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# On bases from perturbed system of exponents in Lebesgue spaces with variable summability exponent

Journal of Inequalities and Applications20142014:495

https://doi.org/10.1186/1029-242X-2014-495

• Received: 26 September 2014
• Accepted: 26 November 2014
• Published:

## Abstract

In this paper the perturbed system of exponents with some asymptotics is considered. Basis properties of this system in Lebesgue spaces with variable summability exponent are investigated.

## Keywords

• system of exponents
• perturbation
• generalized Lebesgue space
• variable exponent

## 1 Introduction

Consider the following system of exponents:
${\left\{{e}^{i{\lambda }_{n}t}\right\}}_{n\in Z},$
(1)
where $\left\{{\lambda }_{n}\right\}\subset R$ is a sequence of real numbers, Z is a set of integer numbers. It is the aim of this paper to investigate basis properties (basicity, completeness, and minimality) of the system (1) in Lebesgue space ${L}_{{p}_{t}}$ with variable summability index $p\left(t\right)$, when $\left\{{\lambda }_{n}\right\}$ has the asymptotics
${\lambda }_{n}=n-\alpha signn+O\left(|n{|}^{-\beta }\right),\phantom{\rule{1em}{0ex}}n\to \mathrm{\infty },$
(2)

where $\alpha ,\beta \in R$ are some parameters.

Many authors have investigated the basicity properties of system of exponents of the form (1), beginning with the well-known result of Paley and Wiener [1] on Riesz basicity. Some of the results in this direction have been obtained by Young [2]. The criterion of basicity of the system (1) in ${L}_{p}\equiv {L}_{p}\left(-\pi ,\pi \right)$, $1, when ${\lambda }_{n}=n-\alpha signn$, has been obtained earlier in [3, 4].

Recently in connection with consideration of some specific problems of mechanics and mathematical physics [5, 6], interest in the study of the various questions connected with Lebesgue ${L}_{{p}_{t}}$ and Sobolev ${W}_{{p}_{t}}^{k}$ spaces with variable summability index $p\left(t\right)$ has increased [59].

Many questions of the theory of operators (for example, theory of singular operators, theory of potentials and etc.) are studied in spaces ${L}_{{p}_{t}}$ [7]. These investigations have allowed one to consider questions of basicity of some system of functions (for example, the classical system of exponents ${\left\{{e}^{int}\right\}}_{n\in Z}$) in ${L}_{{p}_{t}}$. In [9] the basicity of system ${\left\{{e}^{int}\right\}}_{n\in N}$ in ${L}_{{p}_{t}}$ has been established. The special case of the system (1) is considered in [1012], when ${\lambda }_{n}=n-\alpha signn$, $n\in Z$.

In this paper basis properties of the system (1) in ${L}_{{p}_{t}}$ spaces are investigated. Under certain conditions on the parameters α and β equivalence of the basis properties (completeness, minimality, ω-linearly independence, basicity) of the system (2) in ${L}_{{p}_{t}}$ are proved.

## 2 Necessary notion and facts

Let $p:\left[-\pi ,\pi \right]\to \left[1,+\mathrm{\infty }\right)$ be a Lebesgue measurable function. By ${\mathrm{L}}_{0}$ we denote the class of all functions measurable on $\left[-\pi ,\pi \right]$ with respect to Lebesgue measure. We choose the notation
${I}_{p}\left(f\right)\stackrel{\mathrm{def}}{\equiv }{\int }_{-\pi }^{\pi }|f\left(t\right){|}^{p\left(t\right)}\phantom{\rule{0.2em}{0ex}}dt.$
Let $\mathrm{L}\equiv \left\{f\in {\mathrm{L}}_{0}:{I}_{p}\left(f\right)<+\mathrm{\infty }\right\}$. Let ${p}^{-}=inf{vrai}_{\left[-\pi ,\pi \right]}p\left(t\right)$, ${p}^{+}=sup{vrai}_{\left[-\pi ,\pi \right]}p\left(t\right)$. For ${p}^{+}<+\mathrm{\infty }$, with respect to ordinary linear operations of addition of functions and multiplication by number, L turns into a linear space. If we define in ${L}_{{p}_{t}}$ the norm
${\parallel f\parallel }_{{p}_{t}}\stackrel{\mathrm{def}}{\equiv }inf\left\{\lambda >0:{I}_{p}\left(\frac{f}{\lambda }\right)\le 1\right\},$
then L is a Banach space and we denote it by ${L}_{{p}_{t}}$. Denote

