On the orthogonal democratic systems in the $L^p$ spaces

TL;DR

This paper constructs specific orthonormal systems in $L^p$ spaces that exhibit bidemocratic properties depending on a parameter, revealing nuanced behavior of these systems in relation to democratic and bidemocratic criteria.

Contribution

It introduces a family of orthonormal systems with parameter-dependent bidemocratic properties for $L^p$ and $L^{p'}$ spaces, expanding understanding of democratic systems in functional analysis.

Findings

The systems are bidemocratic for $L^p$ and $L^{p'}$ when $l$ is within a certain range.

The systems are not democratic or bidemocratic outside that range.

The properties depend critically on the parameter $l$ and the space $L^p$ considered.

Abstract

The concept of bidemocratic pair for a Banach space was introduced in \cite{KS:18}. We construct a family of orthonormal systems $\mathfrak{F}_{l},$ $l\in (0,\infty)$ of functions defined on $[-1,1]$ such that the pair $(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is bidemocratic for $L^{p}[-1,1]$ and for $L^{p'}[-1,1]$ if $l\in (0, \frac{p}{2(p-2)}]$, where $p>2$ and $p'= \frac{p}{p-1}$. The system $\mathfrak{F}_{l}$ is not democratic for $L^{p'}[-1,1]$ when $l\in (\frac{p}{2(p-2)}, \frac{p}{p-2}). $ When $l> \frac{p}{2(p-2)}$ the pair $(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is not bidemocratic neither for $L^{p}[-1,1]$ nor for $L^{p'}[-1,1]$.

Equations

$$\bigg{\|}\sum_{k\in P}\frac{x_{k}}{\|x_{k}\|}\bigg{\|}\leq D\bigg{\|}\sum_{k\in Q}\frac{x_{k}}{\|x_{k}\|}\bigg{\|}.$$

$$\varphi_{X}(n)=\sup_{|P|\leq n}\bigg{\|}\sum_{k\in P}\frac{x_{k}}{\|x_{k}\|}\bigg{\|}_{\mathbb{B}},\qquad\varphi_{X^{*}}^{*}(n)=\sup_{|P|\leq n}\bigg{\|}\sum_{k\in P}\|x_{k}\|\,x^{*}_{k}\bigg{\|}_{\mathbb{B}^{*}}$$

$${\phi }_X(n){\phi }_{X^{\ast }}^{\ast }(n)\le Cn.$$

$$\Vert \varphi {\Vert }_{L^p(E)}:={\left({\int }_E|\varphi (t){|}^{p}dt\right)}^{\frac{1}{p}}<+\infty .$$

$${\tilde{\phi }}_X(n)=\underset{1\le j\le \nu }{\max }{\phi }_{X_j}(n)n\in \mathbb{N}.$$

$$\frac{1}{\nu }{\phi }_X(n)\le {\tilde{\phi }}_X(n)\le {\phi }_X(n).$$

$${\phi }_X(n)\le \sum _{j=1}^{\nu }{\phi }_{X_j}(n)\le \nu {\tilde{\phi }}_X(n).$$

$${\phi }_X(|P|){\phi }_{X^{\ast }}^{\ast }(|P|)\le \nu {\tilde{\phi }}_X(|P|)\sum _{j=1}^{\nu }{\phi }_{X_j^{\ast }}^{\ast }(|P|)$$

$$\le \nu \sum _{j=1}^{\nu }{C}_j{\phi }_{X_j}(|P|){\phi }_{X_j^{\ast }}^{\ast }(|P|)\le \nu C\sum _{j=1}^{\nu }{C}_j^{\prime }|P|,$$

$$\Vert x_k{\Vert }_{\mathbb{B}}\cdot \Vert x_k^{\ast }{\Vert }_{{\mathbb{B}}^{\ast }}\le C.$$

$$f_j^{(n,l)}(x)=\{\begin{array}{ll}\sqrt{1-2^{-\frac{n}{l}}}{\chi }_j^n(x),& \text{if x∈[−1,0);}\\ 2^{-\frac{n}{2l}}{r}_{k_{n-1}+j}(x)& \text{if x∈[0,1],}\end{array}$$

$$|c_{n,l}|^{p}:=\bigg{\|}f_{j}^{(n,l)}\bigg{\|}^{p}_{L^{p}{[-1,1]}}=2^{n(p-2)}(1-2^{-\frac{n}{l}})^{\frac{p}{2}}+2^{-\frac{np}{2l}}.$$

