \varepsilon_n(f)\, \boldsymbol{x}_n \end{align*}$$

and $g=f-z$ . On the one hand, since $\left \lvert A\setminus B\right \rvert \le \left \lvert B\right \rvert $ , B is a greedy set of g and $g-S_B(g)=f-S_B(f)-z$ , we have

$$\begin{align*}\left\lVert f-S_B(f)-z\right\rVert\le\boldsymbol{L}_m \left\lVert g+z\right\rVert=\boldsymbol{L}_m \left\lVert f\right\rVert. \end{align*}$$

On the other hand, since $\left \lvert \boldsymbol {x}_n^*(f-y)\right \rvert \le \left \lvert \boldsymbol {x}_n^*(f)\right \rvert $ and $\operatorname {\mathrm {sign}}(\boldsymbol {x}_n^*(f-y))=\operatorname {\mathrm {sign}}(\boldsymbol {x}_n^*(f))$ for all $n\in B$ , applying [Reference Albiac, Ansorena, Berná and Wojtaszczyk4, Corollary 2.3] yields

$$\begin{align*}\left\lVert f-S_{B}(f) +y \right\rVert =\left\lVert f-S_{B}(f-y) \right\rVert \le A_p {\boldsymbol{k}^{\boldsymbol{c}}_m} \left\lVert f \right\rVert. \end{align*}$$

Combining both estimates, we obtain

$$\begin{align*}\left\lVert y-z \right\rVert \le \left((A_p\, ({\boldsymbol{k}^{\boldsymbol{c}}_m})^p + (\boldsymbol{L}_m)^p\right)^{1/p} \left\lVert f \right\rVert. \end{align*}$$

Hence, by Lemma 3.4,

$$\begin{align*}\left\lVert \sum_{n\in A\cup B} b_n\, \boldsymbol{x}_n \right\rVert \le {\boldsymbol{\Upsilon}}^{1/p} \left( (A_p\, {\boldsymbol{k}^{\boldsymbol{c}}_m})^p + (\boldsymbol{L}_m)^p\right)^{1/p}\left\lVert f\right\rVert. \end{align*}$$

Applying this inequality with $b_n=0$ for all $n\in B\setminus A$ and $\left \lvert b_n\right \rvert =1$ for all $n\in A$ , we are done.

9.1 Orthogonal systems as bases of $L_p$

Let $(\Omega ,\Sigma ,\mu )$ be a finite measure space, and let $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ be an orthogonal basis of $L_2(\mu )$ . Given $1\le p\le \infty $ such that $ \mathcal {X}\subseteq L_{p^*}(\mu ), $ $\mathcal {X}$ is also a basis of $L_p(\mu )$ (a basis of its closed linear span if $r=\infty $ ).

Lemma 9.1. Let $\mathcal {X}$ be an orthonormal basis of $L_2(\mu )$ , where $(\Omega ,\Sigma ,\mu )$ is a finite measure space. Let $1\le q<2< p\le \infty $ , and suppose that the unit vector system of $\ell _q$ dominates $\mathcal {X}$ regarded as a sequence in $L_p(\mu )$ . Given $0\le \lambda \le 1$ , we define $p_{\lambda }^+\in [2,p]$ , $p_{\lambda }^-\in [p',2]$ , $q_{\lambda }^+\in [q,2]$ and $q_{\lambda }^-\in [2,q']$ by

$$\begin{align*}\frac{1}{p_{\lambda}^+}=(1-\lambda)\frac{1}{p} +\frac{\lambda}{2},\quad p_{\lambda}^-=(p_{\lambda}^+)', \quad \frac{1}{q_{\lambda}^+}=(1-\lambda)\frac{1}{q} +\frac{\lambda}{2},\quad q_{\lambda}^-=(q_{\lambda}^+)'. \end{align*}$$

Then there is a constant C such that, for all $\lambda \in [0,1]$ and $\varepsilon =\pm 1$ ,

$$\begin{align*}\max\lbrace\boldsymbol{k}_m[\mathcal{X},L_{p_{\lambda}^{\varepsilon}}(\mu)], \boldsymbol{\lambda}_m[\mathcal{X}, L_{p_{\lambda}^{\varepsilon}}(\mu)]\rbrace \le C m^{(1-\lambda)(1/q-1/2)}, \quad m\in\mathbb{N}. \end{align*}$$

For uniformly bounded orthogonal systems, we obtain the following.

Lemma 9.2. Let $(\Omega ,\Sigma ,\mu )$ be a finite measure space and $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ be an orthonormal basis of $L_2(\mu )$ with $\sup _n \left \lVert \boldsymbol {x}_n\right \rVert _{\infty }<\infty $ . There is a constant C such that, for all $1\le p \le \infty $ ,

$$\begin{align*}\frac{1}{C}\max\lbrace\boldsymbol{k}_m[\mathcal{X},L_p(\mu)], \boldsymbol{\lambda}_m[\mathcal{X}, L_p(\mu)]\rbrace \le \Phi_p(m):=m^{\left\lvert1/p-1/2\right\rvert}, \quad m\in\mathbb{N}. \end{align*}$$

Let us obtain lower estimates for the parameters.