Throughout this paper, $q\left(t\right)$ denotes the function conjugate to function $p\left(t\right)$, that is, $\frac{1}{p\left(t\right)}+\frac{1}{q\left(t\right)}\equiv 1$.

We have Hölder’s generalized inequality,
${\int }_{-\pi }^{\pi }|f\left(t\right)g\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\le C\left({p}^{-};{p}^{+}\right){\parallel f\parallel }_{{p}_{t}}{\parallel g\parallel }_{{q}_{t}},$

where $C\left({p}^{-};{p}^{+}\right)=1+\frac{1}{{p}^{-}}-\frac{1}{{p}^{+}}$.

For our investigation we need some basic concepts of the theory of close bases, given as follows.

We adopt the standard notation: B-space is a Banach space; ${X}^{\ast }$ is the conjugate to space X; $f\left(x\right)$, $f\in {X}^{\ast }$, and $x\in X$ means the value of functional f on x; $L\left[M\right]$ is a linear span of a set M. The system ${\left\{{x}_{n}\right\}}_{n\in N}\subset X$ is called ω-linear independent in B-space X, if ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}{x}_{n}=0$ true for ${\alpha }_{n}=0$, $\mathrm{\forall }n\in N$.

The following lemma is true.

Lemma 1 Let X be a Banach space with basis ${\left\{{x}_{n}\right\}}_{n\in N}\subset X$ and $F:X\to X$ be a Fredholm operator. Then the following properties of the system ${\left\{{y}_{n}=F{x}_{n}\right\}}_{n\in N}$ in X are equivalent:
1. (1)

${\left\{{y}_{n}\right\}}_{n\in N}$ is complete;

2. (2)

${\left\{{y}_{n}\right\}}_{n\in N}$ is minimal;

3. (3)

${\left\{{y}_{n}\right\}}_{n\in N}$ is ω-linear independent;

4. (4)

${\left\{{y}_{n}\right\}}_{n\in N}$ is isomorphic to ${\left\{{x}_{n}\right\}}_{n\in N}$ basis.

We also need the following easily provable lemma.

Lemma 2 Let X be a Banach space with basis ${\left\{{x}_{n}\right\}}_{n\in N}$ and ${\left\{{y}_{n}\right\}}_{n\in N}\subset X:card\left\{n:{x}_{n}\ne {y}_{n}\right\}<+\mathrm{\infty }$. Then the expression
$Fx=\sum _{n=1}^{\mathrm{\infty }}{x}_{n}^{\ast }\left(x\right){y}_{n}$

generates the Fredholm operator $F:X\to X$, where ${\left\{{x}_{n}^{\ast }\right\}}_{n\in N}\subset {X}^{\ast }$ is conjugate to ${\left\{{x}_{n}\right\}}_{n\in N}$ system.

The following lemma is also true.

Lemma 3 Let ${\left\{{x}_{n}\right\}}_{n\in N}$ be complete and minimal in B-space X and ${\left\{{y}_{n}\right\}}_{n\in N}\subset X:card\left\{n:{x}_{n}\ne {y}_{n}\right\}<+\mathrm{\infty }$. Then the following properties of system ${\left\{{y}_{n}\right\}}_{n\in N}$ in X are equivalent:
1. (1)

${\left\{{y}_{n}\right\}}_{n\in N}$ is complete;

2. (2)

${\left\{{y}_{n}\right\}}_{n\in N}$ is minimal.

These and other results are obtained in [13, 14].

We will use the following statement, which has a proof similar to the proof of Levinson [15].

