On the Frame Properties of System of Exponents with Piecewise Continuous Phase ()
1. Introduction
Consider the following system of exponents
, (1)
where
is a sequence of complex numbers, Z are integers. Systems (1) are model ones while studying spectral properties of differential operators. Under suitable choice of the bounded variation function
on the segment
they are eigenfunctions of first order differential operator
with an integral condition of the form
.
For this reason, many mathematicians appealed to study of basis properties of the systems form (1) in different spaces of functions. If the operator D is considered in the Lebesgue space
, then its natural domain of definition is the Sobolev space
, i.e. the space consisting of absolutely continuous on
functions, whose derivatives belong to
and the relation
, (2)
holds a.e. on all the segment
.
Apparently, the first results for basis properties of the systems of the form (1) in the spaces
,
,
belong to the famous mathematicians Paley P.-N. Wiener [1] and N. Levinson [2]. In sequel, this direction was developed in the investigations of many mathematicians. For more detailed information see the monographs of R. Young [3], A. M. Sedletskii [4], Ch. Heil [5], O. Christensen [6] (and also the papers [7- 9]) and their references. There is also the survey paper [10].
Many problems of mechanics and mathematical physics reduce to discontinuous differential operators, i.e. to the case when the domain of definition of a differential operator is not connected. It should be noted that the systems of the form
(3)
where
has the representation
. (4)
arise as eigen functions of appropriate differential operators while solving many problems of mechanics and mathematical physics by the method of separation of variables. The following system is a trivial example of the case under consideration
![](https://www.scirp.org/html/15-7401348\3be65d0c-980f-4e65-99eb-3fec0d272397.jpg)
Let
,
. It is obvious that ![](https://www.scirp.org/html/15-7401348\723bd840-59c5-4ef6-8574-e2a8cac67f84.jpg)
are the eigen functions of the following spectral problem with a spectrum in boundary conditions
![](https://www.scirp.org/html/15-7401348\c0e5f291-6590-4d44-9f28-322fd8987349.jpg)
Concerning these issues see also the papers [11-14].
Another remarkable example is considered in V. A. Ilin’s paper [15]. Here he considers a mixed problem with conjugation conditions at the inner point
with respect to the wave equation
,
,
with conditions
,
,
,
,
where
![](https://www.scirp.org/html/15-7401348\c3b93cbb-8cbe-4950-ad0f-84e8786dee91.jpg)
(wave velocity in medium) and
(medium density) are positive constants,
are Young modules with additional condition of equality of passage time of wave the segments
and
:
.
The completeness in
of the system of eigenfunctions of an ordinary differential operator that corresponds to this problem is established in the paper [16]. The close class of problems was earlier considered in the paper [17].
These examples very clearly demonstrate expediency of study of frame properties of the systems form (3). The present paper is devoted to investigation of frame property of system (3) in
. Previously some results of this paper were announced without proof in [18].
This work is structured as follows. In Section 2, we present needful information and facts from the theories of bases and close bases that will be used to obtain our main results. This section also contains the main assumptions about the functions
and
which appear in formula (4). In Section 3, we state main results on the basicity of the perturbed system of exponents (3) in Lebesgue spaces
.
2. Necessary Information and Main Assumptions
In sequel we will need the following notion and facts from the theory of bases and frames. We will use the standard notation. N will be the set of all positive integers;
will mean “there exist(s)”;
will mean “it follows”;
will mean “if and only if”;
will mean “there exists unique”;
or
will stand for the set of real or complex numbers, respectively;
is Kroneckers symbol,
. The Banach space will be called a B-space.
is a space conjugate to space X. By
we denote the linear span of the set
, and
will stand for the closure of M.
Definition 1. System
is said to be a basis for X if
,
.
Definition 2. System
is said to be complete in X if
. It is called minimal in X if
.
Definition 3. System
is called
-linearly independent in
-space X, if from
implies
,
.
It holds the following Lemma 1. Let X be a B-space with the basis
and
be a Fredholm operator. Then the following properties of the system
in X are equivalent:
1)
is complete;
2)
is minimal;
3)
is
-linearly independent;
4)
a basis isomorphic to
.
We will need the following notions.
Definition 4. The systems
and
in a B-space X with the norm
are said to be p-close, if
.
Definition 5. The minimal system
in a B-space X with conjugated
is said to be a p-system if for
, where
is an ordinary space of sequences
of scalars with the norm
.
In the case of basicity, such a system will be called a p-basis.
The following lemma is also valid.
