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

- Togrul Muradov
^{1}Email author

**2014**:495

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

© Muradov; licensee Springer. 2014

**Received: **26 September 2014

**Accepted: **26 November 2014

**Published: **12 December 2014

## 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

## 1 Introduction

*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 $p(t)$, when $\{{\lambda}_{n}\}$ has the asymptotics

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 ${L}_{p}\equiv {L}_{p}(-\pi ,\pi )$, $1<p<+\mathrm{\infty}$, 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 $p(t)$ has increased [5–9].

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 ${\{{e}^{int}\}}_{n\in Z}$) in ${L}_{{p}_{t}}$. In [9] the basicity of system ${\{{e}^{int}\}}_{n\in N}$ in ${L}_{{p}_{t}}$ has been established. The special case of the system (1) is considered in [10–12], 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

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

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

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*; $f(x)$, $f\in {X}^{\ast}$, and $x\in X$ means the value of functional *f* on *x*; $L[M]$ is a linear span of a set *M*. The system ${\{{x}_{n}\}}_{n\in N}\subset X$ 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*${\{{x}_{n}\}}_{n\in N}\subset X$

*and*$F:X\to X$

*be a Fredholm operator*.

*Then the following properties of the system*${\{{y}_{n}=F{x}_{n}\}}_{n\in N}$

*in*

*X*

*are equivalent*:

- (1)
${\{{y}_{n}\}}_{n\in N}$

*is complete*; - (2)
${\{{y}_{n}\}}_{n\in N}$

*is minimal*; - (3)
${\{{y}_{n}\}}_{n\in N}$

*is**ω*-*linear independent*; - (4)
${\{{y}_{n}\}}_{n\in N}$

*is isomorphic to*${\{{x}_{n}\}}_{n\in N}$*basis*.

We also need the following easily provable lemma.

**Lemma 2**

*Let*

*X*

*be a Banach space with basis*${\{{x}_{n}\}}_{n\in N}$

*and*${\{{y}_{n}\}}_{n\in N}\subset X:card\{n:{x}_{n}\ne {y}_{n}\}<+\mathrm{\infty}$.

*Then the expression*

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

The following lemma is also true.

**Lemma 3**

*Let*${\{{x}_{n}\}}_{n\in N}$

*be complete and minimal in*

*B*-

*space*

*X*

*and*${\{{y}_{n}\}}_{n\in N}\subset X:card\{n:{x}_{n}\ne {y}_{n}\}<+\mathrm{\infty}$.

*Then the following properties of system*${\{{y}_{n}\}}_{n\in N}$

*in*

*X*

*are equivalent*:

- (1)
${\{{y}_{n}\}}_{n\in N}$

*is complete*; - (2)
${\{{y}_{n}\}}_{n\in N}$

*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 ${\{{e}^{i{\lambda}_{n}t}\}}_{n\in Z}$ 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 $, ${\{{x}_{n}\}}_{n\in N}\subset X$

*be normed basis in*

*X*(

*i*.

*e*. $\parallel {x}_{n}\parallel =1$, $\mathrm{\forall}n\in N$)

*with conjugate system*${\{{x}_{n}^{\ast}\}}_{n\in N}\subset {X}^{\ast}$,

*and*${\{{y}_{n}\}}_{n\in N}\subset X$

*be a system satisfying the inequality*

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

*forms a basis in*${L}_{{p}_{t}}\equiv {L}_{{p}_{t}}(-\pi ,\pi )$,

*then this system is isomorphic to the classical system of exponents*${\{{e}^{int}\}}_{n\in Z}$,

*where the isomorphism is given by*

*where*

*Proof*Consider the operator (4). From the basicity of system ${\{{e}^{int}\}}_{n\in Z}$ 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 $(f,{e}^{inx})=0$, $\mathrm{\forall}n\in Z$. Also, from the basicity of system ${\{{e}^{int}\}}_{n\in Z}$ 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

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

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 $S[{e}^{int}]=A(t){e}^{int}$, $n\ge 0$, and $S[{e}^{-int}]=B(t){e}^{-int}$, $n\ge 1$. 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* $-\frac{1}{2p(\pi )}<\alpha <\frac{1}{2q(\pi )}$, *then the system* $\{{e}^{i{\mu}_{n}t}\}$ *forms a basis in* ${L}_{{p}_{t}}$.