Statement 1 Let system ${\left\{{e}^{i{\lambda }_{n}t}\right\}}_{n\in Z}$ be complete in ${L}_{{p}_{t}}$. If from the system we remove n any functions and add instead of them n other functions ${e}^{i{\mu }_{j}t}$, $j=1,\dots ,n$, where ${\mu }_{1},\dots ,{\mu }_{n}$ are any, mutually different complex numbers not equal to any of numbers ${\lambda }_{k}$, then the new system will be complete.

We shall also need the following theorem of Krein-Milman-Rutman.

Theorem 1 (Krein-Milman-Rutman [13])

Let X be a Banach space with norm $\parallel \cdot \parallel$, ${\left\{{x}_{n}\right\}}_{n\in N}\subset X$ be normed basis in X (i.e. $\parallel {x}_{n}\parallel =1$, $\mathrm{\forall }n\in N$) with conjugate system ${\left\{{x}_{n}^{\ast }\right\}}_{n\in N}\subset {X}^{\ast }$, and ${\left\{{y}_{n}\right\}}_{n\in N}\subset X$ be a system satisfying the inequality
$\sum _{n=1}^{\mathrm{\infty }}\parallel {x}_{n}-{y}_{n}\parallel <{\gamma }^{-1},$

where $\gamma ={sup}_{n}\parallel {x}_{n}^{\ast }\parallel$. Then ${\left\{{y}_{n}\right\}}_{n\in N}$ also forms a basis isomorphic to the basis ${\left\{{x}_{n}\right\}}_{n\in N}$ in X.

## 3 Basic results

Before giving the basic results we will prove the following auxiliary theorem.

Theorem 2 Let $p\in {H}^{ln}$ and ${p}^{-}>1$. If the system
${\left\{{e}^{i\left(n-\alpha signn\right)t}\right\}}_{n\in Z},$
(3)
forms a basis in ${L}_{{p}_{t}}\equiv {L}_{{p}_{t}}\left(-\pi ,\pi \right)$, then this system is isomorphic to the classical system of exponents ${\left\{{e}^{int}\right\}}_{n\in Z}$, where the isomorphism is given by
$Sf={e}^{-i\alpha t}\sum _{0}^{\mathrm{\infty }}\left(f,{e}^{inx}\right){e}^{int}+{e}^{i\alpha t}\sum _{1}^{\mathrm{\infty }}\left(f,{e}^{-inx}\right){e}^{-int},$
(4)
where
$\left(f,g\right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f\left(t\right)\overline{g\left(t\right)}\phantom{\rule{0.2em}{0ex}}dt.$
Proof Consider the operator (4). From the basicity of system ${\left\{{e}^{int}\right\}}_{n\in Z}$ in ${L}_{{p}_{t}}$ it follows that S is a bounded operator on ${L}_{{p}_{t}}$ into itself. It is easy to see that $KerS=0$. Actually, let $Sf=0$. From the basicity of the system (3) in ${L}_{{p}_{t}}$ and from (4) we obtain $\left(f,{e}^{inx}\right)=0$, $\mathrm{\forall }n\in Z$. Also, from the basicity of system ${\left\{{e}^{int}\right\}}_{n\in Z}$ in ${L}_{{p}_{t}}$ it follows that $f=0$. We show that for all $g\in {L}_{{p}_{t}}$, the equation $Sf=g$ in ${L}_{{p}_{t}}$ is solved. Let us assume that
$f=\sum _{n\in Z}{g}_{n}{e}^{int},$

where ${\left\{{g}_{n}\right\}}_{n\in Z}$ are the biorthogonal coefficients of the function g by the system (3).

It is clear that $f\in {L}_{{p}_{t}}$, and so
$\begin{array}{rl}Sf& ={e}^{-i\alpha t}\sum _{n=0}^{\mathrm{\infty }}\left(f,{e}^{inx}\right){e}^{int}+{e}^{i\alpha t}\sum _{n=1}^{\mathrm{\infty }}\left(f,{e}^{-inx}\right){e}^{-int}\\ ={e}^{-i\alpha t}\sum _{n=0}^{\mathrm{\infty }}{g}_{n}{e}^{int}+{e}^{i\alpha t}\sum _{n=1}^{\mathrm{\infty }}{g}_{-n}{e}^{-int}=g,\end{array}$

as by the condition of the theorem, the system (3) forms a basis in ${L}_{{p}_{t}}$.