Lemma 2. Let X be a B-space with q-basis ![](https://www.scirp.org/html/15-7401348\6ac22e8f-6872-4730-874f-63c62fc89e83.jpg)
and the system
be p-close to it:
,
. Then the expression
, generates a Fredholm operator in X, where
is a system conjugated to
.
One can see these or other facts in the monographs [3,19] and also in the papers [7,20-22]. We will need the following Krein-Milman-Rutman’s Theorem [20].
Theorem KMR. X be a B-space with the norm ![](https://www.scirp.org/html/15-7401348\8c578b04-3b63-4b28-97a7-b633ba5fc258.jpg)
and with the normed basis
,
be a system biorthogonal to it. If the system ![](https://www.scirp.org/html/15-7401348\55531bec-9e04-4b00-aefa-f175d3c26c6c.jpg)
satisfies the condition
, where
, then it forms a basis isomorphic to
for X.
While obtaining the basic result, we will use the following easily provable lemma.
Lemma 3. Let X be a B-space with the basis
and
be a system biorthogonal to
. The system
differ from
by a finitely many elements, i.e.
,
. Thenif
the system
is not minimal in X.
Proof. So, X be a B-space with the basis
and
differ from
by finitely many elements, i.e.
. Expand
. by this basis.
(5)
where
. Let
. At first assume that
. Then, it is obvious that
. As a result, it follows from expression (5) that
belongs to the closure of the linear span
, and so the system
is not minimal. Consider the case
, i.e.
(6)
where
. It is obvious that if
for
or
, then the system
is not minimal. Otherwise, excluding xk in (6), we have:
![](https://www.scirp.org/html/15-7401348\12a09343-34bd-4b3b-9898-b0335d2eb075.jpg)
.
It directly follows from these relations that
belongs to the closure of linear span of the remaining elements
, i.e.
is not minimal in X. Consequently, for
the system
doesn’t form a basis. This reasoning is taken to an arbitrary
very easily. ڤ
Before proceeding the main results, we accept the following basic assumptions concerning the functions of
and
.
1)
is a piecewise-Holder function on
,
are its discontinuity points of first kind;
Denote the jumps of the function
at the points
by
.
Let the condition 2)
, be fulfilled.
3) The functions
have the following asymptotic relations
. (7)
3. Basic Results
At first we consider the system of exponents
, (8)
where
,
. For the basicity of system (8) in
, the results of the paper [23] will be used. Represent system (8) in the form
, (9)
(
are non-negative integers). Let the condition 2) be fulfilled. Finding
from the following inequalities
:
, (10)
assume
. (11)
Based on Theorem 1 of the paper [23] we can directly conclude the following Statement 1. Let the conditions 1), 2) be fulfilled for the function
. Suppose that
. The system (9)
forms a basis for
, (for p = 2 a Riesz basis) if and only if it holds the inequality
.
We will use the following statement obtaining from the results of the paper [24].
Statement 2. If system (9) forms a basis for
,
, then it is isomorphic to the classic system of exponents
.
So, let system (8) form a basis for
. Denote by
a system biorthogonal to it. Let
and
be its biorthogonal coefficients by system (8), i.e.
,
, where ![](https://www.scirp.org/html/15-7401348\00c650ca-1425-4dae-b346-dc08a44497ec.jpg)
is complex conjugation. The following theorem can be directly concluded from Statement 2.
Theorem 1. Let system (8) forms a basis for
,
. Then there hold:
1) Let
and
. Then
, and
is fulfilled, where mp is a constant independent of f,
is an ordinary norm in Lp.
2) Let
and the sequence of numbers
belong to
. Then
such that
moreover
, where Mp is a constant independent of
.
Now, study the basicity of system (3) in
. We have
![](https://www.scirp.org/html/15-7401348\f9d138d8-5c6a-4a17-b368-8d3c61a7d4cb.jpg)
where c is a constant independent of n. The last inequality follows from (7).
Consider the different cases.
1) Let
,
. We have
.
Assume that all the conditions of Statement 1 are fulfilled. Then, system (8) forms a basis for
. Thus, by Statement 2 it forms a
-basis for
in this case. Let
be a system biorthogonal to it. Consider the operator
:
, (12)
where
,
. By Lemma 2 operator (12) is Fredholm in Lp. It is easy to see that
,
. Then, the statement of Lemma 1 is valid for system (3).
2) Let
,
. It is clear that
.
Consequently, for
it is valid
, where
depends only on p. Assume that all the conditions of Statement 1 are fulfilled. Consequently, system (8) forms a basis for Lp. It is clear that
and
. Then, from Theorem 1 we obtain that
, where
are the orthogonal coefficients of f by system (8). From the same theorem we obtain:
where the constant Mp is independent of f. Thus, system (8) forms a p-basis in Lp. It is easy to see that systems (3) and (8) q-close in Lp. Consider operator (12). Further, we behave similarly to case I. Hence the validity of the following theorem is proved.