$$C_{1}(l,p)m^{\frac{1}{p}}\leq\bigg{\|}\sum_{k\in A}\frac{1}{|b_{k,l}|}f_{k}^{l}\bigg{\|}_{L^{p}{[-1,1]}}\leq C_{2}(l,p)m^{\frac{1}{p}}.$$

$$2^{-\frac{np}{2l}}\cdot 2^{-n(p-2)}(1-2^{-\frac{n}{l}}{)}^{-\frac{p}{2}}\le 2^{-\frac{p}{2}}{2}^{2-p}{2}^{\frac{p}{2}}=2^{2-p}$$

$$\bigg{\|}\sum_{k\in A}\frac{1}{|b_{k,l}|}f_{k}^{l}\bigg{\|}^{p}_{L^{p}{[-1,0]}}\geq\frac{m}{1+2^{2-p}}>\frac{m}{2}.$$

$$\int_{[0,1]}\bigg{|}\sum_{k\in A}\frac{1}{|b_{k,l}|}f_{k}^{l}(x)\bigg{|}^{p}dx\leq D_{p}\bigg{(}\sum_{n=1}^{\nu}|c_{n,l}|^{-2}2^{-\frac{n}{l}}2^{n}\bigg{)}^{\frac{p}{2}}.$$

$$|c_{n,l}{|}^p\ge \frac{1}{2}{2}^{n(p-2)}\text{for n≥n0}.$$

$$\sum _{n=n_0}^{\nu }|c_{n,l}{|}^{-2}{2}^{-\frac{n}{l}}{2}^n\le 2^{\frac{2}{p}}\sum _{n=n_0}^{\nu }{2}^{-\frac{2n(p-2)}{p}}{2}^{n(1-\frac{1}{l})}.$$

$$\sum _{n=n_0}^{\nu }|c_{n,l}{|}^{-2}{2}^{-\frac{n}{l}}{2}^n\le 2^{\frac{2}{p}}\sum _{n=0}^{\nu }{2}^{\kappa n}\le 2^{\frac{2}{p}}\frac{1}{2^{\kappa }-1}{2}^{\kappa (\nu +1)}\le \frac{2^{2(\kappa +\frac{1}{p})}}{2^{\kappa }-1}{m}^{\kappa }=C_p{m}^{\kappa }.$$

$$\displaystyle\bigg{\|}\sum_{k\in A}\frac{1}{|b_{k,l}|}f_{k}^{l}\bigg{\|}^{p}_{L^{p}{[-1,1]}}\leq m+D_{p}(C^{\prime}_{p}+C_{p}m^{\kappa})^{\frac{p}{2}}$$

$$=m(1+D_p(C_p^{\prime }{m}^{-\frac{2}{p}}+C_p{m}^{\kappa -\frac{2}{p}}{)}^{\frac{p}{2}}).$$

$$|\hat{c}_{n,l}|^{r}:=\bigg{\|}f_{j}^{(n,l)}\bigg{\|}^{r}_{L^{r}{[-1,1]}}=2^{n(r-2)}(1-2^{-\frac{n}{l}})^{\frac{r}{2}}+2^{-\frac{nr}{2l}}.$$

$$\bigg{\|}\sum_{k\in A}\frac{1}{|\hat{b}_{k,l}|}f_{k}^{l}\bigg{\|}^{r}_{L^{r}{[-1,1]}}\geq\frac{m}{2}(1-2^{-\frac{1}{l}})^{\frac{r}{2}}.$$

$$|{\hat{c}}_{n,l}{|}^r\ge \frac{1}{2}{2}^{n(r-2)}\text{for n≥n1}.$$

$$\int_{[0,1]}\bigg{|}\sum_{k\in A}\frac{1}{|\hat{b}_{k,l}|}f_{k}^{l}(x)\bigg{|}^{r}dx\leq D_{r}\bigg{(}\sum_{n\in\Omega_{A}}|\hat{c}_{n,l}|^{-2}2^{-\frac{n}{l}}|A_{n}|\bigg{)}^{\frac{r}{2}}$$

$$\leq 2D_{r}\bigg{(}\sum_{n\in\Omega_{A}}2^{\frac{2n(2-r)}{r}}2^{-\frac{n}{l}}|A_{n}|\bigg{)}^{\frac{r}{2}}\leq 2D_{r}m^{\frac{r}{2}}$$

