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Academic Open Internet Journal |
Volume 13, 2004 |
Riesz
basicity of the system of exponents with degenerated coefficients.
Sadiq G.Veliyev
Nakhcivan Teachers Institute
AZ 7012,Nakhchivan,the
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.
Riesz
basicity of the system of exponents with degenerated coefficients.
Sadiq G.Veliyev
Nakhcivan Teachers Institute
AZ-7012,Nakhchivan,Azerbaiyan,Republic
E-mail:nmi@nakhchivan.az
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), ¹3.
2. Babenko K.I.// DAN SSSR, 1948, v.62, ¹2, p.157-160.
3. Bilalov B.T.// Different. Equations, 1990, v.26, ¹1, p.10-16.
4.
Technical College - Bourgas,