Theorem 2. Let asymptotic Formula (4) hold, the function
satisfy the conditions 1), 2) and for the function
the relations (7) be valid. Assume that it holds
where
,
is defined from expressions (10)(11). Then, the following properties for system (3) in Lp are equivalent:
1) Complete;
2) Minimal;
3)
-linearly independent;
4) Forms a basis isomorphic to
.
In sequel, we will consider a case, when
. In this case, it is obvious that it holds
.
Let all the conditions of Theorem 2 be fulfilled. Then the system
forms a basis for Lp. Denote by
a system biorthogonal to it. Assume
. It is clear that
![](https://www.scirp.org/html/15-7401348\131dfaea-4862-40d0-a591-d1ec368a75b9.jpg)
![](https://www.scirp.org/html/15-7401348\e4ca5be3-a790-4eb1-9fb5-cc301458740a.jpg)
Consider the functions
![](https://www.scirp.org/html/15-7401348\ff8f4163-5994-425e-a0f4-904f30c8be88.jpg)
Thus, it holds
.
Then, as it follows from Theorem KMR, the system
forms a basis isomorphic to
for Lp. System (3) and the basis
differ by a finitely many elements. By
denote a biorthogonal system to this basis. Consider
, (13)
It is obvious that
, n,
. Denote by
the following determinant
. (14)
It is clear that if
, in the expansion (13) the elements
,
may be replaced by the elements
,
. Then the system
forms a basis for
, since
has the expansion
. Hence, it directly follows that if
, then
has an expansion by system (3), i.e. it is complete in
. Consider the operator
. We have
![](https://www.scirp.org/html/15-7401348\7aa972f2-32d3-4d2a-8752-f6c800d4177d.jpg)
where
is an identity operator, and T is an operator generated by the second summand. Fredholm property F in
follows from finite-dimensionality of the operator
. It is clear that
.
Then from Lemma 1 we obtain the basicity of system (3) in Lp. Conversely, if system (3) forms a basis for Lp, then as it follows from Lemma 3,
. Thus, we established that under accepted conditions system (3) forms a basis for Lp if the determinant determined by expression (14) is not zero.
Thus, we proved the following.
Theorem 3. Let all the conditions of Theorem 2, where
, be fulfilled. The determinant
is determined by expression (14). System (3) forms a basis for Lp,
, if and only if
.
Now, consider the case when
. Let for example,
. In this case, as it follows from Theorem 1 of the paper [23], the system
, (15)
forms a basis for
. Consider the system
, (16)
where
is a function. Let the conditions 1), 2) be fulfilled for system (3) and
. Then, it is easy to see that system (16) and basis (15) are
-close in
, where
is determined by the formula
![](https://www.scirp.org/html/15-7401348\7280635d-e0cb-4755-92bb-be222a18048e.jpg)
Consequently, system (3) is not complete in Lp. The remaining cases, when
, are proved in the similar way.
Consider a case, when
, for example,
. In this case, again as it follows from Theorem 1 of the paper [23], the system
, (17)
forms a basis for Lp. If the conditions 1), 2) are fulfilledthen basis (17) and the system
are
-close in Lp. Consequently, system (3) is not minimal in Lp. The remaining cases, when
, are proved similarly.
Therefore, we obtain the following final result for the basicity of system (3) in Lp.
Theorem 4. Let asymptotic formula (4) hold, where the functions
and
satisfy the conditions 1), 2), 3). The variable
be determined from relations
(10), (11) and let
. Then for
system (3) is not minimal in
; for
it is not complete in
. For
the following properties of system (3) in
are equivalent:
1) Complete;
2) Minimal;
3)
-linearly independent;
4) Forms a basis isomorphic to
;
5)
, where
is determined by expression (14).
Indeed, equivalence of properties 1)-4) follows directly from Lemma 1. Equivalence of conditions 4) and 5) is proved.
4. Conclusions
Taking into account the obtained results, we can summarize this work as follows.
Perturbed system of exponents, the phase of which may has different asymptotic behavior in different parts of the basic interval
, is studied in this work. It should be noted that it’s probably the first time the problem of basicity is considered for such a system. Under certain conditions on the functions defining the phase, we prove that this system may have a finite defect in Lp,
. Moreover, it either forms a basis for Lp, or it is not complete and not minimal in Lp.
5. Acknowledgements
The authors express their deepest gratitude to Professor B. T. Bilalov, for his attention and valuable guidance to this article.