1 Introduction
Nonlinear n-term approximations with respect to biorthogonal systems such as the trigonometric system and wavelet bases are frequently used in image or signal processing, PDE solvers and statistic learning (see [1]). The fundamental question of a nonlinear approximation is how to construct good algorithms to realize the best n-term approximation. It turns out the Thresholding Greedy Algorithm (TGA), which was proposed by Konyagin and Temlyakov in [2], in some sense is a suitable method for nonlinear n-term approximation. In this paper, we investigate the efficiency of the Chebyshev Thresholding Greedy Algorithm (CTGA), which is an enhancement of TGA.
Throughout this paper, X is an infinite-dimensional separable Banach space (over \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C}\)) with a norm \(\Vert \cdot \Vert =\Vert \cdot \Vert _{X}\) and its dual space is denoted by \(X^{*}\). A family \(\{e_{i}, e_{i}^{*}\}_{i=1}^{\infty}\subset X \times X^{*}\) is called a bounded biorthogonal system if
1.
\(X=\overline{\operatorname{span}} [ e_{i}: i \in\mathbb{N} ]\).
2.
\(e_{i}^{*}(e_{j})=1\) if \(i=j\), \(e_{i}^{*}(e_{j})=0\) if \(i\neq j\).
3.
\(0< \inf_{i}\min\{ \Vert e_{i}\Vert , \Vert e_{i}^{*}\Vert \} \leq \sup_{i}\max\{ \Vert e_{i}\Vert , \Vert e_{i}^{*}\Vert \} <\infty\).
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For brevity, we refer to \(( e_{i})\) as a basis, and denote it by Ψ. It is known from [1] that, for any \(c>1\), any separable Banach space has a bounded biorthogonal system (a Markushevitch basis) with \(1\leq \Vert e_{i}\Vert \), \(\Vert e_{i}^{*}\Vert \leq c\), and \(X^{*}=\overline{\operatorname{span}}^{w^{*}}[e_{i}^{*}: i\in\mathbb{N}]\).
For any \(x \in X\), we have the formal expansion
It is easy to see that \(\lim_{i\rightarrow\infty} e_{i}^{*}(x)=0\), and \(\sup_{i}\vert e_{i}^{*}(x)\vert >0\), unless \(x=0\). For each \(n\in\mathbb{N}\), let
where \(\#(A)\) denotes the cardinality of the set A. We consider the problem of approximating \(x \in X\) by the elements of \(\Sigma_{n}\) and define the best error of such an approximation as
For any finite set \(A \subset\mathbb{N}\), define the projection operator \(P_{A}x=\sum_{i \in A}e_{i}^{*}(x)e_{i}\). The n-term error of the expansion approximation with respect to Ψ is
It is clear that \(\widetilde{\sigma}_{n}(x) \geq {\sigma}_{n}(x)\).
$$x\sim \sum_{i=1}^{\infty}e_{i}^{*}(x)e_{i}. $$
$$\Sigma_{n}:=\Sigma_{n}(\Psi):=\biggl\{ \sum _{i\in A}a_{i}e_{i}: A\subset \mathbb{N}, \#(A)= n, a_{i} \in\mathbb{K}\biggr\} , $$
$$\sigma_{n}(x):=\sigma_{n}(x, \Psi)= \inf _{y \in\Sigma_{n}}\Vert x-y\Vert . $$
$$\widetilde{\sigma}_{n}(x):=\widetilde{\sigma}_{n}(x, \Psi)= \inf_{\#(A)= n}\Vert x-P_{A}x\Vert . $$
A finite set Γ with \(\min_{i\in\Gamma} \vert e_{i}^{*}(x)\vert \geq \max_{i\in{\Gamma^{c}}}\vert e_{i}^{*}(x)\vert \), is called a greedy set of order n for x if \(\#(\Gamma)=n\), and we write \(\Gamma\in\mathcal{G}(x, n)\). Let \(\rho: \mathbb{N}\rightarrow\mathbb{N}\) be a permutation, it is a greedy ordering if \(\vert e^{*}_{\rho(j)}(x)\vert \geq \vert e^{*}_{\rho(i)}(x)\vert \) for \(j\leq i\). In general, a greedy ordering is not unique. In [2], Konyagin and Temlyakov defined the thresholding greedy operator of x of order n with respect to Ψ by
We write \(\mathcal{G}_{n}(x, \Psi)\) for the set of all thresholding greedy operators of order n with respect to Ψ for x, when there is no confusion about x and Ψ, denote it by \(\mathcal{G}_{n}\). We note that \(G_{n}\in\mathcal{G}_{n}\) is neither continuous nor bounded, but homogeneous, that is, \(G_{n}(\lambda x)=\lambda G_{n}(x)\) for \(\lambda\in\mathbb{K}\), so we can define the norm of \(G_{n}\) as \(\Vert G_{n}\Vert :=\sup_{\Vert x\Vert \leq 1}\Vert G_{n}(x)\Vert \), and we have \(\Vert G_{n}(x)\Vert \leq \Vert G_{n}\Vert \Vert x\Vert \) for any \(x\in X\).
$$G_{n}(x):=G_{n}(x, \Psi):=\sum_{i=1}^{n}e_{\rho(i)}^{*}(x)e_{\rho(i)}. $$
We address the concept of a class of important bases which are called quasi-greedy bases.
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Definition 1.1
([2])
We call a basis Ψ of a Banach space X a quasi-greedy basis if, for any \(G_{n} \in\mathcal{G}_{n}\), the inequality
holds with a constant C independent of n.
$$\Vert G_{n}\Vert \leq C $$
Subsequently, Wojtaszczyk in [3] proved that a basis Ψ is quasi-greedy if and only if for any \(x\in X\),
$$\lim_{n\rightarrow+\infty}G_{n}(x, \Psi)=x. $$
Examples of quasi-greedy bases can be found in the literature [4‐9]. Of course, bases need not to be quasi-greedy, there exists a non-quasi-greedy basis, for these types of bases, TGA may fail to converge for certain vector \(x \in X\). For example, Temlyakov in [10] showed that trigonometric system in Lebesgue spaces is not quasi-greedy. Now we recall this result.
Let d be a natural number, \(\mathbb{T}^{d}\) the d-dimensional torus. For \(1\leq p<\infty\), let \(L_{p}(\mathbb{T}^{d})\) denote the space of all measurable functions for which
For convenience we also set \(L_{\infty}(\mathbb{T}^{d})=C(\mathbb{T}^{d})\), the space of continuous functions with the uniform norm
Let \(\mathcal{T}^{d}\) denote the trigonometric system \(\{e^{i(k,x)}: k\in\mathbb{Z}^{d}\}\) in \(L_{p}(\mathbb{T}^{d})\), where \(\{e^{i(k,x)}\}\) is for the complex exponentials and \(1\leq p\leq\infty\). In [10] Temlyakov proved that there is a positive absolute constant C such that for each n and \(1\leq p \leq\infty\) there exists a nonzero function \(f\in L_{p}(\mathbb{T}^{d})\) for which
So \(\mathcal{T}^{d}\) is not a quasi-greedy basis of \(L_{p}(\mathbb{T}^{d})\) when \(p \neq2\).
$$\Vert f\Vert _{p}:= \biggl(\frac{1}{(2\pi)^{d}} \int_{\mathbb{T}^{d}} \bigl\vert f(x)\bigr\vert ^{p}\,dx \biggr)^{\frac{1}{p}}< \infty. $$
$$\Vert f\Vert _{\infty}:=\max_{x\in\mathbb{T}^{d}}\bigl\vert f(x) \bigr\vert . $$
$$\bigl\Vert G_{n}\bigl(f, \mathcal{T}^{d}\bigr)\bigr\Vert _{p} \geq Cn^{\vert \frac {1}{2}-\frac{1}{p}\vert }\Vert f\Vert _{p}. $$
Now we recall the definition of CTGA. An n-term Chebyshev thresholding greedy approximant of x with respect to Ψ is defined as \(CG_{n}(x):=CG_{n}(x, \Psi) \in \operatorname{span}[e_{\rho(i)}: 1\leq i \leq n]\) such that
We have the following results on the convergence of CTGA in the more general bases.
$$\bigl\Vert x-CG_{n}(x)\bigr\Vert =\min\Biggl\{ \Biggl\Vert x-\sum _{i=1}^{n}a_{i}e_{\rho (i)} \Biggr\Vert : (a_{i})_{i=1}^{n}\in \mathbb{K}^{n}\Biggr\} . $$
Theorem 1.1
Let
\(\Psi=(e_{i})\)
be a basis for a Banach space
X. Then for each
\(x\in X\),
$$\lim_{n\rightarrow+\infty}CG_{n}(x, \Psi)=x. $$
From Theorem 1.1, we know that the convergence property of CTGA is better than TGA for non-quasi-greedy bases. As a consequence, we have the following result, which was firstly pointed out in [11].