*c*is some constant. Let us assume that the following inequalities are satisfied:

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

*f*by the system ${\{{e}^{i{\mu}_{n}t}\}}_{n\in Z}$, and let $g=Tf$. Therefore, ${\{{f}_{n}\}}_{n\in Z}$ are the Fourier coefficients of the function

*g*by the system ${\{{e}^{int}\}}_{n\in Z}$. From (4) and (5), it directly follows that

It follows immediately from (7) that the expression ${\sum}_{n}({e}^{i{\omega}_{n}t}-{e}^{i{\mu}_{n}t}){f}_{n}$ 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 $F[{e}^{i{\mu}_{n}t}]={e}^{i{\omega}_{n}t}$, $\mathrm{\forall}n\in Z$. Hence, the system ${\{{e}^{i{\omega}_{n}t}\}}_{n\in Z}$ forms a basis in ${L}_{{p}_{t}}$ isomorphic to ${\{{e}^{i{\mu}_{n}t}\}}_{n\in Z}$. Systems ${\{{e}^{i{\lambda}_{n}t}\}}_{n\in Z}$ and ${\{{e}^{i{\omega}_{n}t}\}}_{n\in Z}$ differ in a finite number of elements. Therefore, by Statement 1, the system ${\{{e}^{i{\lambda}_{n}t}\}}_{n\in Z}$ 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*

*be fulfilled*,

*where*$\tilde{p}=min\{{p}^{-};2\}$.

*Then the following properties of the system*(1)

*are equivalent in*${L}_{{p}_{t}}$:

- (1)
*the system*(1)*is complete*; - (2)
*the system*(1)*is minimal*; - (3)
*the system*(1)*is**ω*-*linear independent*; - (4)
*the system*(1)*is isomorphic to*${\{{e}^{int}\}}_{n\in N}$*basis*; - (5)
${\lambda}_{i}\ne {\lambda}_{j}$

*for*$i\ne j$.

Let us consider the case $\alpha =-\frac{1}{2p(\pi )}$. In this case, by the results of [11], the system ${\{{e}^{i{\mu}_{n}t}\}}_{n\in Z}$ 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 ${\{{e}^{int}\}}_{n\in Z}$ in ${L}_{{p}_{t}}$, and as a result the system ${\{{e}^{i{\mu}_{n}t}\}}_{n\in Z}$ should form a basis in ${L}_{{p}_{t}}$. This gives a contradiction.

is bounded in ${L}_{{p}_{t}}$. Introducing the new system ${\{{e}^{i{\omega}_{n}t}\}}_{n\in Z}$ 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 $-\frac{1}{2p(\pi )}\le \alpha <\frac{1}{2q(\pi )}$ is satisfied.

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

**Theorem 5**

*We have*:

- (I)
*Let the asymptotics*(2)*occur and the inequalities*(9)*be fulfilled*,*where*$\tilde{p}=min\{{p}^{-};2\}$.*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*${\{{e}^{int}\}}_{n\in N}$*basis*;(1.5) ${\lambda}_{i}\ne {\lambda}_{j}$

*for*$i\ne j$. - (II)
*Let*$\beta >1$*and*$\alpha =-\frac{1}{2p(\pi )}$.*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*${L}_{{p}_{t}}$. - (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*$-\frac{1}{2p(\pi )}\le \alpha <\frac{1}{2q(\pi )}$,*and for*$\alpha <-\frac{1}{2\pi}$*it is not complete*,*but it is minimal*;*and for*$\alpha \ge \frac{1}{2q(\pi )}$*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

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