Proof. If $1\le p< \infty $ , by Kahane–Khintchine’s inequalities and [Reference Albiac, Ansorena, Ciaurri and Varona5, Proposition 2.4], there is a constant $C_1$ such that

Combining these inequalities with Lemma 3.4 yields (i) and the estimate for $\boldsymbol {u}_m$ in (ii). In the case when $2< p<\infty $ , by [Reference Kadets and Pełczyński33, Corollary 7], $\mathcal {X}$ has a subsequence equivalent to the unit vector system of $\ell _2$ . We infer that the estimate for $\boldsymbol {\mu }_m$ in (ii) holds. If $p=\infty $ , since $c=\inf _n \left \lVert \boldsymbol {x}_n\right \rVert _1>0$ (see, e.g., [Reference Albiac, Ansorena, Ciaurri and Varona5, Lemma 2.7]),

$$\begin{align*}\int_{\Omega} \left(\sum_{n\in A} \left\lvert x_n\right\rvert\right)\, d\mu \ge c \left\lvert A\right\rvert, \end{align*}$$

whence

for all $A\subseteq \mathbb {N}$ finite. We deduce that (iii) holds.

9.2 The trigonometric system over $\mathbb {T}^d$

Temlyakov [Reference Temlyakov46, Theorem 2.1 and Remark 2] established the growth of the Lebesgue constants of the trigonometric system in $L_p$ . Later on, Wojtaszczyk [Reference Wojtaszczyk55, Corollary (a)] and Blasco et al. [Reference Berná, Blasco, Garrigós, Hernández and Oikhberg14, Proposition 8.6] revisited this result. Our discussion here uses Theorem 3.5 and, more specifically, the estimates obtained in Sections 9.1. Note that Lemmas 9.2 and 9.3 apply, in particular, to the trigonometric system $\mathcal {T}^d$ over $\mathbb {T}^d:=\mathbb {R}^d/\mathbb {Z}^d$ .

Using Shapiro’s polynomials, we obtain for each $m\in \mathbb {N}$ a set $A_m$ of cardinality m with for a suitable $\varepsilon \in \mathbb {E}^{A_m}$ (see [Reference Rudin43]). Hence, there is a constant $C_1$ such that

$$\begin{align*}\boldsymbol{u}_m[\mathcal{T}^d,L_{\infty}(\mathbb{T}^d)] \ge \frac{1}{C_1} \left\lvert m\right\rvert^{1/2}, \quad m\in\mathbb{N}. \end{align*}$$

In the case when $1<p<\infty $ , Dirichlet’s kernel (see e.g. [Reference Grafakos31]) shows the existence of a constant $C_2$ such that, for the same sets $A_m$ ,

Therefore, for each $1<p< \infty $ there is a constant $C_3$ such that

$$\begin{align*}\min\lbrace \boldsymbol{\mu}_m[\mathcal{T}^d,L_p(\mathbb{T}^d)] , \boldsymbol{u}_m[\mathcal{T}^d,L_p(\mathbb{T}^d)]\rbrace \ge \frac{1}{C_3} \Phi_p(m), \quad m\in\mathbb{N}. \end{align*}$$

In the case when $p=1$ , we have

for all $m\ge 2$ and a suitable constant $C_4$ . Thus, the same argument gives

$$\begin{align*}C_5 \min\lbrace \boldsymbol{\mu}_m[\mathcal{T}^d,L_1(\mathbb{T}^d)], \boldsymbol{u}_m[\mathcal{T}^d,L_1(\mathbb{T}^d)]\rbrace \ge \Psi(m):= \frac{m^{1/2}}{\log m}, \quad m\ge 2, \end{align*}$$

for a suitable constant $C_5$ . This estimate is optimal. Indeed, applying induction on d and using Fubini’s theorem, we infer from [Reference McGehee, Pigno and Smith38, Theorem 2.1] that there is a constant $C_6$ such that

Therefore, using the aforementioned bounded linear map from $\ell _2$ into $L_1(\mathbb {T}^d)$ ,

$$\begin{align*}\boldsymbol{\mu}^{\boldsymbol{s}}_m[\mathcal{T}^d,L_1(\mathbb{T}^d)]\le C_6 \Psi(m), \quad m\ge 2. \end{align*}$$

To obtain sharp estimates for the squeeze-symmetry parameters and the unconditionality parameters in the case $p=1$ , we invoke the De La Vallée–Pousin’s kernel, which yields, for each $m\in \mathbb {N}$ and $s>1$ , a function $v_{m,s}$ with $\left \lVert v_{m,s} \right \rVert _1 \le 1+s$ and

$$\begin{align*}\chi_{A_m} \le \mathcal{F}(v_{m,s}) \le 1, \end{align*}$$

(see, e.g., [Reference Mehta39]). Since there exists a constant $C_7$ such that, for each $m\in \mathbb {N}$ there is $\varepsilon \in \mathbb {E}^{A_m}$ with

, applying Lemma 3.4, we obtain

$$\begin{align*}C_7\min\lbrace {\boldsymbol{\lambda}^{\boldsymbol{d}}_m} [\mathcal{T}^d,L_1(\mathbb{T}^d)] , {\boldsymbol{\Upsilon}} \boldsymbol{q}_m [\mathcal{T}^d,L_1(\mathbb{T}^d)] \rbrace \ge m^{1/2}, \quad m\in\mathbb{N}. \end{align*}$$

Summing up, the democracy, superdemocracy, SLC, disjoint squeeze-symmetry, squeeze-symmetry, almost greediness, Lebesgue, unconditionality, quasi-greediness, truncation quasi-greedy, QGLC and UCC parameters of the trigonometric system in $L_p(\mathbb {T}^d)$ grow as $(\Phi _p(m))_{m=1}^{\infty }$ for all $1\le p \le \infty $ with the following exceptions in the case $p=1$ : The democracy, superdemocracy and the UCC parameters of the trigonometric system in $L_1(\mathbb {T}^d)$ grow as $(\Psi (m))_{m=2}^{\infty }$ , and in the case $p=\infty $ , $\mathcal {T}^d$ is democratic in $L_{\infty }(\mathbb {T}^d)$ .