This means that for all $g\in {L}_{{p}_{t}}$ the equation $Sf=g$ is solved in ${L}_{{p}_{t}}$. Then by the Banach theorem the operator S has a bounded inverse. It is obvious that $S\left[{e}^{int}\right]=A\left(t\right){e}^{int}$, $n\ge 0$, and $S\left[{e}^{-int}\right]=B\left(t\right){e}^{-int}$, $n\ge 1$. This completes the proof. □

Now we study some basis properties of the system (1). Firstly, we recall the following theorem.

Theorem 3 ([11])

Let $p\in {H}^{ln}$ and ${p}^{-}>1$. If parameter $\alpha \in R$ satisfies the condition $-\frac{1}{2p\left(\pi \right)}<\alpha <\frac{1}{2q\left(\pi \right)}$, then the system $\left\{{e}^{i{\mu }_{n}t}\right\}$ forms a basis in ${L}_{{p}_{t}}$.

Let the asymptotics (2) occur. Let us assume ${\mu }_{n}=n-\alpha signn$ and ${\delta }_{n}={\lambda }_{n}-{\mu }_{n}$, $\mathrm{\forall }n\in Z$. It is easy to see that the inequality
$|{e}^{i{\lambda }_{n}t}-{e}^{i{\mu }_{n}t}|\le c|n{|}^{-\beta },\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ne 0,$
(5)
is fulfilled, where c is some constant. Let us assume that the following inequalities are satisfied:
$-\frac{1}{2p\left(\pi \right)}<\alpha <\frac{1}{2q\left(\pi \right)},\phantom{\rule{2em}{0ex}}\beta >\frac{1}{\stackrel{˜}{p}},$
(6)

where $\stackrel{˜}{p}=min\left\{{p}^{-};2\right\}$. Then, from Theorem 3, the system of exponents ${\left\{{e}^{i{\mu }_{n}t}\right\}}_{n\in Z}$ forms a basis in ${L}_{{p}_{t}}$. By Theorem 1, it is isomorphic to the classical system of exponents ${\left\{{e}^{int}\right\}}_{n\in Z}$ in ${L}_{{p}_{t}}$. Therefore the spaces of coefficients of the bases ${\left\{{e}^{i{\mu }_{n}t}\right\}}_{n\in Z}$ and ${\left\{{e}^{int}\right\}}_{n\in Z}$ coincide.