$$2^{-\frac{nr}{2l}}\le |{\hat{c}}_{n,l}{|}^r\le 2\cdot 2^{-\frac{nr}{2l}},$$

$$\displaystyle\bigg{\|}\sum_{k\in B_{n}}\frac{1}{|\hat{b}_{k,l}|}f_{k}^{l}\bigg{\|}^{r}_{L^{r}_{[-1,1]}}=\bigg{\|}\frac{1}{|\hat{c}_{n,l}|}\sum_{k\in B_{n}}f_{k}^{l}\bigg{\|}^{r}_{L^{r}_{[-1,1]}}\geq\frac{1}{2}2^{\frac{nr}{2l}}2^{n(r-1)}(1-2^{-\frac{n}{l}})^{\frac{r}{2}}$$

$$\displaystyle+\frac{1}{2}\bigg{\|}\sum_{j=k_{n-1}+1}^{k_{n}}r_{j}(\cdot)\bigg{\|}^{r}_{L^{r}_{[0,1]}}\geq\frac{1}{2}2^{n\omega}(1-2^{-\frac{n}{l}})^{\frac{r}{2}}+D^{*}_{r}2^{\frac{nr}{2}},$$

$$\frac{r}{2}<\omega<1\quad\text{if $l\in\bigg{(}\frac{r}{2(2-r)},\frac{r}{2-r}\bigg{)}$.}$$

$$\displaystyle\bigg{\|}\sum_{k\in B^{*}_{n}}\frac{1}{|\hat{b}_{k,l}|}f_{k}^{l}\bigg{\|}^{r}_{L^{r}_{[-1,1]}}=\bigg{\|}\frac{1}{|\hat{c}_{n^{2},l}|}\sum_{k\in B^{*}_{n}}f_{k}^{l}\bigg{\|}^{r}_{L^{r}_{[-1,1]}}\leq 2^{\frac{n^{2}r}{2l}}2^{n^{2}(r-2)}2^{n}(1-2^{-\frac{n^{2}}{l}})^{\frac{r}{2}}$$

$$\displaystyle+\bigg{\|}\sum_{j=k_{n^{2}-1}+1}^{k_{n^{2}-1}+2^{n}}r_{j}(\cdot)\bigg{\|}^{r}_{L^{r}_{[0,1]}}\leq\frac{1}{2}2^{n^{2}\omega_{1}+n}(1-2^{-\frac{n^{2}}{l}})^{\frac{r}{2}}+D_{r}2^{\frac{nr}{2}},$$

Full text

1 Introduction

Greedy algorithms have been studied extensively during last two decades. S.V. Konyagin and V.N. Temlyakov [7] gave a characterization of greedy bases: a basis is greedy if and only if it is unconditional and democratic. An infinite system $X=\{x_{k}\}_{k=1}^{\infty}$ in a Banach space $\mathbb{B}$ will be called a democratic system for $\mathbb{B}$ if there exists a constant $D>1$ such that, for any two finite sets of indices $P$ and $Q$ with the same cardinality $|P|=|Q|$, we have

[TABLE]

A pair of systems $X=\{x_{k}\}_{k=1}^{\infty}\subset\mathbb{B},$ $X^{*}=\{x^{*}_{k}\}_{k=1}^{\infty}\subset\mathbb{B}^{*}$ is called biorthogonal if $x^{*}_{k}(x_{m})=\delta_{km}$, where $\delta_{km}$ is the Kronecker symbol. In [1] bidemocratic bases have been studied. Following [1] we put

[TABLE]

and will say that a pair of biorthogonal systems $(X,X^{*})$ is bidemocratic for $\mathbb{B}$ if there exists $C>0$ such that for any $n\in\mathbb{N}$

[TABLE]

Modifying the definition given in [1] we say that $\varphi_{X}(n)$ is the fundamental function and $\varphi_{X^{*}}^{*}(n)$ is the dual fundamental function of the pair of biorthogonal systems $(X,X^{*})$. It is proved in [1] that a bidemocratic basis is a democratic basis. The above definition of bidemocratic system is given for minimal systems which are not necessarily bases. Further we will check that if a pair of biorthogonal systems $(X,X^{*})$ is bidemocratic for $\mathbb{B}$ then the system $X$ is democratic in $\mathbb{B}$. It is clear that if a system is democratic for $\mathbb{B}$ then any its infinite subsystem is also democratic. Using the concept of bidemocratic pair we find conditions for which the inverse assertion is also true. This idea was used in [4] (see also [5]) to give a complete characterization of weight functions $\omega$ for which the higher rank Haar wavelets are bidemocratic systems for $L^{p}(\mathbb{R},\omega),1<p<\infty$.