9.3 The trigonometric system in Hardy spaces

Fix $0<p<1$ . If for $n\in \mathbb {N}\cup \lbrace 0\rbrace $ we set

$$\begin{align*}\tau_n(\theta)=e^{2\pi i \theta}, \quad -1/2\le\theta\le 1/2, \end{align*}$$

the sequence $\mathcal {T}=(\tau _n)_{n=0}^{\infty }$ is a basis of $H_p(\mathbb {T})$ whose biorthogonal functionals are the members of the sequence $(\overline {\tau }_n)_{n=0}^{\infty }$ under the natural dual mapping. Since $H_2(\mathbb {T})\subseteq H_p(\mathbb {T})$ , the unit vector system of $\ell _2$ dominates $\mathcal {T}$ regarded as basis of $H_p(\mathbb {T})$ . In turn, since the dual basis is uniformly bounded (see [Reference Duren, Romberg and Shields25]), $\mathcal {T}$ dominates the unit vector system of $c_0$ . We infer from Lemma 6.1 that there is a constant C such that

$$\begin{align*}\max\lbrace\boldsymbol{k}_m[\mathcal{T},H_p(\mathbb{T})], \boldsymbol{\lambda}_m[\mathcal{T},H_p(\mathbb{T})]\rbrace \le C m^{1/2}, \quad m\in\mathbb{N}. \end{align*}$$

This estimates are optimal. Indeed, the Dirichlet kernel $\sum _{k=0}^{n-1}\tau _k$ , $n\in \mathbb {N}$ , is uniformly bounded in $H_p(\mathbb {T})$ , and Khintchine’s inequalities yield a constant $C_1$ such that

Therefore, for some constant $C_2$ ,

$$\begin{align*}\min\lbrace\boldsymbol{\mu}_m[\mathcal{T},H_p(\mathbb{T})], \boldsymbol{u}_m[\mathcal{T},H_p(\mathbb{T})]\rbrace \ge\frac{1}{C_2} m^{1/2}, \quad m\in\mathbb{N}. \end{align*}$$

9.4 Jacobi polynomials

Given scalars $\alpha $ , $\beta>-1$ , the Jacobi polynomials

$$\begin{align*}\mathcal{J}(\alpha,\beta)=(p_n^{(\alpha,\beta)})_{n=0}^{\infty} \end{align*}$$

appear as the orthonormal polynomials associated with the measure $\mu _{\alpha ,\beta }$ given by

$$ \begin{align*} d\mu_{\alpha,\beta}(x)=(1-x)^{\alpha} (1+x)^{\beta} \chi_{(-1,1)} (x) \, dx. \end{align*} $$

In the case when $\gamma _0:=\min \lbrace \alpha ,\beta \rbrace> -1/2$ , we set $\gamma =\max \lbrace \alpha ,\beta \rbrace $ and

$$\begin{align*}\underline{p}=\underline{p}(\alpha,\beta)=\frac{4(\gamma+1)}{2\gamma+3}, \quad \overline{p}=\overline{p}(\alpha,\beta)=\frac{4(\gamma+1)}{2 \gamma +1}. \end{align*}$$

Notice that $\underline {p}$ and $\overline {p}$ are conjugate exponents. Given $p\in (\underline {p},\overline {p})$ , we define $q(p,\alpha ,\beta )\in (1,\infty )$ by

$$\begin{align*}\frac{1}{q(p,\alpha,\beta)}=\lambda, \end{align*}$$

where $\lambda \in [0,1]$ is such that

$$\begin{align*}p=(1-\lambda)\underline{p}(\alpha,\beta)+\lambda \overline{p}(\alpha,\beta). \end{align*}$$

A routine computation yields

$$\begin{align*}\frac{1}{q(p,\alpha,\beta)}=\frac{2\gamma+3}{2}-\frac{2(\gamma+1)}{p}, \quad \underline{p}(\alpha,\beta) <p< \overline{p}(\alpha,\beta). \end{align*}$$

We also define $r=r(p,\alpha ,\beta )$ by

$$\begin{align*}\frac{1}{r(p,\alpha,\beta)}=\frac{2\gamma_0+3}{2}-\frac{2(\gamma_0+1)}{p}, \quad \underline{p}(\alpha,\beta) <p< \overline{p}(\alpha,\beta). \end{align*}$$

