Lebesgue-Type Inequalities for Quasi-greedy Bases

Abstract

We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N-term approximation ∥x−G _N x∥ and the best N-term approximation σ _N (x) is controlled by max{μ(N),k _N }, where μ(N) and k _N are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is O(logN). We show with various examples that these bounds are actually attained.

Appendix: Proof of (2.1)

The proof suggested in [5] for the inequalities in (2.1) is only valid for real scalars $a_{k}\in {\mathbb{R}}$; we give below a minor modification of their argument that establishes (2.1) also for complex scalars a _k . Below K denotes the quasi-greedy constant in ${\mathbb{X}}$.

The first two lemmas are similar to [24, Proposition 2].

Lemma 10.1

Let $\{{\mathbf{e}}_{j}\}_{j=1}^{\infty}$ be a quasi-greedy basis in a Banach space ${\mathbb{X}}$. For all $\beta_{j}\in {\mathbb{C}}$ with |β _j |=1 and all finite sets A ₁⊂A, it holds that

$$ \biggl\|\sum_{j\in A_1}\beta_j{\mathbf{e}}_j \biggr\| \leq K \biggl\| \sum_{j\in A}\beta_j {\mathbf{e}}_j \biggr\|. $$

(10.1)

Proof

Call A ₂=A∖A ₁. For ε>0, define $x=\sum_{j\in A_{1}}(1+{\varepsilon })\beta_{j}{\mathbf{e}}_{j}+\sum_{j\in A_{2}}\beta_{j}{\mathbf{e}}_{j}$. Then

$$\bigl\|G_{|A_1|}(x)\bigr\|=(1+{\varepsilon }) \biggl\|\sum_{j\in A_1} \beta_j{\mathbf{e}}_j \biggr\| \leq K\|x\|= \biggl\|(1+{\varepsilon })\sum _{j\in A_1}\beta_j{\mathbf{e}}_j+ \sum _{j\in A_2}\beta_j{\mathbf{e}}_j \biggr\|. $$

Letting ε→0, we obtain (10.1). □

Lemma 10.2

Let $\{{\mathbf{e}}_{j}\}_{j=1}^{\infty}$ be a quasi-greedy basis in a Banach space ${\mathbb{X}}$. For all ε _j ∈{±1,±i} and all finite sets A, it holds that

$$ \frac{1}{4K} \biggl\|\sum_{j\in A}{\mathbf{e}}_j \biggr\|\leq \biggl\| \sum_{j\in A}{\varepsilon }_j{\mathbf{e}}_j \biggr\|\leq 4K \biggl\|\sum_{j\in A}{\mathbf{e}}_j \biggr\|. $$

(10.2)

Proof

Call A _k ={j∈A:ε _j =i ^k}, k=1,2,3,4. Then, the triangle inequality and (10.1) (with all β _j =1) give

$$\biggl\|\sum_{j\in A}{\varepsilon }_j{\mathbf{e}}_j \biggr\|\leq \sum_{k=1}^4 \biggl\|\sum _{j\in A_k}{\mathbf{e}}_j \biggr\|\leq 4K \biggl\|\sum _{j\in A}{\mathbf{e}}_j \biggr\|, $$

establishing the right-hand side of (10.2). Arguing similarly,

$$\biggl\|\sum_{j\in A}{\mathbf{e}}_j \biggr\|\leq\sum _{k=1}^4 \biggl\|\sum_{j\in A_k} {\mathbf{e}}_j \biggr\|\leq 4K \biggl\|\sum_{j\in A} {\varepsilon }_j{\mathbf{e}}_j \biggr\|, $$

where we have now used (10.1) with β _j =ε _j . □

Lemma 10.3

For all complex β=a+ib with |a|+|b|≤1 and for all $x,y\in {\mathbb{X}}$, it holds that

$$ \|x+\beta y \|\leq \max \bigl\{\|x\pm y\|,\|x\pm i y\| \bigr\}. $$

(10.3)

Proof

We may assume that a∈[0,1). Then

$$\begin{aligned} \|x+\beta y\| \leq& \|ax + ay\|+ \bigl\|(1-a)x+iby\bigr\| \\ = & a\|x+y\|+(1-a)\|x+i{\gamma} y\|, \end{aligned}$$

(10.4)

where we have set γ=b/(1−a), which is a real number with |γ|≤1. Now

$$\begin{aligned} \|x+i{\gamma}y\| = & \biggl\|\frac{1-{\gamma}}{2}(x-iy)+\frac{1+{\gamma}}{2}(x+iy) \biggr\| \\ \leq &\frac{1-{\gamma}}{2} \|x-iy \|+\frac{1+{\gamma}}{2} \|x+iy \|\leq \max \|x\pm iy \|, \end{aligned}$$

where we have used that −1≤γ≤1. Inserting this into (10.4) easily leads to (10.3). □

We now justify the right-hand bound in (2.1). For a complex number α=a+ib, we shall write |α|₁=|a|+|b|. Then, iterating the previous lemma, we obtain

$$\begin{aligned} \biggl\|\sum_{j\in A}{\alpha }_j{\mathbf{e}}_j \biggr\| \leq& \max_{j\in A}|{\alpha }_j|_1\max _{{\varepsilon }_j\in\{\pm1,\pm i\}} \biggl\|\sum_{j\in A} {\varepsilon }_j{\mathbf{e}}_j \biggr\| \\ \leq& 4\sqrt{2} K\max_{j\in A}|{\alpha }_j| \biggl\|\sum _{j\in A}{\mathbf{e}}_j \biggr\|, \end{aligned}$$

(10.5)

where in the last step we have used Lemma 10.2 and the trivial estimate $|{\alpha }|_{1}\leq\sqrt{2} |{\alpha }|$.