One of the main purposes of the article [1] was the study of the duality properties of the greedy algorithms. For example, if $\mathbb{B}$ is a reflexive Banach space, the pair of biorthogonal systems $(X,X^{*})$ is bidemocratic for $\mathbb{B}$ and $\|x_{j}\|_{\mathbb{B}}\cdot\|x^{*}_{j}\|_{\mathbb{B}^{*}}=\theta,j\in\mathbb{N}$ for some $\theta\geq 1$ then the pair of biorthogonal systems $(X^{*},X)$ is bidemocratic for $\mathbb{B}^{*}$. Of course, we came to the same conclusion if $\varphi_{X}\asymp\varphi^{*}_{X^{*}}$ and $\varphi^{*}_{X}\asymp\varphi_{X^{*}}.$ We say that $\varphi$ and $\psi$ are equivalent, $\varphi\asymp\psi$ if $\varphi$ and $\psi$ defined on $\mathbb{N}$ with values in $\mathbb{R}^{+}=\{t\in\mathbb{R}:t\geq 0\}$ and for some $0<C_{1}<C_{2}$ we have that $C_{1}\varphi(n)\leq\psi(n)\leq C_{2}\varphi(n),n\in\mathbb{N}$.

We construct a family of orthonormal systems such that they are bidemocratic for $L^{p}$ but for a subset of parameters they are not democratic for the dual space $L^{p^{\prime}}$, for another set of parameters those systems are democratic for $L^{p^{\prime}}$ but not bidemocratic for $L^{p}.$ Finally, for another set of parameters they are bidemocratic for $L^{p^{\prime}}$.

The characteristic function of a set $E$ is denoted by $I_{E}$ and $\mathbb{N}_{0}=\mathbb{N}\bigcup\{0\}$. Let $E\subseteq\mathbb{R},|E|>0$ be a measurable set then we write $\phi\in L^{p}(E),1\leq p<\infty$ if $\phi:E\rightarrow\mathbb{R}$ is measurable on $E$ and the norm is defined by

[TABLE]

2 Democratic systems

Let $\mathbb{N}_{j}\subset\mathbb{N},1\leq j\leq\nu$ be such that $\mathbb{N}_{j}\bigcap\mathbb{N}_{i}=\emptyset$ if $i\neq j,$ $\text{card}\,\mathbb{N}_{j}=\infty,1\leq j\leq\nu$ and $\bigcup_{j=1}^{\nu}\mathbb{N}_{j}=\mathbb{N}$. For a given pair of biorthogonal systems $(X,X^{*})$ consider the biorthogonal pairs $(X_{j},X_{j}^{*}),$ where $X_{j}=\{x_{k}\}_{k\in\mathbb{N}_{j}},$ $X^{*}_{j}=\{x^{*}_{k}\}_{k\in\mathbb{N}_{j}},$ $1\leq j\leq\nu.$

Proposition 2.1**.**

Let $(X_{j},X_{j}^{*}),1\leq j\leq\nu$ be pairs of biorthogonal systems defined as above. If pairs $(X_{j},X_{j}^{*}),1\leq j\leq\nu$ are bidemocratic for $\mathbb{B}$ and for any $1\leq i<j\leq\nu$ the functions $\varphi_{X_{i}}(\cdot),\varphi_{X_{j}}(\cdot)$ are equivalent then the pair of biorthogonal systems $(X,X^{*})$ is bidemocratic for $\mathbb{B}$. Moreover, $\varphi_{X}(\cdot)$ and $\varphi_{X_{j}}(\cdot)$ are equivalent for any $1\leq j\leq\nu$.

Proof.

Let

[TABLE]

We have that for any $n\in\mathbb{N}$

[TABLE]

The right hand inequality is obvious. On the other hand

[TABLE]

Let $P\subset\mathbb{N}$ be a finite set. We have that

[TABLE]

where $C=\max_{1\leq j\leq\nu}C_{j}$.

