Basicity of the System of Exponents with a Linear Phase in Sobolev Weight Space

where α(t) and β(t) are continuous or piecewise-continuous functions. Substantiation of themethod requires studying the basis properties of these systems in Lebesgue and Sobolev spaces of functions. In the case when α(t) and β(t) are linear functions, the basis properties of these systems in L p (−π, π), 1 < p < +∞, were completely studied in the papers [1– 9]. The weighted case of the space L p was considered in the papers [10, 11]. The basis properties of these systems in Sobolev spaceswere 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 α(t) = αt, β(t) = αt, where α 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 L p,ρ (−π, π) andW p,ρ (−π, π) be weight spaces with the norms

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.
The following easily provable lemmas play an important role in obtaining the main results.It holds the following.
From (11) it directly follows that  = 0 a.e. on (−, ), and so û = 0. Show that Im  =  1 , (−, ) (Im  is the range of values of the operator ).Let V ∈  1  , (−, ) be an arbitrary function.Let V = (V  ; V(0)).It is clear that V = V and V ∈ L , .Then from the Banach theorem we get that the operator  is boundedly invertible.
The lemma is proved.
The following lemma is also valid.
In obtaining the basic results we need the following main lemma.
From the previous reasonings we get the absolute convergence of the series ∑ ||>|| |  /( +  sign )|.The lemma is proved.