Basicity of the System of Exponents with a Linear Phase in Sobolev Weight Space
In the present paper a criterion for basicity of exponential system with linear phase is obtained in Sobolev weight space .
In solving mathematical physics problems by the Fourier method, there often arise the systems of exponents of the form where and are continuous or piecewise-continuous functions. Substantiation of the method requires studying the basis properties of these systems in Lebesgue and Sobolev spaces of functions. In the case when and are linear functions, the basis properties of these systems in , , were completely studied in the papers [1–9]. The weighted case of the space was considered in the papers [10, 11]. The basis properties of these systems in Sobolev spaces were studied in [12–14]. It should be noted that the close problems were also considered in .
In the present paper we study basis properties of the systems (1) and (2) in Sobolev weight spaces when , , where is a real parameter. Therewith the issue of basicity of system (2) in Sobolev spaces is reduced to the issue of basicity of system (1) in respective Lebesgue spaces.
Let and be weight spaces with the norms respectively, where , . Denote by the direct sum , where is the complex plane. The norm in this space is defined by the expression , where .
The following easily provable lemmas play an important role in obtaining the main results. It holds the following.
Lemma 1. Let , ; . Then the operator realizes an isomorphism between the spaces and ; that is, the spaces and are isomorphic.
Proof. At first prove the boundedness of the operator . We have
Having applied the Holder inequality, hence we get
Let us show that . Put ; that is,
where , . By differentiating this equality, we get , a.e. on . Hence it follows that . From (11) it directly follows that a.e. on , and so . Show that ( is the range of values of the operator ). Let be an arbitrary function. Let . It is clear that and . Then from the Banach theorem we get that the operator is boundedly invertible.
The lemma is proved.
The following lemma is also valid.
Lemma 2. Let and , . Then for all , where
Proof. Let , . We have
Since and , then , . It is easy to see that and moreover .
The lemma is proved.
In obtaining the basic results we need the following main lemma.
Lemma 3. Let , and , , be a real parameter, . Let have the expansion in the space . Then it is valid
Proof. As it follows from Lemma 2, . At first consider the case when , . In this case the system is minimal in (see ). Then from the results of the paper , the Hausdorff-Young inequality is valid for this system; that is,
Applying the Holder inequality, we obtain
If , then , . Then
where , . As a result, we have
Since , then again from the results of the paper  it follows
In the same way we establish the convergence of the series .
Consider the case when . Then . In this case the system is minimal in , and, consequently, from the results of the paper  it holds the Hausdorff-Young inequality; that is, where .
From the previous reasonings we get the absolute convergence of the series . The lemma is proved.
Theorem 4. Let , be a weight function, , , be a real parameter, and the inequalities , ; hold. Then the following statements are equivalent:(1)system (2) forms a basis for ;(2)system (1) forms a basis for .
Proof. At first, suppose that the system (1) forms a basis for . Let us show that the system forms a basis for , where , , .
It is enough to prove that the arbitrary element of has the unique expansion that is, Since the system (1) forms a basis for , then the expansion (23) holds and the coefficients are uniquely determined. By Lemma 3 the series absolutely converges. Then it is clear that the number from (24) is uniquely determined. This means that the system forms a basis for . Consider the system , where , , . It is not difficult to see that
Now, let us prove the converse. Assume that the system (2) forms a basis for . Consider the system , . It is easy to see that the inverse operator is determined as . It is obvious that the system forms a basis for . We have Consequently, has a unique expansion (22) in . As a result, we obtain that each has a unique expansion of the form (23). Indeed, let there exist another expansion for in : The absolute convergence of the series follows from Lemma 3. Put . It is clear that the biorthogonal coefficients of the element are , where . From the basicity of the system in we obtain that , . This contradicts our conjecture. The theorem is proved.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to express their sincere gratitude to Professor Bilal T. Bilalov for his attention to the paper and for valuable advice.
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