Corollary 1.2
Let
\(1\leq p\leq\infty\)
and
\(\mathcal{T}^{d}\)
be trigonometric system in
\(L_{p}(\mathbb{T}^{d})\). Then, for any
\(f\in L_{p}(\mathbb{T}^{d})\),
$$\lim_{n\rightarrow+\infty}\bigl\Vert f-CG_{n}\bigl(f, \mathcal{T}^{d}\bigr)\bigr\Vert _{p}=0. $$
In view of Theorem 1.1, it is natural to study the convergence rate of CTGA for every \(x\in X\). To this end, we will estimate the Chebyshevian Lebesgue constants and its relatives:
and
It is obvious that \(L_{n}^{\mathrm{ch}}\geq\widetilde{L}_{n}^{\mathrm{ch}}\).
$$L_{n}^{\mathrm{ch}}:=L_{n}^{\mathrm{ch}}(\Psi):=\sup _{x \in X, \sigma_{n}(x) \neq 0}\frac{\Vert x-CG_{n}(x)\Vert }{\sigma_{n}(x)} $$
$$\widetilde{L}_{n}^{\mathrm{ch}}:=\widetilde{L}_{n}^{\mathrm{ch}}( \Psi):=\sup_{x \in X, \widetilde{\sigma}_{n}(x) \neq 0}\frac{\Vert x-CG_{n}(x)\Vert }{\widetilde{\sigma}_{n}(x)}. $$
In [6] and [12], the authors got the exact orders for \(L_{n}^{\mathrm{ch}}(\Psi)\) and \(\widetilde{L}_{n}^{\mathrm{ch}}(\Psi)\) with respect to quasi-greedy bases. To state their results, we recall some notions. For \(n\geq1\) and \(A,B \subset\mathbb{N}\), the democracy parameter \(\mu_{n}\) and the disjoint democracy parameter \(\mu_{n}^{d}\) are defined as
where \({\mathbf{1}}_{C}=\sum_{i \in C}e_{i}\) for any finite set C. Clearly, \(\mu_{n}^{d}\leq\mu_{n}\). For a quasi-greedy basis, we define the quasi-greedy constant \(\mathfrak{K}\) to be the least constant such that
$$\mu_{n}=\sup_{\#(A)=\#(B)\leq n}\frac{\Vert {\mathbf{1}}_{A}\Vert }{\Vert {\mathbf{1}}_{B}\Vert }\quad \mbox{and}\quad \mu_{n}^{d}=\sup_{\#(A)=\#(B)\leq n, A\cap B=\emptyset}\frac{\Vert {\mathbf{1}}_{A}\Vert }{\Vert {\mathbf{1}}_{B}\Vert }, $$
$$\bigl\Vert G_{n}(x, \Psi)\bigr\Vert \leq\mathfrak{K}\Vert x \Vert \quad \mbox{and}\quad \bigl\Vert x-G_{n}(x, \Psi)\bigr\Vert \leq \mathfrak{K}\Vert x\Vert \quad \text{for all } x\in X, n\geq1. $$
Theorem 1.3
If Ψ is a
\(\mathfrak{K}\)-quasi-greedy basis in a Banach space (over
\(\mathbb{K}=\mathbb{R}\)
or
\(\mathbb{C}\)), then, for all
\(n\geq1\),
and
$$\frac{\mu_{n}^{d}}{2\mathfrak{K}} \leq\widetilde{L}_{n}^{\mathrm{ch}}(\Psi) \leq L_{n}^{\mathrm{ch}}(\Psi) \leq20\mathfrak{K}^{3} \mu_{n}^{d} \quad \textit{for } \mathbb{K}=\mathbb{R} $$
$$\frac{\mu_{n}^{d}}{2\mathfrak{K}} \leq\widetilde{L}_{n}^{\mathrm{ch}}(\Psi) \leq L_{n}^{\mathrm{ch}}(\Psi) \leq100\mathfrak{K}^{3} \mu_{n}^{d} \quad \textit{for } \mathbb{K}=\mathbb{C}. $$
To compare Theorem 1.3 with the corresponding results of TGA, for a basis Ψ of a Banach space X, we recall the definitions of Lebesgue constants \(L_{n}\) and \(\widetilde{L}_{n}\):
and
$$L_{n}:=L_{n}(\Psi):=\sup_{x \in X, \sigma_{n}(x)\neq 0} \frac{\Vert x-G_{n}(x)\Vert }{\sigma_{n}(x)} $$
$$\widetilde{L}_{n}:=\widetilde{L}_{n}(\Psi):=\sup _{x \in X, \widetilde{\sigma}_{n}(x)\neq 0}\frac{\Vert x-G_{n}(x)\Vert }{\widetilde{\sigma}_{n}(x)}. $$
In what follows, for any two nonnegative sequences \(\{a_{n}\}\) and \(\{b_{n}\}\), the order inequality \(a_{n}\preceq b_{n}\) (\(a_{n}\succeq b_{n}\)) means that there is a number C independent of n such that \(a_{n} \leq Cb_{n}\), (\(a_{n} \geq Cb_{n}\)). The asymptotic relation \(a_{n}\asymp b_{n}\) means \(a_{n}\preceq b_{n}\) and \(a_{n}\succeq b_{n}\).
It is known from Theorem 1.1 in [7] that, for any quasi-greedy basis Ψ,
where \(k_{n}:=\sup_{\# (A)\leq n}\Vert P_{A}\Vert \) is the unconditionality parameter. So together with Theorem 1.3, we have
since \(\mu_{n} \geq1\) for all \(n\geq1\). It is known from [13, Lemma 8.2], that
So we have
From the above inequalities one can see that the order of \(L_{n}^{\mathrm{ch}}\) may have some improvements for some conditional quasi-greedy bases. In fact it is shown in [7], Example 2, that the maximum improvement is lnn. On the other hand, for unconditional bases, CTGA makes no essential improvements.
$$L_{n}(\Psi)\asymp k_{n}+\mu_{n}, $$
$$\frac{L_{n}(\Psi)}{L_{n}^{\mathrm{ch}}(\Psi)}\asymp\frac{k_{n}+\mu_{n}}{\mu_{n}}=1+\frac {k_{n}}{\mu_{n}} \preceq1+k_{n}, $$
$$k_{n}\preceq\ln n\quad \mbox{for any } n=2, 3, \ldots. $$
$$\frac{L_{n}(\Psi)}{L_{n}^{\mathrm{ch}}(\Psi)}\preceq 1+ \ln n. $$
However, as far as we know, there is no result on the estimates of \(L_{n}^{\mathrm{ch}}(\Psi)\) for non-quasi-greedy bases. So in Sect. 2, we study Chebyshevian Lebesgue constants for two examples of such bases, the trigonometric system and the summing basis. We determine the exact order of \(L_{n}^{\mathrm{ch}}(\Psi)\) for these bases. In Sect. 3, we obtain the upper estimates for \(L_{n}^{\mathrm{ch}}(\Psi)\) with respect to the general bases which leads to an improvement of Theorem 1.3 for conditional quasi-greedy bases. In the final section, we present a short survey of the results and questions on the efficiency of CTGA.
2 Chebyshevian Lebesgue constants for two non-quasi-greedy bases
The conclusion of Theorem 1.1 is our motivation to study the efficiency of CTGA for non-quasi-greedy bases. So we begin with the proof of Theorem 1.1.
Proof of Theorem 1.1
Since \(\Psi=(e_{i})\) is a basis for a Banach space X, we have \(X=\overline{\operatorname{span}}\{e_{i}: i\in \mathbb{N}\}\). Let \(x\in X\). For any \(\epsilon>0\), there exist \(M\in \mathbb{N}\) and \(\{x_{i}\}_{i=1}^{M}\subset\mathbb{K}^{M}\), such that
For this M, we can take a positive integer N such that
So for a Chebyshev greedy approximant \(CG_{N}(x)=\sum_{i=1}^{N}c_{i}e_{\rho(i)}\), we have
The proof is completed. □
$$\Biggl\Vert x-\sum_{i=1}^{M}x_{i}e_{i} \Biggr\Vert < \epsilon. $$
$$\{1, 2, \ldots, M\}\subset\bigl\{ \rho(1), \rho(2), \ldots, \rho(N)\bigr\} . $$
$$\Biggl\Vert x-\sum_{i=1}^{N}c_{ i}e_{\rho(i)} \Biggr\Vert \leq \Biggl\Vert x-\sum_{i=1}^{M}x_{i}e_{i} \Biggr\Vert < \epsilon. $$
A basic problem in approximation theory is to represent a given function approximately, and solving this problem is to choose a representation system. Traditionally, a representation system has natural features such as minimality, orthogonality, simple structure and nice computational characteristics. The trigonometric system is one of the most typical representation systems, a very importance feature of the trigonometric system that made it attractive for the representation of periodic functions is orthogonality.
Next we consider \(L_{n}^{\mathrm{ch}}(\Psi)\) for two non-quasi-greedy bases Ψ. The first one is the trigonometric system. We obtain the following result.
Theorem 2.1
For the trigonometric system
\(\mathcal{T}^{d}=\{e^{ikx}\}_{ k \in\mathbb{Z}}\)
in
\(L^{p}(\mathbb{T}^{d})\), \(1< p\leq\infty\), we have, for
\(n\geq1\),
$$L_{n}^{\mathrm{ch}}\bigl(\mathcal{T }^{d}\bigr)\asymp n^{\vert \frac{1}{2}-\frac {1}{p}\vert }. $$
The upper bound of \(L_{n}^{\mathrm{ch}}(\mathcal{T}^{d})\) follows from the known results of \(L_{n}(\mathcal{T}^{d})\) and the proof of the lower bound of \(L_{n}^{\mathrm{ch}}(\mathcal{T}^{d})\) relies on a theorem on the lower estimate of \(L_{n}^{\mathrm{ch}}(\Psi)\) for a general basis Ψ in a Banach space X.
To present this theorem we recall some concepts. For a basis \(\Psi=(e_{i})_{i=1}^{\infty}\) of X, define the partial sum operator \(S_{m}: X\rightarrow X\) by
Then \(S_{m}\) is a continuous linear operator. So we define \(\Vert S_{m}\Vert \) in the usual way. For finite sets \(A, B\subset\mathbb{N}\), we write \(A< B\) if \(\max\{n: n\in A\}< \min\{n: n\in B\}\). Denote by ϒ the set of \(\varepsilon=\{\varepsilon_{i}\}\) with \(\vert \varepsilon _{i}\vert =1\) for all i (where \(\varepsilon_{i}\) could be real or complex).
$$ S_{m}(x):=\sum_{i=1}^{m}e_{i}^{*}(x)e_{i}. $$
(2.1)
Theorem 2.2
Let Ψ be a basis of a Banach space
X. For any
\(A, B\subset\mathbb{N}\)
with
\(\#(A)=\#(B)=n\)
and
\(A< B\), we have, for all
\(\varepsilon\in\Upsilon\)
and
\(n\geq1\),
where
\({\mathbf{1}}_{\varepsilon A}=\sum_{i\in A}\varepsilon_{i} e_{i}\).
