Riesz basicity of the system of exponents with degenerated coefficients.

 

Sadiq G.Veliyev

 

Nakhcivan Teachers Institute

AZ 7012,Nakhchivan,the Azerbaiyan Republic

E-mail;nmi@nakhcivan.az

 

                              

          System of exponents of the form

                                        

where are complex-valued functions,  is degenerated in some points of the segment , is considered in the paper. Necessary and sufficient condition of Riesz basicity for this  system in   is obtained for definite conditions.

          Bibl. 4 names.

 

 

                 

Consider the following system of exponents with degenerated coefficients :

                                                            (1)

where -are complex-valued functions on the segment ;  has the presentation                                                         

                                                                                             (2)

 are the sets of real numbers. Earlier we studied the basicity in  of the system (1) for definite conditions on the functions  and . In offered paper we consider the problem on Riesz basicity of the system (1) in . Earlier the same problem relatively the system of exponents  was investigated in the work of V.F.Gaposhkin [1] and K.I.Babenko [2].

We do the following suppositions relatively the functions  and .

1)      are piecewise-Helder functions on the segment ,  is the set of discontinuity points of the function  on , and moreover

                                  ;

2)      are measurable functions on , and satisfy the condition

                                  

Denote by  the jumps of the function  at the points , i.e. .

The following theorem takes place.

Theorem. Let the conditions 1), 2) take place and the following inequalities

                                           

where  are fulfiled. Then the system (1) forms Riesz basis in  if and only if .

Proof. Consider the system     

                             

Then, as it follows from the results of the work [3], system forms Riesz basis in   if the conditions of theorem are fulfiled. Denote by  biorthogonal system to system (3). It is quite obvious that biorthogonal  to (1) system  has the form: .  Again, according to the results of the work [3], it takes place:  in sufficiently small neighbourhoods of the points . First of all we suppose that . Then from the previous arguments and from the presentation (2) for  it follows that , such that . From Riesz basicity of system (3) in  it follows the divergence of series , where are biorthogonal coefficients of the function  on system (3). Consequently, in this case the system (1) doesn’t form Bessel basis and, also Riesz basis in .

And now let the series  converges for some sequence of numbers . As system (3) forms Riesz basis in , then  for which  are biorthogonal coefficients, i.e.  where

. If , then it is clear that  and according to the conditions of theorem . As a result we obtain, that for  the system (1) forms Hilbert basis in .

          We consider the case, if  for some . Then  such that . Let  are biorthogonal coefficients of the function  on system (3). From Riesz basicity of (3) in  we have: . It is quite obvious that this sequence is biorthogonal coefficient of the functions  on system (1).  From here it follows that the system (1) is not Hilbert basis, as . Really, we can take  such, that  for some , where .

          According to the results of the paper [4] the conjugate system for Hilbert basis is Bessel basis, consequently,  is complete in  and then it is complete in . As a result for such sequence  from  there is no the function , such that  are biorthogonal coefficients of this function on system (1).

Let . We take . It is clear that . From Riesz basicity of system (3) it follows that  where  are biorthogonal coefficients of the function  on system (3), or function  on system (1). Consequently, in this case the system (1) forms Bessel basis in .

Theorem is proved.

The following corollary directly follows from the proof of this theorem.

Corollary. Let all conditions of theorem are fulfiled. Then the system (1) forms Hilbert (Bessel) basis in  if and only if .

Author is very greateful to prof. S.S.Mirzoyev for the attention to the work.

 

 

References:

 

1.     Gaposhkin V.F. // Mat. sbornik, 1958, v.46 (88), №3.

2.      Babenko K.I.// DAN SSSR, 1948, v.62, №2, p.157-160.

3.     Bilalov B.T.// Different. Equations, 1990, v.26, №1, p.10-16.

4.     Bari N.K.// Scientific notices of MSU, 1951, v.4, issue 148.

 

 

 

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