Let $T:{L}_{{p}_{t}}\to {L}_{{p}_{t}}$ be a natural automorphism
$T\left[{e}^{i{\mu }_{n}t}\right]={e}^{int},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in Z.$
For all $f\in {L}_{{p}_{t}}$, let ${\left\{{f}_{n}\right\}}_{n\in Z}$ be biorthogonal coefficients of f by the system ${\left\{{e}^{i{\mu }_{n}t}\right\}}_{n\in Z}$, and let $g=Tf$. Therefore, ${\left\{{f}_{n}\right\}}_{n\in Z}$ are the Fourier coefficients of the function g by the system ${\left\{{e}^{int}\right\}}_{n\in Z}$. From (4) and (5), it directly follows that
$\sum _{n\in Z}{\parallel {e}^{i{\lambda }_{n}t}-{e}^{i{\mu }_{n}t}\parallel }_{{p}_{t}}^{\stackrel{˜}{p}}<+\mathrm{\infty }.$
Consider the following expression:
$\sum _{n}\left({e}^{i{\lambda }_{n}t}-{e}^{i{\mu }_{n}t}\right){f}_{n}.$
We have
$\begin{array}{rl}{\parallel \sum _{n\in Z}\left({e}^{i{\lambda }_{n}t}-{e}^{i{\mu }_{n}t}\right){f}_{n}\parallel }_{{p}_{t}}& \le \sum _{n\in Z}\parallel {e}^{i{\lambda }_{n}t}-{e}^{i{\mu }_{n}t}\parallel |{f}_{n}|\\ \le {\left(\sum _{n}{\parallel {e}^{i{\lambda }_{n}t}-{e}^{i{\mu }_{n}t}\parallel }_{{p}_{t}}^{\stackrel{˜}{p}}\right)}^{1/\stackrel{˜}{p}}{\left(\sum _{n}|{f}_{n}{|}^{\stackrel{˜}{q}}\right)}^{1/\stackrel{˜}{q}},\end{array}$
where $\frac{1}{\stackrel{˜}{p}}+\frac{1}{\stackrel{˜}{q}}=1$. By the Hausdorff-Young theorem [16], we have
${\left(\sum _{n}|{f}_{n}{|}^{\stackrel{˜}{q}}\right)}^{1/\stackrel{˜}{q}}\le {m}_{1}{\parallel g\parallel }_{\stackrel{˜}{p}},$
where ${m}_{1}$ is some constant. From $\stackrel{˜}{p}\le {p}^{-}$ and the continuous embedding ${L}_{{p}_{t}}\subset {L}_{\stackrel{˜}{p}}$, it follows that, $\mathrm{\exists }{m}_{2}>0$,
${\parallel g\parallel }_{\stackrel{˜}{p}}\le {m}_{2}{\parallel g\parallel }_{{p}_{t}}\le {m}_{2}\parallel T\parallel {\parallel f\parallel }_{{p}_{t}}.$
As a result, we obtain
${\parallel \sum _{n}\left({e}^{i{\lambda }_{n}t}-{e}^{i{\mu }_{n}t}\right){f}_{n}\parallel }_{{p}_{t}}\le {m}_{1}{m}_{2}\parallel T\parallel {\left(\sum _{n}{\parallel {e}^{i{\lambda }_{n}t}-{e}^{i{\mu }_{n}t}\parallel }_{{p}_{t}}^{\stackrel{˜}{p}}\right)}^{1/\stackrel{˜}{p}}{\parallel f\parallel }_{{p}_{t}}.$
(7)
Let us take ${n}_{0}\in N$ such that
$\delta ={m}_{1}{m}_{2}\parallel T\parallel {\left(\sum _{|n|>{n}_{0}}{\parallel {e}^{i{\lambda }_{n}t}-{e}^{i{\mu }_{n}t}\parallel }_{{p}_{t}}^{\stackrel{˜}{p}}\right)}^{1/\stackrel{˜}{p}}<1.$
Assume that
${\omega }_{n}=\left\{\begin{array}{ll}{\lambda }_{n},& |n|>{n}_{0},\\ {\mu }_{n},& |n|\le {n}_{0}.\end{array}$
It is clear that the following inequality is satisfied:
${\parallel \sum _{n}\left({e}^{i{\omega }_{n}t}-{e}^{i{\mu }_{n}t}\right){f}_{n}\parallel }_{{p}_{t}}\le \delta {\parallel f\parallel }_{{p}_{t}}.$
(8)

It follows immediately from (7) that the expression ${\sum }_{n}\left({e}^{i{\omega }_{n}t}-{e}^{i{\mu }_{n}t}\right){f}_{n}$ represents a function from ${L}_{{p}_{t}}$ and it can be denoted by ${T}_{0}f$. Drawing attention to (8) we obtain $\parallel {T}_{0}\parallel \le \delta <1$. Thus, the operator $F=I+{T}_{0}$ is invertible, and it is easy to see that $F\left[{e}^{i{\mu }_{n}t}\right]={e}^{i{\omega }_{n}t}$, $\mathrm{\forall }n\in Z$. Hence, the system ${\left\{{e}^{i{\omega }_{n}t}\right\}}_{n\in Z}$ forms a basis in ${L}_{{p}_{t}}$ isomorphic to ${\left\{{e}^{i{\mu }_{n}t}\right\}}_{n\in Z}$. Systems ${\left\{{e}^{i{\lambda }_{n}t}\right\}}_{n\in Z}$ and ${\left\{{e}^{i{\omega }_{n}t}\right\}}_{n\in Z}$ differ in a finite number of elements. Therefore, by Statement 1, the system ${\left\{{e}^{i{\lambda }_{n}t}\right\}}_{n\in Z}$ is complete in ${L}_{{p}_{t}}$, if ${\lambda }_{i}\ne {\lambda }_{j}$ for $i\ne j$. In the following it is necessary to apply Lemmas 1 and 2.

