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Volume 12, 2004



Bases of exponents, which are the eigen functions of two discontinuous

differential operators

 

Sadig Guseinali oglu Veliyev

Nakhchivan Teachers Institute, AZ7012, Nakhchivan, the Azerbaijan Republic

E-mail: nmi@nakhchivan.az

 

In studying approximative properties of eigen functions of some differential operators it is necessary to investigate the basis properties of the system of exponents with degenerated coefficients of the form                                       

                                                  (1)

in spaces , where -are complex-valued on  functions; coefficients  have the following  presentations                                                         

                                               ,                                         (2)

are the sets of real numbers. Systems of the form (1) arise, when we consider discontinuous differential operators of the first order

                                         

on where .

Following V.A.Ilyin [1],  proceed from the generalized treatment of eigen functions of the operator , which allows the consideration quite arbitrary boundary conditions, i.e. under the eigen function of the operator , corresponding the eigen value , we understand any non-zero, piecewize-continuous function with discontinuity points , correspondingly, which is absolutely continuous on  and satisfies the equation  almost everywhere on .

Doing analogously as in the work of V.A.Ilyin and E.I.Moiseyev [2], i.e. taking «a half» of the eigen functions of the operators  and , corresponding the eigen values , we have the system of the form (1)

                             .

For studying the basis properties of the system (1) in   need some statements on basicity of classic system  of exponents  in the weight spaces and weight subspaces. We note that the questions, conñerning the given theme, were considered earlier in the works of V.F.Gaposhkin [3], K.S.Kazaryan, P.I.Lizorkin [4], E.I.Moiseyev [5,6]. Trivial case, when  basicity in  of the system (1) was considered earlier in the works of B.T.Bilalov (see, for example, [7,8]).

So, suppose that the functions satisfy the following conditions:

a)     are measurable on, moreover, it takes place

                                                        ,

where  is the norm in .

b)     are piecewize-Helder functions on ;  is the set of discontinuity points of the function .

Denote by  the jumps of the function  at the points  

                                       .

Further suppose that the function  is continuous from the left side on  and .  

First of all  introduce the necessary in the future denotations. Let .  Re-denote by . Construct the correspondence:  and . Define

                                                                           (3)

                                                                               (4)

                                                                                  (5)

where  is one-point set.

The following theorem is true.

Theorem. Let the complex-valued functions  satisfy the conditions a), b);  have the presentations (2). The numbers  are determined from (3)-(5). If the conditions

                           

                                           

are fulfilled, then the system (1) forms the basis in .

Before proving theorem,  introduce some notions.

Let  be usual Hardi class of analytical functions in the unity circle.  introduce the following weight Hardi class  

                                  

where  is almost everywhere measurable function,  are non-tangent boundary values of the function  at the point . Analogously the class  is introduced. Note, that earlier such classes were introduced in the works of A.P.Soldatov (see, for example [9]).

Let                                                                                                      (6)

                                        .                                                  (7)

Consider the following conjugation problem in classes

                                                           (8)

Under the solution of conjugation problem (8) in classes  we understand any pair of functions  the boundary values of which on unity circle satisfy the equality (8) almost everywhere and  is equal to zero on the infinity. General solution of the problem (8) has the form

                                            

where  is general solution of the corresponding homogeneous problem

                                                                            (9)

 is any particular solution of the problem (8). Define the following analytical functions inside and outside the unity circle:

                                   

                                    

                                      

Let                                                                  

Formally call the function  canonical solution of homogeneous problem (9). In proof of theorem 3 the following lemma plays the essential role.

Lemma. General solution of homogeneous problem (9) in classes , which has the order  on the infinity, can be presented in the form , if all conditions of theorem 3 are fulfilled, where  is polynomial of degree.

Proof. Applying Sokhotsky-Plemel formulas, we have:

                                                                                       (10)

We present  in the form:  where  is continuous part,  is jumps function, which is defined  by the formula:

                       

Let  where .

Denote by  

                        

It is not difficult to note, that . Again applying Sokhotsky-Plemel formula, we obtain:

                           

                               

where                             .

Introducing the function  

                                                          ,

and using the denotations (3)-(5), boundary value  can be presented in the form                           .

 As it follows from the results of the work [10, p.79], the functions  are summable with any degree  on .

Taking into account (10) in (9), we have:                                    

                                                .

Introduce the function :

                                       

As  has no zeroes and poles for , then the functions  and   have the same orders on the infinity. According to the definition . Moreover, as it follows from the results of the work [10, p.74],  for sufficiently small . Consequently,  for some  From the other side  and from the conditions of lemma it follows that . As a result  and thus ,  Consequently,   is a polynomial of degree , i.e. , and as a result                             

                                                  .                                               (11)

According to conditions  and . From (11) and on Smirnov theorem . Thus, according to definition .

Lemma is proved.

Proof of theorem: Consider the function

                                ,                                (12)

where  is canonical solution of homogeneous problem. According to Sokhotsky-Plemel formula the boundary values of the function  satisfy the equation (8) almost everywhere and  i.e.

                          

where  is any function. Let  . It is clear that . Denote by

                                              .

Applying Sokhotsky-Plemel formulas and transforming, we have:

                          .

From this presentation and from theorem 8.4 [10,p.141] in follows that , i.e. . As , then it is clear that             

                                               .

As a result  for some . From  it follows that . Then from Smirnov theorem and so . Inclusion  is proved analogously. From  it follows that the corresponding homogeneous problem has only trivial solution, i.e. non-homogeneous problem has unique solution (12).

Decomposing the functions and  into the systems correspondingly, we obtain that any function from  can be decomposed into the system (1). Further, in the equation (8) as the function  we take  and  then we obtain minimality of the system (1) in . Proof of theorem is completed.

Theorem is proved.

 

 

 

References:

 

1.     Ilyin V.A. // DAN SSSR, 1983, v.273, ¹5, p.1048-1053.

2.      Ilyin V.A., Moiseyev E.I. // Transactions of Math. Inst. Of RAS, 1992, v.201, p.219-230.

3.     Gaposhkin V.F.// Math. sbornik, 1958, v.46(88), ¹3.

4.     Kazaryan K.S., LizorkinP.I.// Transactions of Math. Inst. of RAS, 1989, v.187, p.111-130.

5.     Moiseyev E.I.//Different. Equations, 1998, v.34, ¹1, p.40-44.

6.     Moiseyev E.I. // Different. Equations, 1999, v.35, ¹2, p.200-205.

7.     Bilalov B.T.// Different. Equations, 1990, v.26, ¹1, p.10-16.

8.     Bilalov B.T.// Dokl. RAN, 1994, v.334, ¹4, p.416-419.

9.     Soldatov A.P. One-dimensional singular operators and boundary-value problems of functions theory, M., «Vicshaya Shkola», 1991, 208 p.

10. Danilyuk I.I. Lectures on boundary-value problems for analytical functions and singular integral equations, Novosibirsk, 1964, 226p.

 

 

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