Exponential Atomic Decomposition in Generalized Weighted Lebesgue Spaces

This paper treats the exponential linear phase system which consists of eigenfunctions of the discontinuous differential operator. Frame properties of this system are studied in weighted Lebesgue spaces with the variable order of summability.


Introduction
Perturbed system of exponents   Z n t i n e   plays an important role in the study of spectral properties of discrete differential operators and in the approximation theory.Apparently, the study of basis properties of these systems dates back to the well-known work of (Paley & Wiener, 1934).Since then, a lot of research has been made in this field (more details can be found in Sedletskii, 2005;Duffin & Schaeffer, 1952;Young, 1980;Christensen , 2003;Heil, 2011 ).It should be stressed that similar systems are of great scientific interest in the frame theory as well.
Since recently, there arose a great interest in considering various problems, related to some research fields of mechanics and mathematical physics, in generalized Lebesgue spaces p  (for more information see Kovacik & Rakosnik , 1991;Xianling & Dun , 2001;Kokilashvili & Paatashvili, 2006;Sharapudinov, 2007;Kokilashvili & Samko, 2003).Application of Fourier method to the problems for partial differential equations in generalized Sobolev classes requires a good knowledge of approximative properties of perturbed exponential systems in generalized Lebesgue spaces.Approximation-related issues in these spaces have been first studied by Sharapudinov, 2007.In this work, we consider an exponential linear phase system.The study of frame properties of perturbed exponential systems is closely related to the one of similar properties of perturbed sine and cosine systems.Note that the linear phase sine and cosine systems appear when solving partial differential equations by Fourier method.Basis properties of linear phase trigonometric systems have been studied in (Moiseev, 1984;Moiseev, 1987;Bilalov, 1990;Bilalov, 1999;Bilalov. 2001;Bilalov, 2003;Bilalov, 2004;Moiseev, 1998;Moiseev, 1999).This work is dedicated to the study of frame properties (atomic decomposition, frameness) of the exponential piecewise linear phase system in generalized weighted Lebesgue space.

Needful Information
We will use the usual notations.N  will be a set of all positive integers; Z  will be a set of all integers;   0; Z N R     will be the set of all real numbers; C  will stand for the field of complex numbers;   With respect to the usual linear operations of addition and multiplication by a number, L is a linear space as . The following generalized Hölder inequality is true Directly from the definition we get the property which will be used in sequel.
The following facts play an important role in obtaining the main results.
The validity of the following statement is established in Kokilashvili & Samko, 2003.
. Then, singular operator S is acting boundedly from Let X be some Banach space with a norm X  .Then * X will denote its conjugate with a norm we denote the linear span of the set X M  , and M will stand for the closure of M .
The following criteria of completeness and minimality are available.
, where forms a basis for X , then it is uniformly minimal.
We will also need some facts about an atomic decomposition and frames in Banach spaces.
Definition 1.Let X be a Banach space and K a Banach sequence space indexed by N .Let is an atomic decomposition of X with respect to K if : The concept of the frame is a generalization of the concept of an atomic decomposition.
Definition 2. Let X be a Banach space and K a Banach sequence space indexed by N .Let A and B will be called the frame bounds.
It is true the following Proposition 1.Let X be a Banach space and K a Banach sequence space indexed by N with canonical basis   . Then the following statements are equivalent: is an atomic decomposition of X with respect to K .

Let
   be a unit ball on the complex plane, and let   be a unit circumference.Denote . Applying Hölder's inequality, we get It is not difficult to see that , holds if and only if the following inequality is fulfilled Hence we get the condition on the parameter 0 p : then it follows directly from relations (1) that it is always possible , 1 0 such that the inequalities (3) are satisfied.As a result, we obtain from (2) that , ), ( Using this lemma, we prove the following one.
, and let the weight where C is a constant (and in further too) .Consequently .Lemma is proved.The following theorem is true.
, and let the inequalities (1) be satisfied.If where Proof.First, we consider the necessity.Let . Classical results tell us (see e.g.Kusis, 1984) that a.e. on ) , (
, and C is a constant independent of r and f .Theorem is proved.
Similarly we define the weighted Hardy classes  , and let the inequalities (1) be satisfied.If Vice versa, if , then the function F , defined by ( 5), belongs to the class

The weighted Hardy class
, where  u be non-tangential boundary value on   of u .Then it is known that . Hence directly follows that . Then by Theorem 2 we obtain . Thus, the following theorem is true.
and let the inequalities (1) be satisfied.

