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Academic Open Internet Journal |
Volume 16, 2005 |
Necessary and sufficient condition of the completeness and minimality for one system of exponents with degeneration.
Sadig Guseinali oglu Veliyev
Nakhchivan Teachers Institute,AZ7012, Nakhchivan the Azerbaijan Republic
E-mail:nmi@nakhchivan.az
System of exponents of the form
where 
are complex-valued
functions, 
are
degenerated in some points of the segment 
, is considered in the paper. Necessary
and sufficient condition for this system in 
, is obtained for definite conditions on
these functions.
Bibl. 2 names.
The following system of exponents with “degenerated”
coefficients is
considered:
(1)
where 
-are complex-valued
functions on the segment 
; 
have the
presentations
(2)
are the sets of real
numbers. Earlier we obtained the completeness and minimality of the system (1)
in the space 
for definite conditions on the functions 
and
the coefficients 
. In offered paper we obtain the necessary and
sufficient condition of the completeness and minimality for this system in 
for concrete
conditions on the functions 
. We require the fulfilling of the
following conditions:
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. 
.
Integer numbers 
we define from the
following correlations:
(3)
Let 
. The following theorem
takes place.
Theorem. Let
the functions satisfy the conditions 1), 2); the coefficients have the presentations
(2), moreover, the inequalities
are fulfiled. Then the
system (1) is complete in if and only if 
is minimal in if and only if 
where the value 
is defined from (3).
Before proving this theorem we give some earlier known facts, which will be used further.
Statement 1
[2]. Let the system 
is
minimal in 
and
system 
is
complete and minimal in 
for some 
where 
are Banach spaces, moreover, from the
convergence in it
follows the convergence in . Then if 
then 
is complete in where 
is biorthoqonal to 
system in , 
are conjugate spaces.
Statement 2. Let all conditions of theorem are fulfiled. If the inequalities
take place, then the
system (1) forms the basis in 
.
Proof of theorem. Not
restricting generality, we can consider that the jumps 
satisfy the conditions
Really, otherwise we introduce the following function:
For simplicity we consider
that 
. We
multiply each member of system (1) on this function and consider the new
system:
where 
. It is not difficult to verify that for
this system all conditions of theorem are fulfiled, and all corresponding
values 
are
equal to zero.
We follow the scheme of the work [2].
So, first of
all we suppose that 
.
Denote by 
the
points from the set ,
at which for the corresponding jumps , in the conditions (3) the sign of
equality is reached; i.e. 
. Then it is not difficult to note that
for sufficiently small 
the inequalities
where 
and 
take place. In this case
according to statement 2 system (1) forms basis in 
and, consequently, it is
minimal in .
Further we introduce the following functions:
where
Obviously, the
jumps of the functions 
and at the points are connected by correlations: 
and 
for 
, where 
are the jumps of 
at the points 
We introduce into consideration the new system:
(4)
If we denote
by 
the
value, corresponding to this system, defined from (3), then it will be equal
to: 
for 
and 
for 
. Consequently, 
and
Then for
sufficiently small we
have:
where 
.
Further,
consider the weight Hardi class 
, introduced in 
. Following the work , we consider
conjugation problem in classes 
:
where 
is arbitrary function, 
is usual Lebesque
class with the weight 
. Denote by:
Let
and
We present the
function in the form: 
,
where 
is continuous part, 
is the function of jumps,
which is defined by the formula:
(not restricting
generality, we consider that the function is continuous from the
left side).
Let 
, where 
, 
Denote by
,
Then the
boundary values of the function 
have the following presentations:
Applying these
presentations, taking into account the inequality 
for sufficiently small , and doing analogously
the work [1] we obtain, that the system (4) forms the basis in 
, and in this case
biorthogonal system has the form:
where
are definite coefficients.
Applying the boundary value 
can be presented in the form:
where the function 
in sufficiently small
neighbourhoods of the points 
. From here it follows that the linear
cover 
doesn’t
belong to the space 
.
Then according to the statement 1 the system (1) is complete and minimal in . And now, let 
, for example, 
. Then from the
previous arguments it follows that in this case the system
is complete and minimal,
and, as a result, the system (1) is complete, but is not minimal in . The other cases are
proved analogously.
Theorem is proved.
Author is very qreatful to prof. S.S.Mirzoyev for the attention to the work.
References:
1. Veliyev S.G.// The News of Baku University, 2003, ¹4, p.70-78.
2. Bilalov B.T.// Siberia mathem. Jurnal, 2004, v.45, ¹2, p.264-273.
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