- Open Access
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.
- system of exponents
- generalized Lebesgue space
- variable exponent
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.
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.
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.
- Paley R, Wiener N: Fourier Transforms in the Complex Domain. Am. Math. Soc., Providence; 1934.MATHGoogle Scholar
- Young RM: An Introduction to Nonharmonic Fourier Series. Academic Press, New York; 1980.MATHGoogle Scholar
- Sedletskii AM: Biorthogonal expansions in series of exponents on intervals of the real axis. Usp. Mat. Nauk 1982,37(5(227)):51–95.MathSciNetGoogle Scholar
- Moiseev EI:Basicity of system of exponents, cosines and sines in . Dokl. Akad. Nauk SSSR 1984,275(4):794–798.MathSciNetGoogle Scholar
- Kováčik O, Rákosník J:On spaces and . Czechoslov. Math. J. 1991, 41: 592–618.MATHGoogle Scholar
- Fan X, Zhao D:On the spaces and . J. Math. Anal. Appl. 2001, 263: 424–446. 10.1006/jmaa.2000.7617MathSciNetView ArticleMATHGoogle Scholar
- Kokilashvili V, Samko S: Singular integrals in weighted Lebesgue spaces with variable exponent. Georgian Math. J. 2003,10(1):145–156.MathSciNetMATHGoogle Scholar
- Sharapudinov II:Topology of the space . Mat. Zametki 1979,26(4):613–632.MathSciNetMATHGoogle Scholar
- Sharapudinov II:Some questions of approximation theory in spaces. Anal. Math. 2007,33(2):135–153. 10.1007/s10476-007-0204-0MathSciNetView ArticleGoogle Scholar
- Bilalov BT, Guseynov ZG: Bases from exponents in Lebesgue spaces of functions with variable summability exponent. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 2008,27(1):43–48.MathSciNetMATHGoogle Scholar
- Bilalov BT, Guseynov ZG: Basicity of a system of exponents with a piece-wise linear phase in variable spaces. Mediterr. J. Math. 2012,9(3):487–498. 10.1007/s00009-011-0135-7MathSciNetView ArticleMATHGoogle Scholar
- Bilalov BT, Guseynov ZG: Basicity criterion for perturbed systems of exponents in Lebesgue spaces with variable summability. Dokl. Akad. Nauk, Ross. Akad. Nauk 2011,436(5):586–589.Google Scholar
- Zinger I: Bases in Banach Spaces. I. Springer, Berlin; 1970.Google Scholar
- Bilalov BT: Bases of exponentials, cosines and sines formed by eigenfunctions of differential operators. Differ. Equ. 2003,39(5):652–657. 10.1023/A:1026189819533MathSciNetView ArticleMATHGoogle Scholar
- Levinson N Collog. Publ. 26. In Gap and Density Theorems. Am. Math. Soc., Providence; 1940.Google Scholar
- Zygmund A 1. In Trigonometric Series. Mir, Moscow; 1965.Google Scholar
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