We can now state a slightly more general version of Lemma 10.2.

Lemma 10.4

Let $\{{\mathbf{e}}_{j}\}_{j=1}^{\infty}$ be a quasi-greedy basis in a Banach space ${\mathbb{X}}$. For all ${\varepsilon }_{j}\in {\mathbb{C}}$ with |ε _j |=1 and all finite sets A, it holds that

$$ \frac{1}{4\sqrt{2} K} \biggl\|\sum_{j\in A}{\mathbf{e}}_j \biggr\|\leq \biggl\|\sum_{j\in A}{\varepsilon }_j{\mathbf{e}}_j \biggr\|\leq 4\sqrt{2} K \biggl\|\sum_{j\in A}{\mathbf{e}}_j \biggr\|. $$

(10.6)

Proof

The right-hand side is a special case of (10.5). To obtain the left-hand side, we consider the system $\{{\tilde{{\mathbf{e}}}}_{j}:={\varepsilon }_{j}{\mathbf{e}}_{j}\}$, which is also a quasi-greedy basis in ${\mathbb{X}}$ with the same constant K. Thus, (10.5) for this system (with ${\alpha }_{j}=\bar{{\varepsilon }}_{j}$) gives

$$\biggl\|\sum_{j\in A}\bar{{\varepsilon }}_j{\tilde{{\mathbf{e}}}}_j \biggr\|\leq 4\sqrt{2} K \biggl\|\sum_{j\in A}{\tilde{{\mathbf{e}}}}_j \biggr\|, $$

but this is the same as the left-hand side of (10.6). □

We turn now to the left-hand inequality in (2.1), for which we follow the arguments in [5, p. 579]. We shall prove that, if A is finite, then

$$ \biggl\|\sum_{j\in A}\alpha_j{\mathbf{e}}_j \biggr\| \geq \frac{1}{8\sqrt{2} K^2}\min_{j\in A}| {\alpha }_j| \biggl\|\sum_{j\in A}{\mathbf{e}}_j \biggr\|. $$

(10.7)

Write each scalar α _j =ε _j |α _j |, with ${\varepsilon }_{j}\in {\mathbb{C}}$ such that |ε _j |=1, and consider a permutation {j ₁,…,j _N } of A such that $|{\alpha }_{j_{1}}|\geq|{\alpha }_{j_{2}}|\geq\cdots\geq|{\alpha }_{j_{N}}|$. Let x=∑ _j∈A α _j e _j , and set G ₀(x)=0. Then

$$\begin{aligned} &|{\alpha }_{j_N}| \Biggl\|\sum_{\ell=1}^N {\varepsilon }_{j_\ell} {\mathbf{e}}_{j_\ell} \Biggr\| \\ &\quad {} = |{\alpha }_{j_N}| \Biggl\|\sum _{\ell=1}^N\frac{1}{|{\alpha }_{j_\ell}|} \bigl(G_\ell (x)-G_{\ell-1}(x) \bigr) \Biggr\| \\ &\quad {} = |{\alpha }_{j_N}| \Biggl\|\sum_{\ell=1}^{N-1} \biggl(\frac{1}{|{\alpha }_{j_\ell }|}-\frac{1}{|{\alpha }_{j_{\ell+1}}|} \biggr)G_\ell(x)+ \frac{1}{|{\alpha }_{j_N}|}G_{N}(x) \Biggr\| \\ &\quad {} \leq |{\alpha }_{j_N}| \Biggl[\frac{1}{|{\alpha }_{j_N}|}+ \sum _{\ell=1}^{N-1} \biggl(\frac{1}{| {\alpha }_{j_{\ell+1}}|}-\frac{1}{|{\alpha }_{j_{\ell}}|} \biggr) \Biggr]K \|x \| \leq 2K \|x \|. \end{aligned}$$

(10.8)

On the other hand, by Lemma 10.4, the expression on the left of (10.8) can be estimated from below by $|{\alpha }_{j_{N}}|\|\sum_{j\in A}{\mathbf{e}}_{j}\|/{4\sqrt{2} K}$, from which (10.7) follows.

Thus, putting together (10.5) and (10.7), we have shown:

Proposition 10.5

Let $\{{\mathbf{e}}_{j}\}_{j=1}^{\infty}$ be a quasi-greedy basis in a Banach space ${\mathbb{X}}$. If A is finite and ${\alpha }_{j}\in {\mathbb{C}}$, then

$$\frac{1}{8\sqrt{2} K^2} \min_{j\in A}| {\alpha }_j| \biggl\|\sum_{j\in A}{\mathbf{e}}_j \biggr\| \leq \biggl\|\sum_{j\in A}\alpha_j {\mathbf{e}}_j \biggr\|\leq 4\sqrt{2} K \max_{j\in A}| {\alpha }_j| \biggl\|\sum_{j\in A}{\mathbf{e}}_j \biggr\|. $$