$$ L_{n}^{\mathrm{ch}}(\Psi)\geq\max \biggl\{ \frac{1}{\Vert S_{n}\Vert } \frac{\Vert {\mathbf{1}}_{\varepsilon A}\Vert }{ \Vert {\mathbf{1}}_{B}\Vert }, \frac{1}{1+\Vert S_{n}\Vert }\frac{\Vert {\mathbf{1}}_{B}\Vert }{\Vert {\mathbf{1}}_{\varepsilon A}\Vert } \biggr\} , $$
(2.2)
Proof
Let \(\varepsilon=(\varepsilon_{i})\) be any choice of signs from ϒ. Choose \(\epsilon>0\), we consider
then \(B \in\mathcal{G}(x, n)\) and we can find a Chebyshev greedy approximant \(CG_{n}(x)\) which is supported on B, such that
for some \(\{a_{i}\}_{i\in B}\subset\mathbb{K}\). Thus
Notice that \(A< B\) and \(\Vert S_{n}(x-CG_{n}(x))\Vert \leq \Vert S_{n}\Vert \Vert x-CG_{n}(x)\Vert \), we have
Combining (2.4) and (2.5), we obtain
Hence
Now we consider
then \(A \in\mathcal{G}(y,n)\) and there exists some \(\{b_{i}\}_{i\in A}\subset\mathbb{K}\) such that
Since \(A< B\) and
we have, for \(S_{n}(y-CG_{n}(y))=\sum_{i\in A}b_{i}e_{i}\),
Hence
For any \(z\in\Sigma_{n}\),
Taking the infimum over all such z, and using the symmetry of x and y, we have
From (2.3) and (2.7),
then (2.9) and the above equality imply that
Combining (2.6), (2.8) and (2.10), we obtain
Thus,
and
We complete the proof. □
$$ x=\sum_{i\in A}\varepsilon_{i}e_{i}+(1+ \epsilon)\sum_{i\in B}e_{i}, $$
(2.3)
$$x-CG_{n}(x)=\sum_{i\in A } \varepsilon_{i}e_{i}+\sum_{i\in B}a_{i}e_{i} $$
$$ \bigl\Vert x-CG_{n}(x)\bigr\Vert =\biggl\Vert \sum _{i\in A}\varepsilon_{i}e_{i}+\sum _{i\in B}a_{i}e_{i}\biggr\Vert \leq L_{n}^{\mathrm{ch}} \sigma_{n}(x). $$
(2.4)
$$ \biggl\Vert \sum_{i \in A}\varepsilon_{i}e_{i} \biggr\Vert \leq \Vert S_{n}\Vert \biggl\Vert \sum _{i\in A }\varepsilon_{i}e_{i}+\sum _{i\in B}a_{i}e_{i}\biggr\Vert . $$
(2.5)
$$\biggl\Vert \sum_{i \in A}\varepsilon_{i}e_{i} \biggr\Vert \leq \Vert S_{n}\Vert \biggl\Vert \sum _{i\in A}\varepsilon_{i}e_{i}+\sum _{i\in B}a_{i}e_{i}\biggr\Vert \leq \Vert S_{n}\Vert L_{n}^{\mathrm{ch}} \sigma_{n}(x). $$
$$ \begin{aligned}\sigma_{n}(x) \leq \bigl\Vert x-P_{B}(x)\bigr\Vert =\biggl\Vert \sum_{i \in A} \varepsilon_{i}e_{i}\biggr\Vert \leq \Vert S_{n}\Vert L_{n}^{\mathrm{ch}} \sigma_{n}(x). \end{aligned} $$
(2.6)
$$ y=(1+\epsilon)\sum_{i \in A }\varepsilon_{i}e_{i}+ \sum_{i \in B}e_{i}, $$
(2.7)
$$y-CG_{n}(y)=\sum_{i\in A}b_{i}e_{i}+ \sum_{i\in B}e_{i}. $$
$$\bigl\Vert y-CG_{n}(y)\bigr\Vert =\biggl\Vert \sum _{i\in A}b_{i}e_{i}+\sum _{i\in B}e_{i}\biggr\Vert \leq L_{n}^{\mathrm{ch}} \sigma_{n}(y), $$
$$ \begin{aligned}\biggl\Vert \sum _{i\in B}e_{i}\biggr\Vert &= \biggl\Vert \sum _{i\in A}b_{i}e_{i}+\sum _{i\in B}e_{i}-\sum_{i\in A}b_{i}e_{i} \biggr\Vert \\ &\leq \biggl\Vert \sum_{i\in A}b_{i}e_{i}+ \sum_{i\in B}e_{i}\biggr\Vert +\biggl\Vert \sum_{i\in A}b_{i}e_{i}\biggr\Vert \\ & \leq\biggl\Vert \sum_{i\in A}b_{i}e_{i}+ \sum_{i\in B}e_{i}\biggr\Vert +\Vert S_{n}\Vert \biggl\Vert \sum_{i\in A}b_{i}e_{i}+ \sum_{i\in B}e_{i}\biggr\Vert \\ &= \bigl(1+\Vert S_{n}\Vert \bigr)\biggl\Vert \sum _{i\in A}b_{i}e_{i}+\sum _{i\in B}e_{i}\biggr\Vert \\ &\leq \bigl(1+\Vert S_{n}\Vert \bigr)L_{n}^{\mathrm{ch}} \sigma_{n}(y). \end{aligned} $$
$$ \sigma_{n}(y)\leq\bigl\Vert y-P_{A}(y)\bigr\Vert = \biggl\Vert \sum_{i\in B}e_{i}\biggr\Vert \leq \bigl(1+\Vert S_{n}\Vert \bigr)L_{n}^{\mathrm{ch}} \sigma_{n}(y). $$
(2.8)
$$\sigma_{n}(x)\leq \Vert x-z\Vert \leq \Vert x-y\Vert +\Vert y-z \Vert . $$
$$ \bigl\vert \sigma_{n}(x)-\sigma_{n}(y)\bigr\vert \leq \Vert x-y\Vert . $$
(2.9)
$$\Vert x-y\Vert =\biggl\Vert \epsilon\biggl(\sum _{i \in D}e_{i}-\sum_{i\in A}e_{i} \biggr)\biggr\Vert \rightarrow0\quad \mbox{as } \epsilon\rightarrow 0, $$
$$ \sigma_{n}(x)\asymp\sigma_{n}(y). $$
(2.10)
$$\begin{aligned} \frac{1}{\Vert S_{n}\Vert L_{n}^{\mathrm{ch}}}\biggl\Vert \sum_{i \in A} \varepsilon_{i}e_{i}\biggr\Vert \leq& \biggl\Vert \sum _{i\in B}e_{i}\biggr\Vert \\ \leq&\bigl(1+\Vert S_{n}\Vert \bigr)L_{n}^{\mathrm{ch}}\biggl\Vert \sum _{i \in A}\varepsilon_{i}e_{i}\biggr\Vert . \end{aligned}$$
(2.11)
$$L_{n}^{\mathrm{ch}}\geq \frac{1}{\Vert S_{n}\Vert }\frac{\Vert {\mathbf{1}}_{\varepsilon A}\Vert }{\Vert {\mathbf{1}}_{B}\Vert }\quad \mbox{from the first inequality of (2.11)} $$
$$L_{n}^{\mathrm{ch}}\geq\frac{1}{1+\Vert S_{n}\Vert }\frac{\Vert {\bf 1}_{B}\Vert }{\Vert {\mathbf{1}}_{\varepsilon A}\Vert }\quad \mbox{from the second inequality of (2.11)}. $$
If Ψ is a Schauder basis of a Banach space X, then for any \(x\in X\) there exists a unique expansion
which means the partial sum sequence \(\{S_{m}(x)\}\) defined in (2.1) converges to x in X norm for every \(x\in X\). By the principle of uniform boundedness, we have \(\sup_{m}\Vert S_{m}\Vert <\infty\). The number \(\sup_{m}\Vert S_{m}\Vert \) is called the basis constant of the basis Ψ (see [14]) and denoted by β, which is used frequently for the research of greedy algorithms (see [15]).
$$x=\sum_{i=1}^{\infty}e_{i}^{*}(x)e_{i}, $$
We have the following corollary, which is a direct consequence of Theorem 2.2.
Corollary 2.3
Let Ψ be a Schauder basis for a Banach space
X. For any
\(A, B\subset\mathbb{N}\)
with
\(\#(A)=\#(B)=n\)
and
\(A< B\), we have, for all
\(\varepsilon\in\Upsilon\)
and
\(n\geq1\),
$$L_{n}^{\mathrm{ch}}(\Psi)\geq\max \biggl\{ \frac{1}{\beta} \frac{\Vert {\bf 1}_{\varepsilon A}\Vert }{\Vert {\mathbf{1}}_{B}\Vert }, \frac{1}{1+\beta}\frac{\Vert {\mathbf{1}}_{B}\Vert }{\Vert {\bf 1}_{\varepsilon A}\Vert } \biggr\} . $$
Now we prove Theorem 2.1.
Proof of Theorem 2.1
The upper bounds follows from the obvious relationship \(L_{n}^{\mathrm{ch}}(\mathcal{T})\leq L_{n}(\mathcal{T})\), and the inequality
which was proved by Temlykov in [10].
$$ L_{n}(\mathcal{T}) \leq 1+3n^{\vert \frac{1}{p}-\frac{1}{2}\vert } \quad \mbox{for } 1\leq p\leq \infty, $$
(2.12)
Now we turn to the proof of the lower bounds. It suffices to prove the results for \(d=1\). We consider separately two cases \(1< p <\infty\) and \(p=\infty\).
We first apply Theorem 2.2 to prove the results for \(1< p<\infty\). It is well known that \(e_{k}^{*}(f)\) is the kth Fourier coefficient of f defined by
For notational convenience, we take a particular order
of the trigonometric system \(\mathcal{T}\).
$$e_{k}^{*}(f)=\widehat{f}(k):=\frac{1}{2\pi} \int_{\mathbb{T}}f(x)e^{-ikx}\,dx. $$
$$\bigl\{ 1, e^{ix}, e^{-ix}, e^{i2x}, e^{-i2x}, \ldots\bigr\} $$
We consider two important trigonometric polynomials (see for instance [16]).
