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 [15]. 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 where Consequently where 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 [4]). Then from the results of the paper [16], the Hausdorff-Young inequality is valid for this system; that is,
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