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

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.

1 Introduction

Consider the following system of exponents:

$${\left\{e^{i{\lambda }_{n}t}\right\}}_{n\in Z},$$

(1)

where $\{{\lambda }_n\}\subset R$ 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 $p(t)$, when $\{{\lambda }_n\}$ has the asymptotics

$${\lambda }_n=n-\alpha signn+O(|n{|}^{-\beta }),n\to \mathrm{\infty },$$

(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 $L_p\equiv L_p(-\pi ,\pi )$, $1<p<+\mathrm{\infty }$, 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

Let $p:[-\pi ,\pi ]\to [1,+\mathrm{\infty })$ be a Lebesgue measurable function. By ${\mathrm{L}}_0$ we denote the class of all functions measurable on $[-\pi ,\pi ]$ with respect to Lebesgue measure. We choose the notation

$$I_p(f)\stackrel{\mathrm{def}}{\equiv }{\int }_{-\pi }^{\pi }|f(t){|}^{p(t)}dt.$$

Let $\mathrm{L}\equiv \{f\in {\mathrm{L}}_0:I_p(f)<+\mathrm{\infty }\}$. Let $p^{-}=inf{vrai}_{[-\pi ,\pi ]}p(t)$, $p^{+}=sup{vrai}_{[-\pi ,\pi ]}p(t)$. 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

$${\parallel f\parallel }_{p_t}\stackrel{\mathrm{def}}{\equiv }inf\{\lambda >0:I_p\left(\frac{f}{\lambda }\right)\le 1\},$$

then L is a Banach space and we denote it by $L_{p_t}$. Denote

$$\begin{array}{r}{H}^{ln}\stackrel{\mathrm{def}}{\equiv }\{p:p(\pi )=p(-\pi )\text{ and }\mathrm{\exists }C>0,\mathrm{\forall }{t}_1,t_2\in [-\pi ,\pi ],|t_1-t_2|\le \frac{1}{2}\\ \Rightarrow |p(t_1)-p(t_2)|\le \frac{C}{-ln|t_1-t_2|}\}.\end{array}$$

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

We have Hölder’s generalized inequality,

$${\int }_{-\pi }^{\pi }|f(t)g(t)|dt\le C(p^{-};p^{+}){\parallel f\parallel }_{p_t}{\parallel g\parallel }_{q_t},$$

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

$$Fx=\sum _{n=1}^{\mathrm{\infty }}{x}_n^{\ast }(x)y_n$$

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

$$\sum _{n=1}^{\mathrm{\infty }}\parallel x_n-y_n\parallel <{\gamma }^{-1},$$

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

$${\left\{e^{i(n-\alpha signn)t}\right\}}_{n\in Z},$$

(3)

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

$$Sf=e^{-i\alpha t}\sum _0^{\mathrm{\infty }}(f,e^{inx})e^{int}+e^{i\alpha t}\sum _1^{\mathrm{\infty }}(f,e^{-inx})e^{-int},$$

(4)

where

$$(f,g)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(t)\overline{g(t)}dt.$$

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

$$f=\sum _{n\in Z}{g}_n{e}^{int},$$

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

It is clear that $f\in L_{p_t}$, and so

$$\begin{array}{rl}Sf& =e^{-i\alpha t}\sum _{n=0}^{\mathrm{\infty }}(f,e^{inx})e^{int}+e^{i\alpha t}\sum _{n=1}^{\mathrm{\infty }}(f,e^{-inx})e^{-int}\\ =e^{-i\alpha t}\sum _{n=0}^{\mathrm{\infty }}{g}_n{e}^{int}+e^{i\alpha t}\sum _{n=1}^{\mathrm{\infty }}{g}_{-n}{e}^{-int}=g,\end{array}$$

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 $S[e^{int}]=A(t)e^{int}$, $n\ge 0$, 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 $L_{p_t}$.

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

$$|e^{i{\lambda }_{n}t}-e^{i{\mu }_{n}t}|\le c|n{|}^{-\beta },\mathrm{\forall }n\ne 0,$$

(5)

is fulfilled, where c is some constant. Let us assume that the following inequalities are satisfied:

$$-\frac{1}{2p(\pi )}<\alpha <\frac{1}{2q(\pi )},\beta >\frac{1}{\tilde{p}},$$

(6)

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.