Proof. This result could be derived from [Reference Veprintsev53]. Here, we present an alternative proof. Using Marcinkiewicz’s interpolation theorem (see, e.g., [Reference Grafakos31]) and duality, it suffices to prove that the unit vector system of $\ell _1$ dominates $(p_n^{(\alpha ,\beta )})_{n=0}^{\infty }$ regarded as a sequence in $\mathbb {X}:=L_{\overline {p},\infty } (\mu _{\alpha ,\beta })$ . Since $\mathbb {X}$ is locally convex, we must prove that $\sup _n \left \lVert p_n^{(\alpha ,\beta )}\right \rVert _{\overline {p},\infty } <\infty $ . This can be deduced from classical estimates for Jacobi polynomials (see [Reference Albiac, Ansorena, Ciaurri and Varona5, Theorem 3.2 and Lemma 3.3]) or from the fact that the partial sums of Jacobi–Fourier series $(J_n)_{n=1}^{\infty }$ are uniformly bounded when regarded as operators from $L_{\overline {p},1}(\mu _{\alpha ,\beta })$ into $L_{\overline {p},\infty }(\mu _{\alpha ,\beta })$ (see [Reference Guadalupe, Pérez and Varona32]). Indeed, taking into account that the dual of $L_{\overline {p},1}(\mu _{\alpha ,\beta })$ is $L_{\underline {p},\infty }(\mu _{\alpha ,\beta })$ under the natural dual pairing, the uniform boundedness of the operators $(J_n-J_{n-1})_{n=0}^{\infty }$ yields

$$\begin{align*}\sup_{n\in\mathbb{N}} \left\lVert p_n^{(\alpha,\beta)}\right\rVert_{\underline{p},\infty} \left\lVert p_n^{(\alpha,\beta)}\right\rVert_{\overline{p},\infty}<\infty. \end{align*}$$

Since $L_{\overline {p},\infty }(\mu _{\alpha ,\beta })\subseteq L_1(\mu _{\alpha ,\beta })$ and $\inf _n \left \lVert p_n^{(\alpha ,\beta )}\right \rVert _1>0$ (see, e.g., [Reference Albiac, Ansorena, Ciaurri and Varona5, Equation (3.4)]), we are done.

9.5 Lindenstrauss dual bases

Let $\delta =(d_n)_{n=1}^{\infty }$ be a nondecreasing sequence in $\mathbb {N}$ with $d_n\geq 2$ for all $n\in \mathbb {N}$ . Set

$$ \begin{align*} \quad \sigma(k)=2+\sum_{j=1}^{k-1} d_j, \quad k\in\mathbb{N}. \end{align*} $$

Let $\Gamma \colon \mathbb {N}\to \mathbb {N}\cup \lbrace 0\rbrace $ be the left inverse of the function defined by $n\mapsto \sigma ^{(n)}(1)$ , $n\in \mathbb {N}\cup \lbrace 0\rbrace $ . In [Reference Albiac, Ansorena and Wojtaszczyk8], it was constructed an almost greedy basis $\mathcal {X}_{\delta }$ of a subspace $\mathbb {X}_{\delta }$ of $\ell _1$ with

$$\begin{align*}\frac{1}{4}(1+\Gamma(m)) \le \boldsymbol{k}_m[\mathcal{X}_{\delta},\mathbb{X}_{\delta}] \le 2 (1+\Gamma(m)), \quad m\in\mathbb{N}. \end{align*}$$

In the case when $d_n=2$ for $n\in \mathbb {N}$ , the resulting space is the classical Lindesntrauss space, say $\mathbb {X}$ , built in [Reference Lindenstrauss35]. Moreover, $\mathbb {X}_{\delta }$ is isomporphic to $\mathbb {X}$ regardless the choice of $\delta $ . The dual space of $\mathbb {X}_{\delta }$ is isomorphic to $\ell _{\infty }$ , and the dual basis $\mathcal {X}_{\delta }^*$ spans a space isomorphic to $c_0$ . In [Reference Albiac, Ansorena and Wojtaszczyk8], it is also proved that for each increasing concave function $\phi \colon [0,\infty ) \to [0,\infty )$ with $\phi (0)=0$ , we can choose $\delta $ so that $\Gamma $ grows as $(\phi (\log (m)))_{m=2}^{\infty }$ . By [Reference Albiac, Ansorena and Wojtaszczyk8, Proposition 4.4 and Lemma 7.3],

$$\begin{align*}\boldsymbol{B}_m[\mathcal{X}_{\delta}, \mathbb{X}_{\delta}]\le 2 (1+\Gamma(m)), \quad m\in\mathbb{N}. \end{align*}$$

Thus, by equation (7.2) and Proposition 7.1, there is a constant C such that $\boldsymbol {L}_m[\mathcal {X}_{\delta }^*, \mathbb {X}_{\delta }^*]\le C\Gamma (m)$ for all $m\ge 2$ .

As far as lower bounds is concerned, combining [Reference Albiac, Ansorena and Wojtaszczyk8, Lemma 7.1 and Lemma 7.2] yields

$$\begin{align*}\boldsymbol{\mu}_m[\mathcal{X}_{\delta}^*, \mathbb{X}_{\delta}^*]\ge \frac{1}{2} \Gamma(m), \quad m\in\mathbb{N}, \end{align*}$$

and the proof of [Reference Albiac, Ansorena and Wojtaszczyk8, Lemma 7.1] gives

$$\begin{align*}\boldsymbol{u}_m[\mathcal{X}_{\delta}^*, \mathbb{X}_{\delta}^*]\ge\frac{1}{8} \Gamma(m), \quad m\in\mathbb{N}. \end{align*}$$

9.6 Bases with large greedy-like parameters

The unconditionality constants of truncation quasi-greedy bases grow slowly. Indeed, if $\mathbb {X}$ is a p-Banach space, $0<p\le 1$ ,

