1. Introduction
Basis properties of classical system of exponents
in Lebesgue spaces
, are well studied (see e.g. [1-4]). N. K. Bari in her fundamental work [5] raised the issue of the existence of normalized basis in
, which is not Riesz basis. The first example of this was given by K. I. Babenko [6]. He proved that the degenerate system of exponents 
with 
forms a basis for
, but is not Riesz basis when
. This result has been extended by V. F. Gaposhkin [7]. In [8] the condition on the weight 
was found, which make the system 
form a basis for the weight space 
with a norm
.
Similar problems are considered in [9-13]. Basis properties of a degenerate system of exponents are closely related to the similar properties of an ordinary system of exponents in corresponding weight space. In all the mentioned works the authors consider the cases, when the weight or the degenerate coefficient satisfies the Muckenhoupt condition (see, for example, [14]).
In this paper the basis properties of exponential systems with a degenerate coefficient are studied in the spaces
, when the degenerate coefficient does not satisfy the Muckenhoupt condition. A similar problem was considered earlier in [15].
2. Completeness and Minimality
We consider a system of exponents
, (1)
with a degenerate coefficient
where
, are different points.
It is clear that
, if and only if
. Assume that the function
cancels the system 
out, that is
, (2)
where 
is a complex conjugate. It is clear that
, where
. Then, it follows directly from (2) that 
a.e. on 
and, consequently, 
a.e. on
. Thus, if
then system 
is complete in
.
Now consider the minimality of system 
in
. If
then it is minimal in 
and system
is biorthogonal to it. So in this case
. Let
. Consider the system
. (3)
We have
(4)
It is clear that in the neighborhood of zero it holds
.
Consequently, the following representation
on 
is true.
From this representation directly follows that if
then the system (3) belongs to the space
. Then from the relation (4) we obtain that the system 
is minimal in
.
Consider the completeness of system
, (5)
in
. Let for some function 
we have
.
Since
, then from this relation follows that
where 
is some constant. It is clear that
so
. Consequently, 
, hence
. As a result we obtain that under the following conditions
, (6)
system (5) is complete and minimal in
. Thus, the system (1) is complete, but it is not minimal in
.
Consider the basicity of system (5) in
. If the conditions
satisfies, then it is known that (see. e.g. [9-13]) system (1) forms basis for
, and in the case 
it is complete and minimal in
. Then it is clear that system (5) is minimal, but is not complete in
.
Now, let the condition (6) holds. It is easy to see that the system
is biorthogonal to the system (5) in
. Let us show that in this case the system (5) does not form a basis for
. Let it forms a basis for
.
At first consider the case
. Then it is known that (see e.g. [16]) should fulfilled the following conditions
where 
is an arbitrary norm for
. We have
.
Regarding biorthogonal system we get the following condition
. (7)
So
where 
is a constant depending only on 
( in sequel also). Choose 
as small as the interval 
does not contain the points
. Then it is absolutely clear that
:
.
Consequently
It is clear that for sufficiently great 
we have
. Thus
so
. And it contradict the condition (7).
Consider the case
. In the absolutely same way as in the previous case, we get
.
Hence it directly follows that
.
Consider the case
. We have
.
Let
. Take
. Consequently
.
In the sequel we should pay attention to the following identities
From these relations and from the fact that the product of cosines expressed in terms of cosines, it directly follows
where 
, are some constants. It is easy to see that
. Taking into account the expressions above we have
.
It is clear that integrals 
converge. Then from the previous inequality follows that
. Thus, the following theorem is true.
Theorem 1. Let the following condition be satisfied
.
Then the system 
forms a basis for
. If the relation (6) holds, then this system is complete, but is not minimal in 
. In this case system (5) is complete and minimal in 
but does not form a basis for it.
The following theorem is also true.
Theorem 2. Let the conditions
, (8)
be satisfied. Then the system 
is complete and minimal in
, but does not form a basis in it. If the conditions
, (9)
hold, then system (5) is complete and minimal in
, but does not form a basis for it.
Proof. If the conditions (8) holds then the system 
is complete and minimal in
. Indeed, it is clear that
.
Consequently, the system 
forms a basis for
, and as a result it is complete in
.
is a biorthogonal system to
.
It is clear that the system 
belongs to
, and consequently, it is minimal in
. So the singular operator with the Hilbert kernel is not bounded in
, then as a result it follows that this system does not form a basis for
. If the conditions (9) hold then in the absolutely same way as in the previous case we establish that the system (5) is complete and minimal in
, but does not form a basis for it. Consequently, the system
is complete, but is not minimal in
and has a defect equal to 1.
The theorem is proved.
3. Acknowledgements
The authors are thankful to the referees for their valuable comments.