As a result we obtain the following theorem.

Theorem 4 Let the asymptotics (2) occur and the inequalities
$-\frac{1}{2p\left(\pi \right)}<\alpha <\frac{1}{2q\left(\pi \right)},\phantom{\rule{2em}{0ex}}\beta >\frac{1}{\stackrel{˜}{p}},$
(9)
be fulfilled, where $\stackrel{˜}{p}=min\left\{{p}^{-};2\right\}$. Then the following properties of the system (1) are equivalent in ${L}_{{p}_{t}}$:
1. (1)

the system (1) is complete;

2. (2)

the system (1) is minimal;

3. (3)

the system (1) is ω-linear independent;

4. (4)

the system (1) is isomorphic to ${\left\{{e}^{int}\right\}}_{n\in N}$ basis;

5. (5)

${\lambda }_{i}\ne {\lambda }_{j}$ for $i\ne j$.

Let us consider the case $\alpha =-\frac{1}{2p\left(\pi \right)}$. In this case, by the results of [11], the system ${\left\{{e}^{i{\mu }_{n}t}\right\}}_{n\in Z}$ is complete and minimal in ${L}_{{p}_{t}}$, but it does not form a basis in it. Then from the previous considerations it follows that the system (1) cannot form a basis in ${L}_{{p}_{t}}$. Because otherwise, by Theorem 2, it will be isomorphic to system ${\left\{{e}^{int}\right\}}_{n\in Z}$ in ${L}_{{p}_{t}}$, and as a result the system ${\left\{{e}^{i{\mu }_{n}t}\right\}}_{n\in Z}$ should form a basis in ${L}_{{p}_{t}}$. This gives a contradiction.

By ${\left\{{v}_{n}\right\}}_{n\in Z}\subset {L}_{{q}_{t}}$ we denote the system biorthogonal to ${\left\{{e}^{i{\mu }_{n}t}\right\}}_{n\in Z}$. It is clear that using the estimates from [4], for ${v}_{n}$, $n\in Z$, we establish that the following relation is true:
$\gamma =\underset{n}{sup}{\parallel {v}_{n}\parallel }_{{q}_{t}}<+\mathrm{\infty }.$
Let $\beta >1$. Then it is clear that the following inequality is satisfied:
$\sum _{n}{\parallel {e}^{i{\lambda }_{n}t}-{e}^{i{\mu }_{n}t}\parallel }_{{p}_{t}}<+\mathrm{\infty }.$
Similarly to the previous case, we can show that the operator
$\stackrel{˜}{T}f=\sum _{n}{v}_{n}\left(f\right)\left({e}^{i{\lambda }_{n}t}-{e}^{i{\mu }_{n}t}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }f\in {L}_{{p}_{t}},$

is bounded in ${L}_{{p}_{t}}$. Introducing the new system ${\left\{{e}^{i{\omega }_{n}t}\right\}}_{n\in Z}$ in the same manner we establish the completeness of the system (1) in ${L}_{{p}_{t}}$, if ${\lambda }_{i}\ne {\lambda }_{j}$ for $i\ne j$. Minimality of the system (1) in ${L}_{{p}_{t}}$ follows from Lemma 3. Thus, if ${\lambda }_{i}\ne {\lambda }_{j}$ for $i\ne j$ and $\beta >1$, then the system (1) is complete and minimal in ${L}_{{p}_{t}}$ if the condition $-\frac{1}{2p\left(\pi \right)}\le \alpha <\frac{1}{2q\left(\pi \right)}$ is satisfied.