(9.2) $$ \begin{align} \boldsymbol{k}_m \le C (\log m)^{1/p},\quad m\ge 2, \end{align} $$

(see [Reference Dilworth, Kalton and Kutzarova18, Lemma 8.2], [Reference Garrigós, Hernández and Oikhberg29, Theorem 5.1] and [Reference Albiac, Ansorena and Wojtaszczyk8, Theorem 5.1]). Hence, if $\mathcal {X}$ is quasi-greedy and democratic there is a constant C such that

$$ \begin{align*} \boldsymbol{L}_m \le C (\log m)^{1/p},\quad m\ge 2. \end{align*} $$

In general, the unconditionality parameters of a basis $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ of a p-Banach space $\mathbb {X}$ can grow much faster. Notice that $\mathcal {X}$ satisfies equation (2.2), then

$$\begin{align*}\max\lbrace \boldsymbol{k}_m[\mathcal{X},\mathbb{X}], \boldsymbol{\lambda}_m[\mathcal{X},\mathbb{X}]\rbrace \le (C[\mathcal{X}])^2 m^{1/p}, \quad m\in\mathbb{N}. \end{align*}$$

Consequently, there is a constant C such that $\boldsymbol {L}_m[\mathcal {X},\mathbb {X}]\le C m^{1/p}$ for all $m\in \mathbb {N}$ . There are bases for which $\boldsymbol {k}_m\approx m^{1/p}$ so that this estimate is optimal. Take, for instance, the difference basis $\mathcal {D}=(\boldsymbol {d}_n)_{n=1}^{\infty }$ in $\ell _p$ given by

$$\begin{align*}\boldsymbol{d}_n=\boldsymbol{e}_n-\boldsymbol{e}_{n-1},\quad n\in \mathbb{N}, \end{align*}$$

where $(\boldsymbol {e}_n)_{n=1}^{\infty }$ is the unit vector system and $\boldsymbol {e}_0=0$ . For $0<p\le 1$ , $\mathcal {D}$ is a Schauder basis of $\ell _p$ whose dual basis is (naturally identified with) the summing basis $\mathcal {S}=(\boldsymbol {s}_n)_{n=1}^{\infty }$ of $c_0$ given by

$$\begin{align*}\boldsymbol{s}_n=\sum_{k=1}^n \boldsymbol{e}_k,\quad n\in \mathbb{N}. \end{align*}$$

Since $\left \lVert \sum _{n=1}^m \boldsymbol {d}_n\right \rVert _p=1$ and $\left \lVert \sum _{n=1}^{m} \boldsymbol {d}_{2n} \right \rVert _p=(2m)^{1/p}$ for all $m\in \mathbb {N}$ , we have

$$\begin{align*}\boldsymbol{\mu}_m[\mathcal{D},\ell_p]\ge (2m)^{1/p},\quad m\in \mathbb{N}, \end{align*}$$

and

$$\begin{align*}\boldsymbol{u}_m[\mathcal{D},\ell_p]\ge m^{1/p}, \quad m\in \mathbb{N}. \end{align*}$$

As for the dual basis, we have

$$\begin{align*}\left\lVert \sum_{n=1}^m \boldsymbol{s}_n\right\rVert_{\infty}=m,\;\mbox{ and }\; \left\lVert \sum_{n=1}^m (-1)^n\boldsymbol{s}_n\right\rVert_{\infty}=1, \end{align*}$$

whence $\boldsymbol {u}_m[\mathcal {S},c_0]\ge m$ , for all $m\in \mathbb {N}$ . Notice that the summing basis of $c_0$ is democratic.

Another classical basis with large greedy-like parameters is the $L_1$ -normalized Haar system $\mathcal {H}$ . It is essentially known that

$$\begin{align*}\boldsymbol{\mu}_m[\mathcal{H},L_1([0,1])]\ge m, \quad m\in\mathbb{N}, \end{align*}$$

and

$$\begin{align*}\boldsymbol{u}_{2m}[\mathcal{H},L_1([0,1])\ge m/4,\quad m\in\mathbb{N} \end{align*}$$

(see [Reference Dilworth, Kutzarova and Wojtaszczyk21]). Nonetheless, certain subbases of $\mathcal {H}$ are quasi-greedy basic sequences in $L_1([0,1])$ [Reference Dilworth, Kutzarova and Wojtaszczyk21, Reference Gogyan30].

9.7 Tsirelson’s space

The space $\mathcal {T}^*$ constructed by Tsirelson [Reference Tsirelson52] to prove the existence of a Banach space that contains no copy of $\ell _p$ or $c_0$ is the dual of the Tsirelson space $\mathcal {T}$ defined by Figiel and Johnson [Reference Figiel and Johnson26]. The unit vector system $\mathcal {B}=(\boldsymbol {e}_n)_{n=1}^{\infty }$ is a greedy basis of $\mathcal {T}$ whose fundamental function is equivalent to the fundamental function of the unit vector system of $\ell _1$ (see [Reference Dilworth, Odell, Schlumprecht and Zsák22]). Although $\mathcal {T}$ contains no copy of $\ell _1$ , its unit vector system contains finite subbases uniformly equivalent to the unit vector system of $\ell _1^n$ for all $n\in \mathbb {N}$ (see [Reference Casazza and Shura16, Proposition I.2]). Thus, the unit vector system of the original Tsirelson space $\mathcal {T}^*$ contains finite subbases uniformly equivalent to the unit vector system of $\ell _{\infty }^n$ for all $n\in \mathbb {N}$ . In particular, $\underline {\boldsymbol {\varphi }}[\mathcal {B},\mathcal {T}^*]$ is bounded. With an eye to studying the TGA with respect to the canonical basis of the original Tsirelson space $\mathcal {T}^*$ we give a general lemma.

