Riesz basicity of the system of exponents with degenerated coefficients.

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  and K.I.Babenko .

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 , 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 , 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  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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