Consider the case $\alpha \notin \left[-\frac{1}{2p\left(\pi \right)},\frac{1}{2q\left(\pi \right)}\right)$. Let, for example, $\alpha \in \left[\frac{1}{2q\left(\pi \right)},\frac{1}{2q\left(\pi \right)}+\frac{1}{2}\right)$. Multiplication of each member of the system (1) by ${e}^{i\frac{t}{2}}$ does not affect its basis properties in ${L}_{{p}_{t}}$. After appropriate transformations we obtain the system
${e}^{i\left[\stackrel{˜}{\alpha }+{\stackrel{˜}{\alpha }}_{0}\right]t}\bigcup {\left\{{e}^{i{\stackrel{˜}{\lambda }}_{n}t}\right\}}_{n\in Z},$
(10)
where $\stackrel{˜}{\alpha }=\alpha -\frac{1}{2}$ and
${\stackrel{˜}{\lambda }}_{n}=n-\stackrel{˜}{\alpha }signn+O\left(|n{|}^{-\beta }\right),\phantom{\rule{1em}{0ex}}n\to \mathrm{\infty }.$

Denote by ${\stackrel{˜}{\alpha }}_{0}$ the member of $O\left(|n{|}^{-\beta }\right)$ in (2), corresponding to $n=0$. It is easy to see that condition ${\lambda }_{i}\ne {\lambda }_{j}$ is equivalent to ${\stackrel{˜}{\lambda }}_{i}\ne {\stackrel{˜}{\lambda }}_{j}$. It is clear that $-\frac{1}{2p\left(\pi \right)}\le \stackrel{˜}{\alpha }<\frac{1}{2q\left(\pi \right)}$. Then, by the previous results, the system ${\left\{{e}^{i{\stackrel{˜}{\lambda }}_{n}t}\right\}}_{n\in Z}$ is complete and minimal in ${L}_{{p}_{t}}$, and therefore the system (10), and at the same time the system (1), is complete, but it is not minimal in ${L}_{{p}_{t}}$. Continuing this process we find that the system (1) is not complete, but it is minimal for $\alpha <-\frac{1}{2p\left(\pi \right)}$; and the system (1) is complete, but it is not minimal in ${L}_{{p}_{t}}$ for $\alpha \ge \frac{1}{2q\left(\pi \right)}$. Thus, the following theorem is proved.

Theorem 5 We have:
1. (I)

Let the asymptotics (2) occur and the inequalities (9) be fulfilled, where $\stackrel{˜}{p}=min\left\{{p}^{-};2\right\}$. Then the following properties of the system (1) are equivalent in ${L}_{{p}_{t}}$:

(1.1) the system (1) is complete;

(1.2) the system (1) is minimal;

(1.3) the system (1) is ω-linear independent;

(1.4) the system (1) is isomorphic to ${\left\{{e}^{int}\right\}}_{n\in N}$ basis;

(1.5) ${\lambda }_{i}\ne {\lambda }_{j}$ for $i\ne j$.

2. (II)

Let $\beta >1$ and $\alpha =-\frac{1}{2p\left(\pi \right)}$. Then the following properties of the system (1) in ${L}_{{p}_{t}}$ are equivalent:

(2.1) the system (1) is complete;

(2.2) the system (1) is minimal;

(2.3) ${\lambda }_{i}\ne {\lambda }_{j}$, for $i\ne j$.

Moreover, in this case the system (1) does not form a basis in ${L}_{{p}_{t}}$.

3. (III)

Let $\beta >1$ and ${\lambda }_{i}\ne {\lambda }_{j}$, for $i\ne j$. Then the system (1) is complete and minimal in ${L}_{{p}_{t}}$ for $-\frac{1}{2p\left(\pi \right)}\le \alpha <\frac{1}{2q\left(\pi \right)}$, and for $\alpha <-\frac{1}{2\pi }$ it is not complete, but it is minimal; and for $\alpha \ge \frac{1}{2q\left(\pi \right)}$ it is complete, but it is not minimal in ${L}_{{p}_{t}}$.

## Declarations

### Acknowledgements

I wish to expresses my thanks to Prof. Bilal T Bilalov, Institute of Mathematics and Mechanics of National Academy of Sciences, Baku, Azerbaijan, for his kind help, careful reading, and making useful comments on the earlier version of the paper.

## Authors’ Affiliations

(1)
Department of Non-harmonic Analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, 9 B. Vahabzadeh Str., Baku, Azerbaijan

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