1. The Dirichlet kernel of order n is defined as
It is well known that
and
$$ \mathcal{D}_{n}(x)=\sum_{\vert k\vert \leq n}e^{ikx},\quad x\in \mathbb{T}. $$
(2.13)
$$ \Vert \mathcal{D}_{n}\Vert _{p}\asymp n^{1-\frac{1}{p}},\quad n=1,2,\ldots, 1< p\leq\infty, $$
(2.14)
$$ \Vert \mathcal{D}_{n}\Vert _{1} \asymp\ln n,\quad n=1, 2, 3 \ldots. $$
(2.15)
2. The Rudin–Shapiro polynomial \(\mathcal{R}_{N}(x)\), which is defined recursively by pairs of trigonometric polynomials \(P_{j}(x)\) and \(Q_{j}(x)\) of order \(2^{j}-1\):
From the definition of \(P_{n}\), it is clear that
Let N be a natural number, and let it have the binary representation
$$\begin{aligned}&P_{0}=Q_{0}=1, \\ &P_{j+1}(x)=P_{j}(x)+e^{i2^{j}x}Q_{j}(x), \qquad Q_{j+1}(x)=P_{j}(x)-e^{i2^{j}x}Q_{j}(x). \end{aligned} $$
$$P_{n}(x)=\sum_{k=0}^{2^{n}-1} \varepsilon_{k}e^{ikx},\quad \varepsilon_{k}=\pm1. $$
$$N=\sum_{j=1}^{m}2^{n_{j}},\quad n_{1}>n_{2}>\cdots>n_{m}\geq0. $$
We set
$$\begin{aligned}&\widetilde{\mathcal{R}}_{N}(x)=P_{n_{1}}(x)+\sum _{j=2}^{m}P_{n_{j}}(x)e^{i(2^{n_{1}}+\cdots+2^{n_{j-1}})x}, \\ &\mathcal{R}_{N}(x)=\widetilde{\mathcal{R}}_{N}(x)+ \widetilde{\mathcal{R}}_{N}(-x)-1. \end{aligned}$$
Then \(\mathcal{R}_{N}(x)\) has the form
and it is known that
Notice that
which implies \(\Vert \mathcal{R}_{N}\Vert _{1}\succeq N^{\frac{1}{2}}\). It is trivial that \(\Vert \mathcal{R}_{N}\Vert _{1}\preceq \Vert \mathcal{R}_{N}\Vert _{p}\preceq \Vert \mathcal{R}_{N}\Vert _{\infty}\). Hence, we have
$$ \mathcal{R}_{N}(x)=\sum_{\vert k\vert \leq N} \varepsilon_{k}e^{ikx},\quad \varepsilon_{k}=\pm1, $$
(2.16)
$$\Vert \mathcal{R}_{N}\Vert _{\infty}\leq CN^{\frac{1}{2}} \quad \mbox{for } 1\leq p\leq\infty. $$
$$ \begin{aligned} 2N+1&=\langle \mathcal{R}_{N}, \mathcal{R}_{N}\rangle \\ &\leq \Vert \mathcal{R}_{N}\Vert _{1}\cdot \Vert \mathcal{R}_{N}\Vert _{\infty} \\ &\leq \Vert \mathcal{R}_{N}\Vert _{1}\cdot CN^{\frac{1}{2}}, \end{aligned} $$
$$ \Vert \mathcal{R}_{N}\Vert _{p}\asymp N^{\frac{1}{2}} \quad \mbox{for } 1\leq p\leq\infty. $$
(2.17)
Firstly, we assume that n is even. Define the sets
It is clear that \(\#(A)=\#(B)=n\) and \(A< B\). So from (2.13) we have
where \({\mathbf{1}}_{B}(x)=\sum_{k\in B}e^{ikx}\), hence the triangle inequality and (2.14) give
which is equivalent to
And if we choose in ε the signs of the corresponding Rudin–Shapiro polynomial, according to (2.16) and the definition of the set A, we have
where \({\bf 1}_{\varepsilon A}(x)=\sum_{k\in A}\varepsilon_{k}e^{ikx}\), by (2.17) and the triangle inequality, we obtain
which can be rewritten as
$$A=\{k: 2\leq k\leq n+1\} \quad \mbox{and}\quad B=\{k: n+1< k\leq2n+1\}. $$
$${\mathbf{1}}_{B}(x)=\mathcal{D}_{n}(x)-\mathcal{D}_{\frac{n}{2}}(x), $$
$$n^{1-\frac{1}{p}}\succeq \Vert \mathcal{D}_{n}\Vert _{p}+ \Vert \mathcal{D}_{\frac{n}{2}}\Vert _{p}\geq \Vert { \mathbf{1}}_{B}\Vert _{p}\geq \Vert \mathcal{D}_{n} \Vert _{p}-\Vert \mathcal{D}_{\frac {n}{2}}\Vert _{p} \succeq n^{1-\frac{1}{p}}, $$
$$ \Vert {\mathbf{1}}_{B}\Vert _{p}\asymp n^{1-\frac{1}{p}}. $$
(2.18)
$${\mathbf{1}}_{\varepsilon A}(x)=\mathcal{R}_{\frac{n}{2}}(x)-\varepsilon_{0}, $$
$$n^{\frac{1}{2}}-1 \preceq \Vert \mathcal{R}_{\frac{n}{2}} \Vert _{p}-1\leq \Vert {\mathbf{1}}_{\varepsilon A}\Vert _{p}\leq \Vert \mathcal{R}_{\frac {n}{2}}\Vert _{p}+1 \preceq n^{\frac{1}{2}}, $$
$$ \Vert {\mathbf{1}}_{\varepsilon A}\Vert _{p}\asymp n^{\frac{1}{2}}. $$
(2.19)
Combining (2.2), (2.18) and (2.19), we obtain
and
which imply that
Here we use the fact that the trigonometric system \(\mathcal{T}^{d}=\{e^{i(k,x)}\}_{k \in\mathbb{Z}^{d}}\) is a Schauder basis for \(L_{p}(\mathbb{T}^{d})\), which is equivalent to
i.e., the partial sum operator \(S_{n}\) is uniformly bounded.
$$L_{n}^{\mathrm{ch}}(\mathcal{T})\geq\frac{1}{1+ \Vert S_{n}\Vert _{L_{p}\rightarrow L_{p}}} \frac{\Vert {\mathbf{1}}_{B}\Vert _{p}}{\Vert {\bf 1}_{\varepsilon A}\Vert _{p}} \asymp\frac{n^{1-\frac{1}{p}}}{n^{\frac{1}{2}}}=n^{\frac{1}{2}-\frac {1}{p}} \quad \mbox{for } 2< p< \infty $$
$$L_{n}^{\mathrm{ch}}(\mathcal{T})\geq\frac{1}{ \Vert S_{n}\Vert _{L_{p}\rightarrow L_{p}}} \frac{\Vert {\mathbf{1}}_{\varepsilon A}\Vert _{p}}{\Vert {\mathbf{1}}_{B}\Vert _{p}} \asymp\frac{n^{\frac{1}{2}}}{n^{1-\frac{1}{p}}}=n^{\frac{1}{p}-\frac {1}{2}} \quad \mbox{for } 1< p \leq2, $$
$$ L_{n}^{\mathrm{ch}}(\mathcal{T})\succeq n^{\vert \frac{1}{2}-\frac{1}{p}\vert } \quad \mbox{for } 1< p< \infty. $$
(2.20)
$$\sup_{n}\Vert S_{n}\Vert < \infty, $$
Secondly, if n is an odd number. For \(n>1\), we define
An argument similar to (2.19) gives
where we choose ε as above.
$$A=\{k: 1\leq k\leq n\} \quad \mbox{and}\quad B=\{k: n< k\leq2n+1\}. $$
$$ \Vert {\mathbf{1}}_{\varepsilon A}\Vert _{p}=\Vert \mathcal{R}_{\frac{n-1}{2}} \Vert _{p}\asymp (n-1)^{\frac{1}{2}}\asymp n^{\frac{1}{2}}, $$
(2.21)
On the other hand, it is clear that
and from (2.14) and the triangle inequality
we have
Combining (2.2), (2.21), (2.22) and using a similar argument to above, we also prove the inequality (2.20). So we complete the proof for \(1< p<\infty\).
$${\bf 1}_{B}(x)=\mathcal{D}_{n}(x)-\mathcal{D}_{\frac{n-1}{2}}(x), $$
$$ \begin{aligned} (n-1)^{1-\frac{1}{p}}&\preceq \Vert \mathcal {D}_{n-1}\Vert _{p}-\Vert \mathcal{D}_{\frac{n-1}{2}} \Vert _{p} \\ &\leq \Vert {\mathbf{1}}_{B}\Vert _{p} \\ &\leq \Vert \mathcal{D}_{n-1}\Vert _{p}+\Vert \mathcal{D}_{\frac {n-1}{2}}\Vert _{p} \\ &\preceq (n-1)^{1-\frac{1}{p}}, \end{aligned} $$
$$ \Vert {\mathbf{1}}_{B}\Vert _{p}\asymp(n-1)^{1-\frac {1}{p}}\asymp n^{1-\frac{1}{p}}. $$
(2.22)
For \(p=\infty\), if we take \(A, B\) and ε as above, then from Theorem 2.2, (2.14) and (2.17), we obtain
where we use the well-known relation \(\Vert S_{n}\Vert _{L_{\infty}\rightarrow L_{\infty}}\preceq \ln n\) for all \(n\geq2\) (see [17]).
