On bases from perturbed system of exponents in Lebesgue spaces with variable summability exponent
© Muradov; licensee Springer. 2014
Received: 26 September 2014
Accepted: 26 November 2014
Published: 12 December 2014
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.
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  on Riesz basicity. Some of the results in this direction have been obtained by Young . 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 . These investigations have allowed one to consider questions of basicity of some system of functions (for example, the classical system of exponents ) in . In  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
Throughout this paper, denotes the function conjugate to function , that is, .
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.
is ω-linear independent;
is isomorphic to basis.
We also need the following easily provable lemma.
generates the Fredholm operator , where is conjugate to system.
The following lemma is also true.
We will use the following statement, which has a proof similar to the proof of Levinson .
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 )
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.
where 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 .
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 ()
Let and . If parameter satisfies the condition , then the system forms a basis in .
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.
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.
the system (1) is complete;
the system (1) is minimal;
the system (1) is ω-linear independent;
the system (1) is isomorphic to basis;
Let us consider the case . In this case, by the results of , 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.
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.
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.
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 .
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 .
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 .
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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