Let $T:L_{p_t}\to L_{p_t}$ be a natural automorphism

$$T\left[e^{i{\mu }_{n}t}\right]=e^{int},\mathrm{\forall }n\in Z.$$

For all $f\in L_{p_t}$, let ${\{f_n\}}_{n\in Z}$ be biorthogonal coefficients of 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

$$\sum _{n\in Z}{\parallel e^{i{\lambda }_{n}t}-e^{i{\mu }_{n}t}\parallel }_{p_t}^{\tilde{p}}<+\mathrm{\infty }.$$

Consider the following expression:

$$\sum _n(e^{i{\lambda }_{n}t}-e^{i{\mu }_{n}t})f_n.$$

We have

$$\begin{array}{rl}{\parallel \sum _{n\in Z}(e^{i{\lambda }_{n}t}-e^{i{\mu }_{n}t})f_n\parallel }_{p_t}& \le \sum _{n\in Z}\parallel e^{i{\lambda }_{n}t}-e^{i{\mu }_{n}t}\parallel |f_n|\\ \le {\left(\sum _n{\parallel e^{i{\lambda }_{n}t}-e^{i{\mu }_{n}t}\parallel }_{p_t}^{\tilde{p}}\right)}^{1/\tilde{p}}{(\sum _n|f_n{|}^{\tilde{q}})}^{1/\tilde{q}},\end{array}$$

where $\frac{1}{\tilde{p}}+\frac{1}{\tilde{q}}=1$. By the Hausdorff-Young theorem [16], we have

$${(\sum _n|f_n{|}^{\tilde{q}})}^{1/\tilde{q}}\le m_1{\parallel g\parallel }_{\tilde{p}},$$

where $m_1$ is some constant. From $\tilde{p}\le p^{-}$ and the continuous embedding $L_{p_t}\subset L_{\tilde{p}}$, it follows that, $\mathrm{\exists }{m}_2>0$,

$${\parallel g\parallel }_{\tilde{p}}\le m_2{\parallel g\parallel }_{p_t}\le m_2\parallel T\parallel {\parallel f\parallel }_{p_t}.$$

As a result, we obtain

$${\parallel \sum _n(e^{i{\lambda }_{n}t}-e^{i{\mu }_{n}t})f_n\parallel }_{p_t}\le m_1{m}_2\parallel T\parallel {\left(\sum _n{\parallel e^{i{\lambda }_{n}t}-e^{i{\mu }_{n}t}\parallel }_{p_t}^{\tilde{p}}\right)}^{1/\tilde{p}}{\parallel f\parallel }_{p_t}.$$

(7)

Let us take $n_0\in N$ such that

$$\delta =m_1{m}_2\parallel T\parallel {\left(\sum _{|n|>n_0}{\parallel e^{i{\lambda }_{n}t}-e^{i{\mu }_{n}t}\parallel }_{p_t}^{\tilde{p}}\right)}^{1/\tilde{p}}<1.$$

Assume that

$${\omega }_n=\{\begin{array}{ll}{\lambda }_n,& |n|>n_0,\\ {\mu }_n,& |n|\le n_0.\end{array}$$

It is clear that the following inequality is satisfied:

$${\parallel \sum _n(e^{i{\omega }_{n}t}-e^{i{\mu }_{n}t})f_n\parallel }_{p_t}\le \delta {\parallel f\parallel }_{p_t}.$$

(8)

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

$$-\frac{1}{2p(\pi )}<\alpha <\frac{1}{2q(\pi )},\beta >\frac{1}{\tilde{p}},$$

(9)

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.

By ${\{v_n\}}_{n\in Z}\subset L_{q_t}$ we denote the system biorthogonal to ${\{e^{i{\mu }_{n}t}\}}_{n\in Z}$. 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

$$\tilde{T}f=\sum _n{v}_n(f)(e^{i{\lambda }_{n}t}-e^{i{\mu }_{n}t}),\mathrm{\forall }f\in L_{p_t},$$

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.

Consider the case $\alpha \notin [-\frac{1}{2p(\pi )},\frac{1}{2q(\pi )})$. Let, for example, $\alpha \in [\frac{1}{2q(\pi )},\frac{1}{2q(\pi )}+\frac{1}{2})$. 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

$$e^{i[\tilde{\alpha }+{\tilde{\alpha }}_0]t}\bigcup {\left\{e^{i{\tilde{\lambda }}_{n}t}\right\}}_{n\in Z},$$

(10)

where $\tilde{\alpha }=\alpha -\frac{1}{2}$ and

$${\tilde{\lambda }}_n=n-\tilde{\alpha }signn+O(|n{|}^{-\beta }),n\to \mathrm{\infty }.$$

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 $L_{p_t}$.