$$L_{n}^{\mathrm{ch}}(\mathcal{T})\geq\frac{1}{1+\Vert S_{n}\Vert _{L_{\infty }\rightarrow L_{\infty}}} \frac{\Vert {\mathbf{1}}_{B}\Vert _{\infty}}{ \Vert {\mathbf{1}}_{\varepsilon A}\Vert _{\infty}}\succeq \frac{n^{\frac{1}{2}}}{\ln n}, $$
Now we adopt a different approach to remove the factor lnn. Let \(T(n)\) denote the space of trigonometric polynomials of degree n, which consists of functions of the form
For \(n\geq1\), we define the class of sparse trigonometric polynomials
and the function
which can be rewritten as
It is clear that \(\{1, 2, \ldots, n\}\in\mathcal{G}(f_{n}, n)\). Let \(CG_{n}(f_{n})\) be the best approximant to \(f_{n}\) in \(L_{\infty}(\mathbb{T})\) from \(\operatorname{span}\{\cos x, \cos2x, \ldots, \cos nx\}\). From the relationship
which holds for any \(1< q<\infty\), and the Hölder inequality, we have
where \(\frac{1}{q}+\frac{1}{q'}=1\). It is easy to check that
Thus, we obtain
Letting \(q'\rightarrow\infty\), we have
To estimate the upper bound of \(\sigma_{n}(f_{n})_{\infty}\), we invoke the following inequality, which holds for any \(m>n\) and \(g\in A^{1}( T(m))\):
which was proved by DeVore and Temlyakov in [18] with the help of Gluskin’s theorem [19]. Notice that \(f_{n}\in A^{1}(T(2n))\), we have by (2.23), with \(m=2n\),
Hence
Thus the proof of Theorem 2.1 is completed. □
$$T(x)=\sum_{\vert k\vert \leq n}c_{k}(T)e^{ikx},\quad k\in \mathbb{Z}. $$
$$A^{1}\bigl( T(n)\bigr):=\biggl\{ T\in T(n): \sum _{\vert k\vert \leq n}\bigl\vert c_{k}(T)\bigr\vert \leq1\biggr\} $$
$$f_{n}(x):=\frac{1}{4n}\sum_{1\leq \vert k\vert \leq2n} \biggl(1-\frac {\vert k\vert }{4n} \biggr)e^{ikx}, $$
$$f_{n}(x)=\frac{1}{2n}\sum_{k=1}^{2n} \biggl(1-\frac{k}{4n} \biggr)\cos kx. $$
$$\Biggl\Vert \sum_{k=n+1}^{2n}\cos kx\Biggr\Vert _{q}\asymp n^{1-\frac{1}{q}}, $$
$$ \begin{aligned} \Biggl\langle f_{n}, \sum _{k=n+1}^{2n}\cos kx\Biggr\rangle &= \Biggl\langle f_{n}-CG_{n}(f_{n}), \sum _{k=n+1}^{2n}\cos kx\Biggr\rangle \\ &\leq \bigl\Vert f_{n}-CG_{n}(f_{n})\bigr\Vert _{q'}\cdot\Biggl\Vert \sum_{k=n+1}^{2n} \cos kx\Biggr\Vert _{q } \\ &\asymp n^{ \frac{1}{q'}} \bigl\Vert f_{n}-CG_{n}(f_{n}) \bigr\Vert _{q'}, \end{aligned} $$
$$\Biggl\langle f_{n}, \sum_{k=n+1}^{2n} \cos kx\Biggr\rangle \asymp1. $$
$$\bigl\Vert f_{n}-CG_{n}(f_{n})\bigr\Vert _{q'} \succeq\frac{1}{n^{\frac{1}{q'}}}. $$
$$\bigl\Vert f_{n}-CG_{n}(f_{n})\bigr\Vert _{\infty} \succeq1. $$
$$ \sigma_{n}(g, \mathcal{T})_{\infty}\leq Cn^{-\frac{1}{2}} \biggl(1+\ln\frac{m}{n} \biggr)^{\frac{1}{2}}, $$
(2.23)
$$\sigma_{n}(f_{n}, \mathcal{T})_{\infty}\leq C n^{-\frac{1}{2}}(1+\ln 2)^{\frac{1}{2}}\preceq n^{-\frac{1}{2}}. $$
$$L_{n}^{\mathrm{ch}}(\mathcal{T})\geq \frac{\Vert f_{n}-CG_{n}(f_{n})\Vert _{\infty}}{{\sigma_{n}(f_{n}},\mathcal {T})_{\infty}}\succeq {n^{\frac{1}{2}}}. $$
Remark
When \(p=1\), we have, for \(n\geq2\),
As in the proof of Theorem 2.1, we can derive the inequality
Choosing A, B and \(\{\varepsilon_{i}\}\) as above, from (2.15) and (2.17) we have
Moreover it is well known that \(\Vert S_{n}\Vert _{L_{1}\rightarrow L_{1}} \preceq \ln n\), see for instance [16], Theorem 7.4.1, so we get the inequality (2.24).
$$ L_{n}^{\mathrm{ch}}(\mathcal{T})\succeq \frac{n^{\frac{1}{2}}}{(\ln n)^{2}}. $$
(2.24)
$$L_{n}^{\mathrm{ch}}\geq \frac{\Vert \sum_{i \in A}\varepsilon_{i}e_{i}\Vert _{1}}{\Vert S_{n}\Vert _{L_{1}\rightarrow L_{1}} \Vert \sum_{i\in B}e_{i}\Vert _{1}}. $$
$$\biggl\Vert \sum_{i \in A}\varepsilon_{i}e_{i} \biggr\Vert _{1}\asymp n^{\frac{1}{2}},\quad \mbox{and} \quad \biggl\Vert \sum _{i\in B}e_{i}\biggr\Vert _{1} \preceq\ln n. $$
An interesting and challenging problem is whether the inequality (2.24) can be improved to
$$L_{n}^{\mathrm{ch}}(\mathcal{T}) \succeq n^{\frac{1}{2}}. $$
The second example is the summing basis. Let X be the real Banach space of all sequences \({\boldsymbol{\alpha}} = (a_{n})_{n \in \mathbb{N}}\) with
The standard canonical basis \(\{e_{n}, e_{n}^{*}\}\) satisfies \(\Vert e_{m}\Vert \equiv1, \Vert e_{1}^{*}\Vert =1\) and \(\Vert e_{n}^{*}\Vert =2\) if \(n \geq2\), which is called the summing basis. It is known from [20], Proposition 5.1, for this basis, that \(g_{n}=2n\). So this is a non-quasi-greedy basis of X. For this basis, we can give the estimate for \(L_{n}^{\mathrm{ch}}(\Psi)\) by directly computing.
$$\Vert {\boldsymbol{\alpha}} \Vert :=\sup_{M \geq1}\Biggl\vert \sum _{n=1}^{M}a_{n}\Biggr\vert \leq \infty. $$
Theorem 2.4
For the summing basis
\(\Psi=\{e_{n}\}\)
of the real Banach space
X, we have, for
\(n \geq1\),
$$1+2n\leq\widetilde{L}_{n}^{\mathrm{ch}}\leq1+4n \leq L_{n}^{\mathrm{ch}} \leq1+6n. $$
Proof
We begin with the lower estimate of \(\widetilde{L}_{n}^{\mathrm{ch}}\). We first pick the vector
Then \(\Gamma=\{n: x_{n}=-2\} \in\mathcal{G}(x, n)\). Let \(CG_{n}(x)\) be the Chebyshev approximant of x which is supported on Γ, we have, for some \(\{a_{i}\}_{i=1}^{n}\subset\mathbb{R}^{n}\),
Notice that
and
we have
$$x=(\overbrace{0, 2, 0}; \overbrace{0, 2, 0}; \ldots; \overbrace{0, 2, 0}; 1; \overbrace{-2, 2}, \overbrace{-2, 2}, \ldots, \overbrace{-2, 2}, 0, 0, \ldots). $$
$$x-CG_{n}(x)=(\overbrace{0, 2, 0}; \overbrace{0, 2, 0}; \ldots; \overbrace{0, 2, 0}; 1; a_{1}, 2, a_{2}, 2, \ldots, a_{n}, 2, 0, 0, \ldots). $$
$$ \begin{aligned}\bigl\Vert x-CG_{n}(x)\bigr\Vert &= \bigl\Vert (\overbrace{0, 2, 0}; \overbrace{0, 2, 0}; \ldots; \overbrace {0, 2, 0}; 1; a_{1}, 2, a_{2}, 2, \ldots, a_{n}, 2, 0, 0,\ldots)\bigr\Vert \\ &\geq \bigl\Vert (\overbrace{0, 2, 0}; \overbrace{0, 2, 0}; \ldots; \overbrace {0, 2, 0}; 1; 0, 0, 0, 0, \ldots, 0, 0, 0, 0,\ldots)\bigr\Vert \\ & = 2n+1 \end{aligned} $$
$$ \begin{aligned}\widetilde{\sigma}_{n}(x)&\leq \bigl\Vert x-(\overbrace{0, 2, 0}; \ldots;\overbrace{0, 2, 0}; 0, 0, \ldots) \bigr\Vert \\ &=\Vert(0, 0, 0; \ldots; 0, 0, 0;1, \overbrace{-2, 2}, \overbrace{-2, 2}, \ldots,\overbrace{-2,2},0,0,\ldots) \\ &=1, \end{aligned} $$
$$\widetilde{L}_{n}^{\mathrm{ch}}\geq\frac{\Vert x-CG_{n}(x)\Vert }{\widetilde {\sigma}_{n}(x)} \geq 2n+1. $$
For the upper estimate of \(\widetilde{L}_{n}^{\mathrm{ch}}\), we use the notation and the method in the proof of Theorem 3.5 of the next section and notice that (3.6) can be bounded by
which implies \(k_{n}\leq2n\), we have
where \(g_{n}^{c}=\sup_{G \in\bigcup_{k \leq n}\mathcal{G}_{k}}\Vert I-G\Vert \) and \(k_{n}^{c}=\sup_{\#(A)\leq n}\Vert I-P_{A}\Vert \).