Lemma 9.7. Let $\mathcal {X}$ be a basis of a quasi-Banach space $\mathbb {X}$ . Suppose that $\underline {\boldsymbol {\varphi }}[\mathcal {X}^*,\mathbb {X}^*]$ is bounded and that $\mathcal {X}^{**}$ is equivalent to $\mathcal {X}$ . Then

$$ \begin{align*} \boldsymbol{B}_m[\mathcal{X},\mathbb{X}] &\approx{\boldsymbol{L}^{\boldsymbol{a}}_m}[\mathcal{X}^*,\mathbb{X}^*] \approx \boldsymbol{\lambda}_m[\mathcal{X}^*,\mathbb{X}^*] \approx {\boldsymbol{\lambda}^{\boldsymbol{d}}_m}[\mathcal{X}^*,\mathbb{X}^*] \approx\boldsymbol{\varphi_u}[\mathcal{X}^*,\mathbb{X}^*]\\ &\approx \boldsymbol{\mu}_m[\mathcal{X}^*,\mathbb{X}^*] \approx\boldsymbol{\delta}_m[\mathcal{X}^*,c_0] \approx \boldsymbol{\delta}_m[\ell_1,\mathcal{X}]. \end{align*} $$

Proof. The equivalence $\boldsymbol {\delta }_m[\mathcal {X}^*,c_0]\approx \boldsymbol {\delta }_m[\ell _1,\mathcal {X}]$ follows by duality. The inequality

$$\begin{align*}\boldsymbol{B}_m[\mathcal{X},\mathbb{X}]\le \boldsymbol{\varphi_u}[\mathcal{X}^*,\mathbb{X}^*](m) \sup_n\left\lVert \boldsymbol{x}_n\right\rVert \end{align*}$$

holds for any basis of any Banach space. Combining Lemma 3.3, Propositions 5.2 and 7.1 and inequalities (6.1) and (9.1) concludes the proof.

Loosely speaking, Lemma 9.7 says that, for bases close to canonical $\ell _1$ -basis, the squeeze-symmetry parameters of their dual basis measure how far the basis is from the unit vector system of $\ell _1$ . We note that this applies in particular to the Lindenstrauss bases we considered in §9.5.

By Theorem 7.6, the dual basis $\mathcal {X}^*$ of any squeeze-symmetric basis $\mathcal {X}$ of as quasi-Banach space $\mathbb {X}$ satisfies $\boldsymbol {L}_m[\mathcal {X}^*,\mathbb {X}^*]=O(\log m)$ . For the canonical basis of the original Tsirelson space (which is not greedy), this general estimate is far from being optimal. To write down a precise statement of this estimate, we recursively define $\log ^{(k)}\colon (e^{k-1},\infty ) \to (0,\infty )$ by $\log ^{(1)}=\log $ and

$$\begin{align*}\log^{(k)}=\log^{(k-1)} \circ \log. \end{align*}$$

Since $\mathcal {B}$ is an unconditional basis of $\mathcal {T}^*$ , applying Lemma 9.7 to the canonical basis of $\mathcal {T}$ yields $\boldsymbol {L}_m[\mathcal {B},\mathcal {T}^*]\approx \boldsymbol {\delta }_m[\ell _1,\mathcal {T}]$ . Moreover, by [Reference Casazza and Shura16, Proposition I.9.3],

$$\begin{align*}\boldsymbol{\delta}_m[\ell_1,\mathcal{T}]\approx\sup\left\lbrace\sum_{n=1}^m \left\lvert a_n\right\rvert \colon \left\lVert \sum_{n=1}^m a_n \boldsymbol{e}_n \right\rVert_{\mathcal{T}} \le 1\right\rbrace, \quad m\in\mathbb{N}. \end{align*}$$

Then, by [Reference Casazza and Shura16, Proposition IV.b.3],

$$\begin{align*}\lim_m \frac{\boldsymbol{L}_m[\mathcal{B},\mathcal{T}^*]}{\log^{(k)}(m)}=0 \end{align*}$$

for all $k\in \mathbb {N}$ .