∎

The condition (1.2) yields

Remark 2.1**.**

If a pair of biorthogonal systems $(X,X^{*})$ is bidemocratic for $\mathbb{B}$ then there exists $C>0$ such that for any $k\in\mathbb{N}$

[TABLE]

It is proved in [1] that a bidemocratic basis is a democratic basis. The proof given in [1] for bases also works for the proof of the following

Proposition 2.2**.**

Let $(X,X^{*})$ be a pair of biorthogonal systems bidemocratic for $\mathbb{B}$. Then the system $X$ is democratic for $\mathbb{B}$.

We are going to construct a family of orthonormal systems in order to clarify some duality properties of orthonormal systems if it is democratic for the $L^{p},1<p<\infty$ spaces.

Let $\chi$ be an orthonormal system of functions defined on $[-1,1]$ as follows:

For any $n\in\mathbb{N}$ we divide the interval $(-2^{-n+1},-2^{-n}]$ into $2^{n}$ equal intervals $\Delta_{j}^{n},1\leq j\leq 2^{n}$ such that $\Delta_{j}^{n}\bigcap\Delta_{i}^{n}=\emptyset$ if $i\neq j$.

Set $\chi_{j}^{n}(x)=2^{n}I_{\Delta_{j}^{n}}(x):1\leq j\leq 2^{n},n\in\mathbb{N}$, where $I_{E}(\cdot)$ is the characteristic function of the set $E\subset[-1,1]$. It is clear that the system $\chi=\{\chi_{j}^{n}(x):1\leq j\leq 2^{n},n\in\mathbb{N}\},$ is an orthonormal system of functions on $[-1,1]$.

We put $k_{0}=0$ and $k_{n}=k_{n-1}+2^{n}=2(2^{n}-1),n\in\mathbb{N}$. In our construction we use the Rademacher system $\{r_{k}(t)\}_{k=1}^{\infty},$ which is an orthonormal system of functions defined on $[0,1]$(see [2],[3]). Let

[TABLE]

$1\leq j\leq 2^{n},n\in\mathbb{N}$ and $l\in(0,\infty)$.

Let $f^{l}_{k}(x)=f_{j}^{(n,l)}(x)$ if $k=k_{n-1}+j$ and $1\leq j\leq 2^{n}$. For any fixed $l\in(0,\infty)$ the system $\mathfrak{F}_{l}=\{f_{k}^{l}(x)\}_{k=1}^{\infty}$ is an orthonormal system of functions defined on $[-1,1]$.

Proposition 2.3**.**

For any $l\in(0,\infty)$ he system $\mathfrak{F}_{l}$ is a democratic system for $L^{p}{[-1,1]},$ $2\leq p<\infty,$ and $\varphi_{\mathfrak{F}_{l}}(m)\asymp m^{\frac{1}{p}}$.

Proof.

The proposition is obviously true if $p=2$. Thus we only will consider the case $p>2$. We have that

[TABLE]

Let $|b_{k,l}|^{p}:=|c_{n,l}|^{p}$ if $k=k_{n-1}+j$ and $1\leq j\leq 2^{n}$.

We prove that there exists $0<C_{1}(l,p)<C_{2}(l,p)$ such that for any finite set $A\subset\mathbb{N},|A|=m$

[TABLE]

Let $n_{0}=[l]+1,$ where $[l]\in\mathbb{N}$ and $[l]\leq l<[l]+1$. Thus $2^{-\frac{np}{2l}}\leq 2^{-\frac{p}{2}}$ if $n\geq n_{0}$. We have that

[TABLE]

when $n\geq n_{0}$. Then for any $A\subset[n_{0},\infty)\bigcap\mathbb{N}$, $|A|=m$ we have that

[TABLE]

The supports of functions on $[-1,0)$ are not empty and do not coincide. Thus changing the constant we easily get the left side inequality in (2.3) for the general case.

Let $m>2^{n_{0}},m\in\mathbb{N}$ and $\nu\in\mathbb{N}$ be such that $2^{\nu-1}<m\leq 2^{\nu}$. By the Khintchine inequality (see [2]) it follows that

[TABLE]

Clearly

[TABLE]

Thus it follows that

[TABLE]

Let $\kappa=\frac{4}{p}-1-\frac{1}{l}$. If $\kappa>0$ we write

[TABLE]

Thus it follows

[TABLE]

Whence we obtain the right hand inequality in (2.3) because $\kappa-\frac{2}{p}<0$. If $\kappa\leq 0$ then the proof is obvious. ∎

Proposition 2.4**.**

The system $\mathfrak{F}_{l}$ is a democratic system for $L^{r}{[-1,1]},$ $1\leq r<2,$ $l\in(0,\frac{r}{2(2-r)}]$ and $\varphi_{\mathfrak{F}_{l}}(m)\asymp m^{\frac{1}{r}}$.