$$\begin{aligned} \begin{aligned} \biggl\Vert \sum _{i\in A\setminus\Gamma}e_{i}^{*}(x)e_{i}\biggr\Vert &\leq \sum_{i\in A\setminus\Gamma}\bigl\vert e_{i}^{*}(x)\bigr\vert \Vert e_{i}\Vert \\ &\leq\sup_{i\in A\setminus\Gamma} \Vert e_{i}\Vert \sum _{\Gamma \setminus A}\bigl\vert e_{i}^{*}(x)\bigr\vert \\ &\leq\sup_{i}\Vert e_{i}\Vert \sum _{\Gamma\setminus A}\bigl\vert e_{i}^{*}(x-P_{A}x) \bigr\vert \\ &\leq\sup_{i}\Vert e_{i}\Vert \bigl\Vert e_{i}^{*}\bigr\Vert \vert \Gamma\setminus A\vert \Vert x-P_{A}x\Vert \\ &\leq 2n\Vert x-P_{A}x\Vert , \end{aligned} \end{aligned}$$
$$\widetilde{L}_{n}^{\mathrm{ch}}\leq g_{n}^{c}+2n \leq k_{n}^{c}+2n\leq1+k_{n}+2n\leq 1+4n, $$
Now we turn to the estimate of \(L_{n}^{\mathrm{ch}}\). It is known from Proposition 5.1 in [20] that \(L_{n} \leq1+6n\), hence the upper bound follows from this inequality and the trivial inequality \(L_{n}^{\mathrm{ch}}\leq L_{n}\). For the estimate of the lower bound, let
Similar to the proof of the first inequality, for \(\Gamma=\{n: x_{n}=-1\} \in\mathcal{G}(x, n)\), there exists some \(\{b_{i}\}_{i=1}^{n}\subset\mathbb{R}^{n}\) such that, for the Chebyshev approximant \(CG_{n}(x)\) of x which is supported on Γ,
Using the definition of the norm, we have
Notice that
so we conclude
$$x=\biggl(\overbrace{\frac{1}{2}, 1, \frac{1}{2}}; \overbrace{ \frac{1}{2}, 1, \frac{1}{2}}; \ldots; \overbrace{\frac{1}{2}, 1, \frac{1}{2}}; \frac{1}{2}; \overbrace{-1, 1}, \overbrace{-1, 1}, \ldots, \overbrace{-1, 1}, 0, 0, \ldots\biggr). $$
$$x-CG_{n}(x)=\biggl(\overbrace{\frac{1}{2}, 1, \frac{1}{2}}; \overbrace{\frac{1}{2}, 1, \frac{1}{2}}; \ldots; \overbrace{\frac{1}{2}, 1, \frac{1}{2}}; \frac{1}{2}; \overbrace{b_{1}, 1}, \overbrace{b_{2}, 1}, \ldots, \overbrace{b_{n}, 1}, 0, 0, \ldots\biggr). $$
$$\bigl\Vert x-CG_{n}(x)\bigr\Vert \geq2n+\frac{1}{2}. $$
$$\sigma_{n}(x)\leq\bigl\Vert x-2(\overbrace{0, 1, 0}; \ldots; \overbrace {0,1, 0}; 0, 0, \ldots)\bigr\Vert =\frac{1}{2}, $$
$$L_{n}^{\mathrm{ch}}\geq \frac{\Vert x-CG_{n}(x)\Vert }{\sigma_{n}(x)} \geq1+4n. $$
The proof of Theorem 2.4 is completed. □
3 Chebyshevian Lebesgue constants for general bases
In this section, we obtain some upper bounds for \(L_{n}^{\mathrm{ch}}(\Psi)\) and \(\widetilde{L}_{n}^{\mathrm{ch}}(\Psi)\) with respect to a general basis Ψ in a Banach space X. These bounds are given in terms of the following quantities, which have been defined in [20].
Let \(\mathcal{G}=\bigcup_{n\geq1}\mathcal{G}_{n}\). Given \(G, G^{'}\in \mathcal{G}\) we shall write \(G^{'}< G\) whenever \(G\in\mathcal{G}_{n}\) and \(G^{'}\in\mathcal{G}_{m}\) with \(m< n\) and \(\{\rho(1), \ldots, \rho(m)\}\subset \{\rho(1), \ldots, \rho(n)\}\). Now we introduce the following parameters. where \(\vert x\vert _{\infty} =\sup_{i}\vert e_{i}^{*}(x)\vert \), \(\operatorname{supp}{x}=\{i\in \mathbb{N}: e_{i}^{*}(x)\neq0\}\), and \(A\mathbin{\dot{\cup}}B\mathbin{\dot{\cup}}x\) means that A, B and suppx are pairwise disjoint.
-
Quasi-greedy parameters:$$\widetilde{g}_{n}=\sup_{{G \in\bigcup_{k \leq n}\mathcal{G}_{k}}, G^{'}< G}\bigl\Vert G-G'\bigr\Vert . $$
-
Super-democracy parameters and their counterparts for disjoint sets:$$\widetilde{\mu}_{n}=\sup_{{\#(A)=\#(B)\leq n, {\boldsymbol{\varepsilon}},{\boldsymbol{\eta}}\in \Upsilon}}\frac{\Vert {\mathbf{1}}_{{\boldsymbol{\varepsilon}}A} \Vert }{\Vert {\mathbf{1}}_{{\boldsymbol{\eta}}B} \Vert } \quad \mbox{and}\quad \widetilde{\mu}_{n}^{d}=\sup_{{\#(A)=\#(B)\leq n, A\cap B=\emptyset, {\boldsymbol{\varepsilon}},{\boldsymbol{\eta}}\in \Upsilon}} \frac{\Vert {\mathbf{1}}_{{\boldsymbol{\varepsilon}}A} \Vert }{\Vert {\mathbf{1}}_{{\boldsymbol{\eta}}B} \Vert }. $$
-
A-property parameters:$$\nu_{n}=\sup\biggl\{ \frac{\Vert {\mathbf{1}}_{\varepsilon A}+x\Vert }{\Vert {\mathbf{1}}_{\eta B}+x\Vert }: \#(A)=\#(B)\leq n, \varepsilon, \eta\in \Upsilon, \vert x\vert _{\infty}\leq1, A\mathbin{\dot{\cup}}B \mathbin{\dot{\cup}}x\biggr\} , $$
To prove our results, we shall develop the technique used in the proof of Theorem 1.3. We will make use of the properties of the truncation operators defined below.
For any \(z \in\mathbb{C}\), we set
And for each \(\alpha>0\), we define the α-truncation of z by
We extend \(T_{\alpha}\) to an operator in X by
where \(\Lambda_{\alpha}=\{n: \vert e_{i}^{*}(x)\vert >\alpha\}\). The sum above converges, since \(\Lambda_{\alpha}\) is finite. Moreover, we notice that \(T_{\alpha}(x)\) has the property, for every \(i, e_{i}^{*}(T_{\alpha}(x))=T_{\alpha}(e_{i}^{*}(x))\).
$$\operatorname{sign}z = \textstyle\begin{cases} z/\vert z\vert & \mbox{if }\vert z\vert \neq0,\\ 0 & \mbox{if }z=0. \end{cases} $$
$$T_{\alpha}(z) = \textstyle\begin{cases} \alpha \operatorname{sign}(z) & \mbox{if }\vert z\vert > \alpha,\\ z & \mbox{if }\vert z\vert \leq\alpha. \end{cases} $$
$$T_{\alpha}(x)=\sum_{i}T_{\alpha} \bigl(e_{i}^{*}(x)\bigr)e_{i}=\sum _{i \in \Lambda_{\alpha}}\alpha \frac{e_{i}^{*}(x)}{\vert e_{i}^{*}(x)\vert }e_{i}+\sum _{i \notin\Lambda_{\alpha}}e_{i}^{*}(x)e_{i}, $$
We also need the following lemmas.
Lemma 3.1
([20])
If
\(x \in X\)
and
\({\boldsymbol{\varepsilon}}=\{\operatorname{sign}e_{i}^{*}(x)\}\), then
$$\min _{\Lambda}\bigl\vert e_{i}^{*}(x)\bigr\vert \Vert {\mathbf{1}}_{\boldsymbol{\varepsilon}\Lambda} \Vert \leq\widetilde{g}_{n} \Vert x \Vert ,\quad \forall\Lambda\in\mathcal{G}(x, n). $$
Lemma 3.2
([20])
Let
\(x \in X\)
and
\(\alpha\geq \max \vert e_{i}^{*}(x)\vert \). Then
and for all
B
and
z
such that
\(\#(\operatorname{supp}z)\leq\#(B) \leq n\), \(B\mathbin{\dot{\cup}} x\mathbin{\dot{\cup}} z\)
and
\(\vert z\vert _{\infty}\leq\alpha\).
$$\Vert x+z\Vert \leq\nu_{n} \Vert x+\alpha{\mathbf{1}}_{\eta B} \Vert ,\quad \forall \eta\in\Upsilon, $$
The following lemma is a special case of Lemma 3.2 for \(x=0\).
Lemma 3.3
Let
\(z \in X\)
and
\(B \subset \mathbb{N}\)
such that
\(\#(\operatorname{supp}z) \leq\#(B) \leq n, B\mathbin{\dot{\cup}}z\). Then
$$\Vert z\Vert \leq\widetilde{\mu}_{n}^{d} \max\bigl\vert e_{i}^{*}(z)\bigr\vert \Vert {\mathbf{1}}_{{\boldsymbol{\eta}}B} \Vert , \quad \forall \boldsymbol{\eta} \in\Upsilon. $$
Now we present our results.
Theorem 3.4
If Ψ is a basis for a Banach space
X, then for all
\(n \geq1\)
we have
$$L_{n}^{\mathrm{ch}}(\Psi)\leq g_{2n}^{c}+2 \nu_{n}\widetilde{g}_{n} \quad \textit{and}\quad \widetilde{L}_{n}^{\mathrm{ch}}( \Psi)\leq g_{n}^{c}+\nu_{n}\widetilde{g}_{n}. $$
Proof
For \(x \in X\) let \(a_{i}=e_{i}^{*}(x)\), and \(\Gamma\in \mathcal{G}(x, n)\). Fix \(\epsilon>0\), pick \(z=\sum_{A}b_{i}e_{i} \in \Sigma_{n}\) with \(\operatorname{supp}(z) \subset A\) and \(\#(A)=\#(\Gamma)=n\) such that \(\Vert x-z\Vert <\sigma_{n}(x)+\epsilon\). Set \(y:=x-z=\sum_{i}y_{i}e_{i}\), for which
$$y_{i} = \textstyle\begin{cases} a_{i}, & i\notin A,\\ a_{i}-b_{i}, & i \in A. \end{cases} $$
Let \(\alpha=\max_{\Gamma^{c}}\vert e_{i}^{*}(x)\vert \). We define
For the first term of the above equality, we have
Note that, for \(i \notin A\), \(y_{i}=a_{i}\), and for \(i \notin\Gamma\), \(\vert a_{i}\vert \leq\alpha\), we have, for \(i \in A^{c} \cap\Gamma^{c}\), \(T_{\alpha}(y_{i})=T_{\alpha}(a_{i})=a_{i}\). hence \(\vert X\vert _{\infty }=0\). And for \(Z=P_{A \backslash\Gamma}(x-T_{\alpha}(y))\), we have
and \(\#(\operatorname{supp}Z)\leq\#(A\backslash\Gamma)=\#(\Gamma\backslash A)\leq n\), applying Lemma 3.2 with \(\eta=\{\operatorname{sign}e_{i}^{*}(y)\}\), to obtain
where we choose Γ̃ as a set of cardinality \(\#(\widetilde{\Gamma})=\#(\Gamma\backslash A)=\#(A \backslash \Gamma)\) in which \(x-z\) attains the largest coefficients, and hence \(\widetilde{\Gamma} \in\mathcal{G}(y, n)\). Using (3.2) and Lemma 3.1, we conclude
For the second term of (3.1), [20], Lemma 2.3, gives
where \(\Lambda_{\alpha}=\{i: \vert e_{i}^{*}(y)\vert >\alpha\}\).
