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.
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1 Introduction
Consider the following system of exponents:
where 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 with variable summability index , when has the asymptotics
where 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 , , when , 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 and Sobolev spaces with variable summability index 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 [7]. These investigations have allowed one to consider questions of basicity of some system of functions (for example, the classical system of exponents ) in . In [9] the basicity of system in has been established. The special case of the system (1) is considered in [10–12], when , .
In this paper basis properties of the system (1) in 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 are proved.
2 Necessary notion and facts
Let be a Lebesgue measurable function. By we denote the class of all functions measurable on with respect to Lebesgue measure. We choose the notation
Let . Let , . For , with respect to ordinary linear operations of addition of functions and multiplication by number, L turns into a linear space. If we define in the norm
then L is a Banach space and we denote it by . Denote
Throughout this paper, denotes the function conjugate to function , that is, .
We have Hölder’s generalized inequality,
where .
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; is the conjugate to space X; , , and means the value of functional f on x; is a linear span of a set M. The system is called ω-linear independent in B-space X, if true for , .
The following lemma is true.
Lemma 1 Let X be a Banach space with basis and be a Fredholm operator. Then the following properties of the system in X are equivalent:
-
(1)
is complete;
-
(2)
is minimal;
-
(3)
is ω-linear independent;
-
(4)
is isomorphic to basis.
We also need the following easily provable lemma.
Lemma 2 Let X be a Banach space with basis and . Then the expression
generates the Fredholm operator , where is conjugate to system.
The following lemma is also true.
Lemma 3 Let be complete and minimal in B-space X and . Then the following properties of system in X are equivalent:
-
(1)
is complete;
-
(2)
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 be complete in . If from the system we remove n any functions and add instead of them n other functions , , where are any, mutually different complex numbers not equal to any of numbers , 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 , be normed basis in X (i.e. , ) with conjugate system , and be a system satisfying the inequality
where . Then also forms a basis isomorphic to the basis in X.
3 Basic results
Before giving the basic results we will prove the following auxiliary theorem.
Theorem 2 Let and . If the system
forms a basis in , then this system is isomorphic to the classical system of exponents , where the isomorphism is given by
where
Proof Consider the operator (4). From the basicity of system in it follows that S is a bounded operator on into itself. It is easy to see that . Actually, let . From the basicity of the system (3) in and from (4) we obtain , . Also, from the basicity of system in it follows that . We show that for all , the equation in is solved. Let us assume that
where are the biorthogonal coefficients of the function g by the system (3).
It is clear that , and so
as by the condition of the theorem, the system (3) forms a basis in .
This means that for all the equation is solved in . Then by the Banach theorem the operator S has a bounded inverse. It is obvious that , , and , . This completes the proof. □
Now we study some basis properties of the system (1). Firstly, we recall the following theorem.
Theorem 3 ([11])
Let and . If parameter satisfies the condition , then the system forms a basis in .
Let the asymptotics (2) occur. Let us assume and , . It is easy to see that the inequality
is fulfilled, where c is some constant. Let us assume that the following inequalities are satisfied:
where . Then, from Theorem 3, the system of exponents forms a basis in . By Theorem 1, it is isomorphic to the classical system of exponents in . Therefore the spaces of coefficients of the bases and coincide.
Let be a natural automorphism
For all , let be biorthogonal coefficients of f by the system , and let . Therefore, are the Fourier coefficients of the function g by the system . From (4) and (5), it directly follows that
Consider the following expression:
We have
where . By the Hausdorff-Young theorem [16], we have
where is some constant. From and the continuous embedding , it follows that, ,
As a result, we obtain
Let us take such that
Assume that
It is clear that the following inequality is satisfied:
It follows immediately from (7) that the expression represents a function from and it can be denoted by . Drawing attention to (8) we obtain . Thus, the operator is invertible, and it is easy to see that , . Hence, the system forms a basis in isomorphic to . Systems and differ in a finite number of elements. Therefore, by Statement 1, the system is complete in , if for . 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 . Then the following properties of the system (1) are equivalent in :
-
(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 basis;
-
(5)
for .
Let us consider the case . In this case, by the results of [11], the system is complete and minimal in , 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 . Because otherwise, by Theorem 2, it will be isomorphic to system in , and as a result the system should form a basis in . This gives a contradiction.
By we denote the system biorthogonal to . It is clear that using the estimates from [4], for , , we establish that the following relation is true:
Let . Then it is clear that the following inequality is satisfied:
Similarly to the previous case, we can show that the operator
is bounded in . Introducing the new system in the same manner we establish the completeness of the system (1) in , if for . Minimality of the system (1) in follows from Lemma 3. Thus, if for and , then the system (1) is complete and minimal in if the condition is satisfied.
Consider the case . Let, for example, . Multiplication of each member of the system (1) by does not affect its basis properties in . After appropriate transformations we obtain the system
where and
Denote by the member of in (2), corresponding to . It is easy to see that condition is equivalent to . It is clear that . Then, by the previous results, the system is complete and minimal in , and therefore the system (10), and at the same time the system (1), is complete, but it is not minimal in . Continuing this process we find that the system (1) is not complete, but it is minimal for ; and the system (1) is complete, but it is not minimal in for . Thus, the following theorem is proved.
Theorem 5 We have:
-
(I)
Let the asymptotics (2) occur and the inequalities (9) be fulfilled, where . Then the following properties of the system (1) are equivalent in :
(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 basis;
(1.5) for .
-
(II)
Let and . Then the following properties of the system (1) in are equivalent:
(2.1) the system (1) is complete;
(2.2) the system (1) is minimal;
(2.3) , for .
Moreover, in this case the system (1) does not form a basis in .
-
(III)
Let and , for . Then the system (1) is complete and minimal in for , and for it is not complete, but it is minimal; and for it is complete, but it is not minimal in .
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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.
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Muradov, T. On bases from perturbed system of exponents in Lebesgue spaces with variable summability exponent. J Inequal Appl 2014, 495 (2014). https://doi.org/10.1186/1029-242X-2014-495
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DOI: https://doi.org/10.1186/1029-242X-2014-495