L
. Take . If the inequalities (1) hold, then . Then, as is known More information about this fact can be found in Kusis,1984;Privalov, 1950.If the inequalities (1) hold, then it follows from the results in Danilyuk, 1975 that the system of exponents E forms a basis for . Taking into account (6), we obtain that f can be expanded in a series in , where n f are the biorthogonal coefficients of f with respect to the system E .It is absolutely clear that such an expansion is unique.Consequently, the system . Similarly it can be proved that if the inequalities (1) hold, then the system . Thus, the following theorem is true.
and let the inequalities (1) be satisfied.Then the system

H Classes
Consider the following Riemann problem in is some function.By the solution of problem (7) we mean a pair of analytic functions , boundary values of which satisfy the relation ( 7) almost everywhere.
Introduce the following functions , which are analytic inside (with the + sign) and outside (with thesign) the unit circle, respectively: the canonical solution of the problem (7).Substituting the ( 8) , and define the piecewise analytic function , because the rest will immediately follow from the Smirnov theorem (Danilyuk, 1975) .
We will suppose that the coefficient ) ( G satisfies the following conditions: As is known (see Danilyuk, 1975 ), the boundary values It follows directly from the Sokhotski-Plemelj formula that Thus, the following representation is true for By the definition of solution, we have , then we obtain directly from the Hölder inequality that ) L .
We will need the following easy-to-prove lemma that follows directly from Statement 1.
Then the function , and k  is defined by the relation Taking into account Lemma 3, we obtain that if the inequalities are true, then the product

H
. We have , and hence are true, then the general solution of the homogenous problem and suppose The Sokhotski-Plemelj formulas imply that the boundary values F satisfy a.e. the equality (7).Moreover, it follows from Statement 2 that . As a result, we get the validity of the following theorem.
 be defined by ( 10) and the inequalities ( 1), ( 12) be satisfied.Then the general solution of the Riemann problem ( 7) in classes is the particular solution of non-homogenous problem (7) defined by ( 13), and

Atomic Decomposition
Consider the following exponential linear phase system where C   is a complex parameter.To explore the decomposition with regard to the system (14), we will follow the scheme of Bilalov & Guseynov, 2012 .Consider the conjugate problem where   of the problem (15) has the following form Proceeding as in the previous section, we get   . Then from Theorem 7 we obtain that if the inequalities are true, then the conjugate problem (15) has a unique solution . Also, it can be proved (in the same way as in Bilalov & Guseynov, 2012;Bilalov & Guseinov, February, 2011) that, with the inequalities ( 16) and ( 17) fulfilled, the system (14) forms a basis for Thus, the following theorem is true.
K is an isomorphism between , because the system ( 14) forms a basis for and, moreover, be the system conjugate to E .Then, by definition (see, e.g.Christensen, 2003), the pair  

L
, where , and where . If the inequalities ( 16), ( 19) are true, then this problem is uniquely solvable in classes . It is obvious that if the conditions (1) are satisfied, then the system   It is absolutely clear that the system is linearly independent when 1 0  k .We denote the linear span of this system by   Obviously, in this case the system E forms a basis for  16),( 17) be fulfilled, and the set  P be defined by (18).Then, the inequalities ( 21), ( 22) hold with regard to the space of coefficients of the system (14,) usual Hardy class, where coefficients of the system (14).It is known that the canonical system  

K
is some B -space of the sequences of scalars furnished with the norm d Without loss of generality, we will assume that the operator S is defined on that the B -space of the sequences of scalars d is not difficult to see that the following direct decompositions hold: It is not difficult to see that the function   So we have arrived at the conclusion that the following theorem is true.
The author would like to express her deepest gratitude to Professor B.T. Bilalov for his encouragement and valuable guidance throughout this research.