$$ \omega=x-P_{\Gamma}\bigl(x-T_{\alpha}(y)\bigr)=(I-P_{\Gamma}) \bigl(x-T_{\alpha}(y)\bigr)+T_{\alpha}(y). $$
(3.1)
$$ \begin{aligned}(I-P_{\Gamma}) \bigl(x-T_{\alpha}(y) \bigr)&=(I-P_{\Gamma \cup A}) \bigl(x-T_{\alpha}(y)\bigr)+P_{A \backslash \Gamma} \bigl(x-T_{\alpha}(y)\bigr):=X+Z. \end{aligned} $$
$$\vert Z\vert _{\infty}=\max_{i\in A \backslash \Gamma}\bigl\vert e_{i}^{*}\bigl(x-T_{\alpha}(y)\bigr)\bigr\vert =\max _{i\in A \backslash \Gamma}\bigl(\vert a_{i}\vert +\bigl\vert T_{\alpha}(y_{i})\bigr\vert \bigr)\leq2\alpha $$
$$\begin{aligned} \bigl\Vert (I-P_{\Gamma}) \bigl(x-T_{\alpha}(y)\bigr)\bigr\Vert \leq& \nu_{n}\bigl\Vert (I-P_{\Gamma\cup A}) \bigl(x-T_{\alpha}(y)\bigr)+2\alpha{ \mathbf{1}}_{\eta \widetilde{\Gamma}}\bigr\Vert \\ =&2\nu_{n} \max_{\Gamma^{c}}\bigl\vert e_{i}^{*}(x)\bigr\vert \Vert {\mathbf{1}}_{\eta\widetilde {\Gamma}} \Vert \\ \leq& 2\nu_{n} \min_{\Gamma\backslash A}\bigl\vert e_{i}^{*}(y)\bigr\vert \Vert {\mathbf{1}}_{\eta\widetilde{\Gamma }}\Vert \\ \leq&2\nu_{n} \min_{\widetilde{\Gamma}}\bigl\vert e_{i}^{*}(y)\bigr\vert \Vert {\mathbf{1}}_{\eta\widetilde{\Gamma}} \Vert , \end{aligned}$$
(3.2)
$$ \bigl\Vert (I-P_{\Gamma}) \bigl(x-T_{\alpha}(y)\bigr)\bigr\Vert \leq 2\nu_{n} \widetilde{g}_{n} \Vert y\Vert . $$
(3.3)
$$\bigl\Vert T_{\alpha}(y)\bigr\Vert \leq g_{\#(\Lambda_{\alpha})}^{c} \Vert y\Vert , $$
Since
we have
$$ \Lambda_{\alpha}=(\Lambda_{\alpha}\cap A) \cup \bigl( \Lambda_{\alpha}\cap A^{c}\bigr)=\bigl\{ i\in A: \vert y_{i}\vert > \alpha\bigr\} \cup \bigl\{ i \in A^{c}: \vert a_{i}\vert > \alpha\bigr\} \subset A \cup\Gamma, $$
(3.4)
$$\Vert \omega \Vert \leq\bigl(g_{\#(A \cup\Gamma)}^{c}+2\nu_{n} \widetilde{g}_{n}\bigr) \Vert y\Vert \leq\bigl(g_{2n}^{c}+2 \nu_{n} \widetilde{g}_{n}\bigr) \Vert y\Vert . $$
Since \(x-\omega=P_{\Gamma}(x-T_{\alpha}(y))\), clearly \(\operatorname{supp}(x-\omega)\subset\Gamma\). We have
Letting \(\epsilon\rightarrow0\), we conclude that
$$\bigl\Vert x-CG_{n}(x)\bigr\Vert \leq \Vert \omega \Vert \leq \bigl(g_{2n}^{c}+2\nu _{n} \widetilde{g}_{n} \bigr) \Vert x-z\Vert \leq \bigl(g_{2n}^{c}+2 \nu_{n}\widetilde{g}_{n}\bigr) \bigl(\sigma_{n}(x)+ \epsilon\bigr). $$
$$L_{n}^{\mathrm{ch}}\leq g_{2n}^{c}+2 \nu_{n}\widetilde{g}_{n}. $$
The estimate for \(\widetilde{L}_{n}^{\mathrm{ch}}\) is similar: we choose an index set A and let \(\widetilde{\Gamma}=\Gamma\backslash A\), for which \(\Vert x-P_{A}x\Vert \leq\widetilde{\sigma}_{n}(x)+\epsilon\). For \(y=x-P_{A}x\),
Now we estimate ω. On one hand, we have
hence
$$y_{i} = e_{i}^{*}(y)=\textstyle\begin{cases} a_{i}, & i\notin A,\\ 0, & i \in A. \end{cases} $$
$$\Lambda_{\alpha}=\bigl\{ i:\bigl\vert e_{i}^{*}(y)\bigr\vert > \alpha\bigr\} \subset\bigl\{ i:\vert a_{i}\vert >\alpha\bigr\} \subset \Gamma, $$
$$ \bigl\Vert T_{\alpha}(y)\bigr\Vert \leq g_{\#(\Lambda_{\alpha})}^{c} \Vert y\Vert \leq g_{n}^{c} \Vert y\Vert . $$
(3.5)
On the other hand, note that \(\widetilde{\Gamma}=\Gamma\backslash A \in\mathcal{G}(x-P_{A}x, \#(\Gamma\backslash A))\). We have the same inequalities as (3.2) and (3.3) for \(\eta=\{\operatorname{sign}e_{i}^{*}(x-P_{A}x)\}\). Thus,
The proof of Theorem 3.4 is completed. □
$$\widetilde{L}_{n}^{\mathrm{ch}} \leq g_{n}^{c}+ \nu_{n} \widetilde{g}_{n}. $$
Next, we replace \(\nu_{n}\) by \(\widetilde{\mu}_{n}^{d}\) in Theorem 3.4 and get another estimate for the upper bounds.
Theorem 3.5
If Ψ is a basis for a Banach space
X, then for all
\(n \geq1\)
we have
$$L_{n}^{\mathrm{ch}}(\Psi)\leq g_{2n}^{c}+2 \widetilde{\mu}_{n}^{d}\widetilde{g}_{n} \quad \textit{and}\quad \widetilde{L}_{n}^{\mathrm{ch}}(\Psi) \leq g_{n}^{c}+\widetilde{\mu}_{n}^{d} \widetilde{g}_{n}. $$
Proof
With the same notation as in Theorem 3.4, we now deal with \(P_{\Gamma^{c}}(x-T_{\alpha}(y))\). It is clear that
Now choose \(\eta=\{\operatorname{sign}e_{i}^{*}(x-z)\}\), \(B=\Gamma\setminus A\). Notice that \(\operatorname{supp}(\sum_{A\backslash \Gamma}(a_{i}-T_{\alpha}(y_{i}))e_{i})\mathbin{\dot{\cup}}B\),
by Lemma 3.3, we can continue the estimate
Then inequality (3.6) and Lemma 3.1 give
Using (3.4), (3.5) and (3.7), we have
Letting \(\epsilon\rightarrow0\), we conclude \(L_{n}^{\mathrm{ch}}\leq g_{2n}^{c}+2 \widetilde{\mu}_{n}^{d} \widetilde{g}_{n}\). The estimate of \(\widetilde{L}_{n}^{\mathrm{ch}}\) can be obtained in a similar way. We complete the proof. □
$$P_{\Gamma^{c}}\bigl(x-T_{\alpha}(y)\bigr)=\sum _{A\backslash\Gamma}\bigl(a_{i}-T_{\alpha}(y_{i}) \bigr)e_{i}. $$
$$\#\biggl(\operatorname{supp}\biggl(\sum_{A\backslash\Gamma} \bigl(a_{i}-T_{\alpha}(y_{i})\bigr)e_{i} \biggr)\biggr) \leq\#(A \backslash\Gamma)=\#(B)\leq n\quad \mbox{and}\quad B \in \mathcal{G}\bigl(P_{A^{c}}(x-z), n\bigr), $$
$$\begin{aligned} \biggl\Vert \sum _{A \backslash\Gamma}\bigl(a_{i}-T_{\alpha}(y_{i})\bigr)e_{i}\biggr\Vert \leq&\widetilde{ \mu}_{n}^{d} \max_{A \backslash\Gamma}\bigl\vert a_{i}-T_{\alpha}(y_{i})\bigr\vert \Vert {\mathbf{1}}_{\eta B}\Vert \\ \leq&\widetilde{\mu}_{n}^{d} \max_{A \backslash \Gamma} \bigl(\vert a_{i}\vert +\bigl\vert T_{\alpha}(y_{i}) \bigr\vert \bigr) \Vert {\mathbf{1}}_{\eta B}\Vert \\ \leq &2 \alpha \widetilde{\mu}_{n}^{d} \Vert { \mathbf{1}}_{\eta B}\Vert \\ \leq&2 \max_{\Gamma^{c}}\vert a_{i}\vert \widetilde{ \mu}_{n}^{d} \Vert {\mathbf{1}}_{\eta B}\Vert . \end{aligned}$$
(3.6)
$$\begin{aligned} \bigl\Vert P_{\Gamma^{c}} \bigl(x-T_{\alpha}(y)\bigr)\bigr\Vert \leq&2 \max_{\Gamma^{c}} \vert a_{i}\vert \widetilde{\mu}_{n}^{d} \Vert {\mathbf{1}}_{\eta B}\Vert \\ \leq&2 \min_{\Gamma\setminus A}\bigl\vert e_{i}^{*}(x-z)\bigr\vert \widetilde{\mu}_{n}^{d} \Vert {\mathbf{1}}_{\eta B} \Vert \\ \leq& 2 \widetilde{\mu}_{n}^{d} \widetilde{g}_{n} \bigl\Vert P_{A^{c}}(x-z)\bigr\Vert \\ \leq &2 \widetilde{\mu}_{n}^{d} \widetilde{g}_{n} \Vert x-z\Vert . \end{aligned}$$
(3.7)
$$\bigl\Vert x-CG_{n}(x)\bigr\Vert \leq \Vert \omega \Vert \leq \bigl(g_{2n}^{c}+2 \widetilde{\mu}_{n}^{d} \widetilde{g}_{n}\bigr) \Vert x-z\Vert \leq\bigl(g_{2n}^{c}+2 \widetilde{\mu}_{n}^{d} \widetilde{g}_{n}\bigr) \bigl(\sigma_{n}(x)+\epsilon\bigr). $$
The following corollary shows the upper bounds in Theorem 3.5 are asymptotically optimal for quasi-greedy bases from the viewpoint of convergence rate.