9.8 The dual basis of the Haar system in $\mathrm {BV}(\mathbb {R}^d)$

Given $d\in \mathbb {N}$ , $d\ge 2$ , let $\mathscr {D}$ denote the set consisting of all dyadic cubes in the Euclidean space $\mathbb {R}^d$ . If $Q\in \mathscr {D}$ there is a unique $k=k(Q)\in \mathbb {Z}$ such that $\left \lvert Q\right \rvert =2^{-kd}$ . Given $P\in \mathscr {D}$ and $k\in \mathbb {Z}$ , we define

$$\begin{align*}\mathscr{R}[P,k]=\lbrace Q\in \mathscr{D} \colon Q\subseteq P,\; k(Q)=k\rbrace. \end{align*}$$

Of course, $\mathscr {R}[P,k]=\emptyset $ for all $k<k(P)$ . Set also

$$\begin{align*}\mathscr{D}[P,k]=\bigcup_{j=k(P)}^{k-1} \mathscr{R}[P,j], \quad k>k(P). \end{align*}$$

Given an interval $J\subseteq \mathbb {R}$ , we denote by $J_{l}$ its left half and by $J_{r}$ its right half, and we set $h_I^0=\chi _I$ and $h_J^1=-\chi _{J_{l}}+\chi _{J_{r}}$ . For $\theta =(\theta _i)_{i=1}^d\in \Theta _d:=\lbrace 0,1\rbrace ^d\setminus \lbrace 0\rbrace $ and $Q\in \prod _{i=1}^d J_i\in \mathscr {D}$ put

$$\begin{align*}h_{Q,\theta}=\left\lvert Q\right\rvert^{(1-d)/d}\prod_{j=1}^d h_{J_i}^{\theta_i}, \quad h_{Q,\theta}^*=\left\lvert Q\right\rvert^{-1/d}\prod_{i=1}^d h_{J_i}^{\theta_i}, \end{align*}$$

and we denote by $\mathbb {X}$ be the subspace of $\mathrm {BV}(\mathbb {R}^d)$ spanned by

$$\begin{align*}\mathcal{H}=(h_{Q,\theta})_{(Q,\theta)\in\mathscr{D}\times\Theta_d}. \end{align*}$$

Let $\operatorname *{\mathrm {Ave}}(f;Q)$ denote average value of $f\in L_1(\mathbb {R}^d)$ over the cube Q. For every $f\in \mathrm {BV}(\mathbb {R}^d)$ , $P\in \mathscr {D}$ and $k\in \mathbb {Z}$ , $k> k(P)$ , we have

$$ \begin{align*} T_{P,k}(f)&:= \sum_{Q\in\mathscr{D}[P,k]}\sum_{\theta\in\Theta_d} h_{Q,\theta} \int_{\mathbb{R}^d} f(x)\, h_{Q,\theta}^*(x)\, dx\\ &=-\operatorname*{\mathrm{Ave}}(f;P)\chi_P+ \sum_{Q\in\mathscr{R}[P,k]} \operatorname*{\mathrm{Ave}}(f;Q) \chi_Q. \end{align*} $$

Hence, if $\pi \colon \mathbb {N}\to \mathscr {D}\times \Theta _d$ is a bijection such that the sets

$$\begin{align*}\pi^{-1}(\lbrace p\rbrace\times \Theta_d) \;\mbox{ and }\; \pi^{-1}(\mathscr{R}[P,k(P)+1]\times \Theta_d) \end{align*}$$

are integer intervals for every $P\in \mathscr {D}$ , applying [Reference Wojtaszczyk56, Corollary 12] gives that $(h_{\pi (n)})_{n=1}^{\infty }$ is a seminormalized Schauder basis of $\mathbb {X}$ . By [Reference Cohen, DeVore, Petrushev and Xu17, Theorem 8.1 and Remark 8.1], $\mathcal {H}$ is a squeeze-symmetric basis whose fundamental function is of the same order as $(m)_{m=1}^{\infty }$ . By [Reference Wojtaszczyk56, Theorem 10], $\mathcal {H}$ is a quasi-greedy basis of $\mathbb {X}$ . Pick a sequence $(Q_j)_{j=1}^{\infty }$ of pairwise disjoint dyadic cubes such that $k(Q_{j+1})=1+k(Q_j)$ , and pick an arbitrary sequence $(\theta _j)_{j=1}^{\infty }$ in $\Theta _d$ . By [Reference Dilworth, Kalton and Kutzarova18, Corollary 8.6], $(h_{Q_j,\theta _j})_{j=1}^{\infty }$ is equivalent to the unit vector system of $\ell _1$ . By Lemma 9.7 and Theorem 7.6, the dual basis

$$\begin{align*}\mathcal{H}^*=(h_{Q,\theta}^*)_{(Q,\theta)\in\mathscr{D}\times\Theta_d} \end{align*}$$

of $\mathcal {H}$ satisfies

$$\begin{align*}{\boldsymbol{L}^{\boldsymbol{a}}_m}[\mathcal{H}^*,\mathbb{X}^*]\approx \boldsymbol{\lambda}_m[\mathcal{H}^*,\mathbb{X}^*] \approx {\boldsymbol{\lambda}^{\boldsymbol{d}}_m}[\mathcal{H}^*,\mathbb{X}^*]\approx \boldsymbol{\mu}_m[\mathcal{H}^*,\mathbb{X}^*] \end{align*}$$

and $\boldsymbol {L}_m[\mathcal {H}^*,\mathbb {X}^*] =O( \log m)$ . We will prove that

$$\begin{align*}\boldsymbol{L}_m[\mathcal{H}^*,\mathbb{X}^*]\approx {\boldsymbol{L}^{\boldsymbol{a}}_m}[\mathcal{H}^*,\mathbb{X}^*]\approx \boldsymbol{g}_m[\mathcal{H}^*,\mathbb{X}^*] \approx \log m. \end{align*}$$

To that end, it suffices to show that $\log m=O( \boldsymbol {\varphi _u}[\mathcal {H}^*,\mathbb {X}^*](m))$ and $\log m=O( \boldsymbol {u}[\mathcal {H}^*,\mathbb {X}^*](m))$ .

