%0 Journal Article
%T Basicity of Systems of Sines with Linear Phase in Weighted Sobolev Spaces
%A V. F. Salmanov
%A A. R. Safarova
%J International Journal of Engineering Mathematics
%D 2013
%R 10.1155/2013/612472
%X The perturbed systems of sines, which appear when solving some partial differential equations by the Fourier method, are considered in this paper. Basis properties of these systems in weighted Sobolev spaces of functions are studied. 1. Introduction When solving many problems in mathematical physics by Fourier method (see e.g., [1¨C4]), there appear perturbed systems of sines and cosines of the following form: where , are real parameters. Using Fourier method requires the study of basis properties of the above systems in Lebesgue and Sobolev spaces of functions. Relevant investigations date back to the well-known works by Paley and Wiener [5] and Levinson [6]. For , basis properties of these systems in spaces , , are completely studied in [7¨C12]. The case of weighted was considered by E. I. Moiseev in [13, 14]. Basis properties of some perturbed systems of exponents in Sobolev spaces are studied in [15¨C19]. Further references include [20¨C23]. Our paper is devoted to the study of basis properties of these systems in weighted Sobolev spaces. Unlike previous works, we offer a different method of investigation. 2. Auxiliary Facts Let and be weighted Lebesgue and Sobolev spaces with the following norms: where , , . Denote by the following direct sum: where is a complex plane. The norm in this space is defined as follows: , where . The following easily provable lemmas play an important role in obtaining our main result. Lemma 1. Let . Then the operator performs an isomorphism between the spaces and ; that is, the spaces and are isomorphic. Proof. First we show the boundedness of this operator. We have the following: Applying Hlder's inequality, we obtain the following: Consequently, where . Let us show that . Let ; that is, where , . Differentiating this equation, we obtain a.e. on . It follows that . From (9) it directly follows that a.e. on and this implies that . Show that . Let be an arbitrary function. Assume . It is clear that and . Then by Banach theorem we find that the operator has a bounded inverse. This proves Lemma 1. Now let us prove the following lemma. Lemma 2. Let and . Then for all , where . Proof. Let ,£¿£¿ . We have the following: As and , then . Similarly, we find that and . It is easy to see that and, moreover, This proves the lemma. From results of the paper [24] it follows the validity of the following lemma. Lemma 3. Let , , , and in . Then the series converges absolutely. 3. Main Result Theorem 4. Let , . Then system (2) forms a basis for if and only if system (1) forms a basis for , where , . Proof. First let us assume that the system
%U http://www.hindawi.com/journals/ijem/2013/612472/