Corollary 3.6
If Ψ is a quasi-greedy basis of a Banach space
X
with the quasi-greedy constant
\(\mathfrak{K}\), then
with
\(\kappa=1\)
or 2 for the real or complex spaces, respectively.
$$L_{n}^{\mathrm{ch}}(\Psi) \leq20\kappa^{2} \mathfrak{K}^{2}\mu_{n}^{d} \quad \textit{and}\quad \widetilde{L}_{n}^{\mathrm{ch}}(\Psi) \leq12\kappa^{2} \mathfrak{K}^{2}\mu_{n}^{d}, $$
Proof
We need the additional inequality
where \(\gamma_{n} =\sup\{\frac{\Vert {\mathbf{1}}_{\varepsilon B}\Vert }{ \Vert {\mathbf{1}}_{\varepsilon A}\Vert }: B \subset A, \#(A) \leq n, \varepsilon\in\Upsilon\}\), to pass from \(\widetilde{\mu}_{n}^{d}\) to \(\mu_{n}^{d}\). The proof of this inequality is almost the same as Lemma 3.4 in [20]. It is clear that, for a quasi-greedy basis \((e_{i})\) with the quasi-greedy constant \(\mathfrak{K}\), \(\gamma_{n} \leq g_{n} \leq\mathfrak{K}\).
$$\widetilde{\mu}_{n}^{d} \leq 4\kappa^{2} \gamma_{n} \mu_{n}^{d}, $$
By Theorems 3.5 and the above inequality, we have
where we have used \(\widetilde{g}_{n} \leq2\mathfrak{K}\) in the last inequality. Note that \(1\leq\mu_{n}\leq2\mathfrak{K}\mu_{n}^{d}\) and \(1\leq\mathfrak{K}\), thus \(\mathfrak{K}\leq2\mathfrak{K}^{2}\mu_{n}^{d}\) and we get the first conclusion
$$ \begin{aligned} L_{n}^{\mathrm{ch}}&\leq g_{2n}^{c}+2\widetilde{\mu}_{n}^{d} \widetilde {g}_{n} \\ &\leq1+\mathfrak{K}+8\kappa^{2}\gamma_{n} \mu_{n}^{d} \widetilde{g}_{n} \\ &\leq 1+\mathfrak{K}+16\kappa^{2} {\mathfrak{K}}^{2} \mu_{n}^{d}, \end{aligned} $$
$$ \begin{aligned} L_{n}^{\mathrm{ch}}&\leq 1+ \mathfrak{K}+16\kappa^{2} {\mathfrak{K}}^{2} \mu_{n}^{d} \\ &\leq 2\mathfrak{K}+16\kappa^{2} {\mathfrak{K}}^{2} \mu_{n}^{d} \\ &\leq20\kappa^{2} {\mathfrak{K}}^{2}\mu_{n}^{d}. \end{aligned} $$
Similarly for \(\widetilde{L}_{n}^{\mathrm{ch}}\), we have
The proof of this corollary is completed. □
$$ \begin{aligned} \widetilde{L}_{n}^{\mathrm{ch}}& \leq g_{n}^{c}+\widetilde{\mu}_{n}^{d} \widetilde{g}_{n} \\ &\leq1+\mathfrak{K}+4\kappa^{2}\gamma_{n} \mu_{n}^{d} \widetilde{g}_{n} \\ &\leq 1+\mathfrak{K}+8\kappa^{2} {\mathfrak{K}}^{2} \mu_{n}^{d} \\ &\leq12\kappa^{2} {\mathfrak{K}}^{2}\mu_{n}^{d}. \end{aligned} $$
From the definition of quasi-greedy constant, we know \(\mathfrak{K}\geq1\). Moreover, it is shown in [5] that if the quasi-greedy constant \(\mathfrak{K}= 1\), then Ψ is an unconditional basis. So compared to \(O(\mathfrak{K}^{3})\) in the upper bounds of Theorem 1.3, our results improve the implicit constants for all conditional quasi-greedy bases since in this case \(\mathfrak{K}>1\).
Using Lemma 3.3 and the approach we adopt in the proof of Theorem 3.5, we obtain the following results for TGA.
Theorem 3.7
For all
\(n \geq1\),
$$L_{n}(\Psi) \leq k_{2n}^{c}+\widetilde{g}_{n} \widetilde{\mu}_{n}^{d} \quad \textit{and}\quad \widetilde{L}_{n}( \Psi)\leq g_{n}^{c}+\widetilde{g}_{n}\widetilde{ \mu}_{n}^{d}. $$
Corollary 3.8
If Ψ is a quasi-greedy basis of a Banach space
X
with the quasi-greedy constant
\(\mathfrak{K}\), then
and
with
\(\kappa=1\)
or 2 for the real or complex spaces, respectively.
$$\max\bigl\{ k_{n}^{c}, \mu_{n}^{d}\bigr\} \leq L_{n}(\Psi)\leq k_{2n}^{c}+8 \kappa^{2}\mathfrak {K}^{2}\mu_{n}^{d} $$
$$\mu_{n}^{d}\leq\widetilde{L}_{n}(\Psi)\leq 12 \kappa^{2}\mathfrak{K}^{2}\mu_{n}^{d}, $$
Remark
In [20], Berná, Garrigós and Óscar established the inequalities:
Note that the bounds in Theorem 3.7 are slightly better since \(\widetilde{\mu}_{n}^{d}\leq \widetilde{\mu}_{n}\).
$$L_{n}(\Psi)\leq k_{2n}^{c}+\widetilde{g}_{n} \widetilde{\mu}_{n} \quad \mbox{and} \quad \widetilde{L}_{n}(\Psi)\leq g_{n}^{c}+\widetilde{g}_{n}\widetilde{ \mu}_{n}. $$
Compared Theorem 3.5 with Theorem 3.7, we replace the unconditionality parameter \(k_{n}^{c}\) by quasi-greedy parameter \(g_{n}^{c}\) in the first part of the addictive bound. In general, \(g_{n}^{c}\) is essentially smaller than \(k_{n}^{c}\), thus the estimate for \(L_{n}^{\mathrm{ch}}\) is better than the estimate for \(L_{n}\). In particular, for some conditional quasi-greedy bases, a faster convergence rate is obtained.
4 Status of the research on CTGA
In this section, we present a short overview of some results and questions on the efficiency of CTGA. We will compare the approximation properties of CTGA with TGA.
We first consider quasi-greedy bases. It is well known that, for some conditional quasi-greedy bases, CTGA has a better convergence rate than TGA, in particular, for democratic, conditional quasi-greedy basis Ψ, it follows from Theorem 3.5 that the inequality
holds for any \(x\in X\), \(n\geq1\), while for TGA with respect to these bases, from Theorem 1.5 in [20], we know that, for any \(x\in X\), \(n\geq1\), the best inequality we can get is
On the other hand, for unconditional bases, CTGA has the same rate of convergence as TGA.
$$\bigl\Vert x-CG_{n}(x,\Psi)\bigr\Vert \leq C\sigma_{n}(x, \Psi) $$
$$\bigl\Vert x-G_{n}(x,\Psi)\bigr\Vert \leq C \ln x \sigma_{n}(x,\Psi). $$
Secondly we consider non-quasi-greedy bases. From Theorem 1.1, we know that, for these bases, CTGA has better convergence properties than TGA. However, as for the convergence rate, the results obtained so far show CTGA does not make essential improvements over TGA. For example, we prove that the orders of the Lebesgue constants of CTGA with respect to the trigonometric system and the summing basis are the same as those of TGA. Moreover, for the sparse classes \(A^{1}(\mathcal{T})\) defined by
the error performance of CTGA is also the same as TGA. In fact, we know from [11] and [12] that, for \(1\leq p\leq\infty\),
$$A^{1}(\mathcal{T}):=\biggl\{ T: \sum_{k\in\mathbb{Z}} \bigl\vert c_{k}(T)\bigr\vert \leq 1\biggr\} , $$
$$\sup_{f\in A^{1}(\mathcal{T})}\bigl\Vert f-CG_{n}(f,\mathcal{T})\bigr\Vert _{p}\asymp\sup_{f\in A^{1}(\mathcal{T})}\bigl\Vert f-G_{n}(f,\mathcal {T})\bigr\Vert _{p}. $$
For the Lebesgue constant \(L_{n}^{\mathrm{ch}}\) of CTGA with respect to non-quasi-greedy bases, many problems are worthy to be studied: Can one establish some new estimates of the upper and lower bounds for the general basis, and then obtain the exact rate of \(L_{n}^{\mathrm{ch}}\) for more non-quasi-greedy bases? One more interesting question is the following: Can one find a non-quasi-greedy basis for which CTGA has an essentially smaller Lebesgue constant than TGA?
Acknowledgements
This work is supported by the National Natural Science Foundation of China [No. 11671213].
Competing interests
The authors declare that they have no competing interests.
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