For each $k\in \mathbb {N}$ , we define $f_k^*\in (\mathrm {BV}(\mathbb {R}^d))^*$ by

$$\begin{align*}f_k^*(f)= \frac{\partial}{\partial x_1} \left(T_{[0,1]^d,k}(f)\right)\left( \left[ \frac{1}{3},\infty\right)\times\mathbb{R}^{d-1}\right). \end{align*}$$

It is clear that $\left \lVert f_k^*\right \rVert =\left \lVert f_k^*|_{\mathbb {X}}\right \rVert $ for all $k\in \mathbb {N}$ , and $\sup _k \left \lVert f_k^*\right \rVert <\infty $ . Note that for each $j\in \mathbb {N}\cup \lbrace 0\rbrace $ there is a unique dyadic interval $I_j$ with $\left \lvert I_j\right \rvert =2^{-j}$ and $1/3\in I_j$ . Let $A_k$ (resp. $B_k$ ) be the subset of $\mathscr {D}\times \Theta _d$ defined by $(Q,\theta )\in A_k$ (resp. $(Q,\theta )\in B_k$ ) if and only if $\theta =(1,0,\dots ,0)$ , $Q=\prod _{i=1}^d J_i\subseteq [0,1)^d$ and $J_1=I_j$ for some even (resp. odd) integer $j\in [0,k-1]$ . A routine computation yields

$$\begin{align*}f_k^*(h_{Q,\theta})= \begin{cases} 0 & \mbox{ if } (Q,\theta)\notin A_k\cup B_k,\\ 1 & \mbox{ if } (Q,\theta)\in A_k \mbox{ and} \\ -1 & \mbox{ if } (Q,\theta)\in B_k \\ \end{cases} \end{align*}$$

(cf. [Reference Garrigós, Hernández and Oikhberg29, Example 2]). In other words,

. Set $f=\chi _{[0,1/3)\times [0,1)^{d-1}}$ . The arguments in [Reference Garrigós, Hernández and Oikhberg29, Example 2] also give

Since

$$\begin{align*}\left\lvert A_k\cup B_k\right\rvert=\frac{ 2^{(d-1)k}-1}{2^{d-1}-1}, \end{align*}$$

we are done. Note that this yields $\boldsymbol {k}_m[\mathcal {H},\mathbb {X}]=\boldsymbol {k}_m[\mathcal {H}^*,\mathbb {X}^*]\approx \log m$ .

9.9 The Franklin system as a basis of $\mathrm {VMO}$

As in §9.8, we denote by $\mathscr {D}$ the set consisting of all d-dimensional dyadic cubes, $d\in \mathbb {N}$ . The homogeneous Triebel–Lizorkin sequence space $\skew6\mathring {\boldsymbol {f}}_{p,q}^{d}$ of indices $p,q\in (0,\infty )$ consists of all scalar sequences $f=(a_Q)_{Q\in \mathscr {D}}$ for which

$$\begin{align*}\left\lVert f \right\rVert_{ \boldsymbol{f}_{p,q}^{d}}= \left\lVert \left(\sum_{Q\in\mathscr{D}} \left\lvert Q\right\rvert^{-q/p} \left\lvert a_Q\right\rvert^q \chi_{Q}\right)^{1/q}\right\rVert_p<\infty. \end{align*}$$

By definition, the unit vector system $\mathcal {B}=(\boldsymbol {e}_Q)_{Q\in \mathscr {D}}$ is a normalized unconditional basis of $\skew6\mathring {\boldsymbol {f}}_{p,q}^{d}$ . Moreover, it is a democratic (hence greedy) basis whose fundamental function is of the same order as $(m^{1/p})_{m=1}^{\infty }$ (see [Reference Albiac, Ansorena, Berná and Wojtaszczyk4, Section 11.3]). Let $\mathscr {D}_0$ denote the set consisting of all dyadic cubes contained in $[0,1]^d$ , and consider the subbasis $\mathcal {B}_0=(\boldsymbol {e}_Q)_{Q\in \mathscr {D}_0}$ of $\mathcal {B}$ . It is known that certain wavelet bases of homogeneous (resp. inhomogenous) Triebel–Lizorkin function spaces $\skew6\mathring {F}_{p,q}^{s}(\mathbb {R}^d)$ (resp. $F_{p,q}^{s}(\mathbb {R}^d)$ ) of smoothness $s\in \mathbb {R}$ are equivalent to $\mathcal {B}$ (resp. $\mathcal {B}_0$ ) regarded as a basis (resp. basic sequence) of $\skew6\mathring {\boldsymbol {f}}_{p,q}^{d}$ (see [Reference Frazier, Jawerth and Weiss28, Theorem 7.20] for the homogeneous case and [Reference Triebel51, Theorem 3.5] for the inhomogenous case). In the particular case that $p=1$ , $q=2$ and $d=1$ , $\mathcal {B}_0$ is equivalent to both the Franklin system in the Hardy space $H_1$ and the Haar system in the dyadic Hardy space $H_1(\delta )$ (see [Reference Maurey37, Reference Wojtaszczyk54]). Consequently, the dual basis of $\mathcal {B}_0$ is equivalent to both the Franklin system regarded as a basis of $\mathrm {VMO}$ and the Haar system regarded as a basic sequence in dyadic $\mathrm {BMO}$ .