Proof.

We have that

[TABLE]

As above we put $|\hat{b}_{k,l}|^{r}:=|\hat{c}_{n,l}|^{r}$ if $k=k_{n-1}+j$ and $1\leq j\leq 2^{n}$.

If $l\in(0,\frac{r}{2(2-r)}]$ then $2^{n(r-2)}\geq 2^{-\frac{nr}{2l}}$ and it follows that

[TABLE]

On the other hand we have that there exists $n_{1}\in\mathbb{N}$ such that

[TABLE]

Let $A_{n}=[k_{n-1}+1,k_{n}]\bigcap A,n\in\mathbb{N}$ and $\Omega_{A}=\{n\in\mathbb{N}:A_{n}\neq\emptyset\}$.

Let $m>2^{n_{1}},m\in\mathbb{N}$ and $\nu\in\mathbb{N}$ be such that $2^{\nu-1}<m\leq 2^{\nu}$ then by the Khintchine inequality it follows that

[TABLE]

[TABLE]

if $l\leq\frac{r}{2(2-r)}.$ Afterwards we proceed as in the proof of Proposition 2.3 and easily finish the proof.

∎

Proposition 2.5**.**

The system $\mathfrak{F}_{l}$ is not a democratic system for $L^{r}{[-1,1]},$ $1\leq r<2$ if $l\in(\frac{r}{2(2-r)},\frac{r}{2-r})$.

Proof.

If $l>\frac{r}{2(2-r)}$ then $2^{n(r-2)}<2^{-\frac{nr}{2l}}$. Thus for any $n\in\mathbb{N}$

[TABLE]

where $\hat{c}_{n,l}$ is defined by (2.5). Let $B_{n}=[k_{n-1}+1,k_{n}]\bigcap\mathbb{N},n\in\mathbb{N}$. Then it follows that

[TABLE]

where $\omega=\frac{r}{2l}+r-1$ and $D^{*}_{r}>0$. Observe that

[TABLE]

Afterwards we consider $B^{*}_{n}=[k_{n^{2}-1}+1,k_{n^{2}-1}+2^{n}]\bigcap\mathbb{N},n\in\mathbb{N}$. In this case we have

[TABLE]

where $D_{r}>0$ and $\omega_{1}=\frac{r}{2l}+r-2=\omega-1<0$ if l\in\bigg{(}\frac{r}{2(2-r)},\frac{r}{2-r}\bigg{)}. The inequality $\frac{r}{2}<\omega$ yields that the system $\mathfrak{F}_{l}$ is not a democratic system. ∎

Proposition 2.6**.**

The system $\mathfrak{F}_{l}$ is a democratic system for $L^{r}{[-1,1]},$ $1\leq r<2,$ if $l\in[\frac{r}{2-r},\infty)$. Moreover, $\varphi_{\mathfrak{F}_{l}}(n)\asymp n^{\frac{1}{2}}$.

Proof.

If $l\in[\frac{r}{2-r},\infty)$ we have that the inequalities (2.6) hold. Hence, for any $A\subset\mathbb{N},|A|=m$

[TABLE]

where the last inequality follows by the Khintchine inequalities and $C_{r}>0$. Let $\nu\in\mathbb{N}$ be such that $2^{\nu-1}\leq m<2^{\nu}$. Then

[TABLE]

By Khintchine’s inequalities we obtain

[TABLE]

∎

Resuming the propositions proved above we easily obtain the following theorem.

Theorem 2.1**.**

Let $p>2$ and $p^{\prime}=\frac{p}{p-1}$. Then the pair $(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is bidemocratic for $L^{p}[-1,1]$ and for $L^{p^{\prime}}[-1,1]$ if $l\in(0,\frac{p}{2(p-2)}].$ The system $\mathfrak{F}_{l}$ is not democratic for $L^{p^{\prime}}[-1,1]$ when $l\in(\frac{p}{2(p-2)},\frac{p}{p-2}).$

When $l>\frac{p}{2(p-2)}$ the pair $(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is not bidemocratic neither for $L^{p}[-1,1]$ nor for $L^{p^{\prime}}[-1,1]$.