Consider the following system of exponents

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

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  

where has the representation

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

Let,. It is obvious that

are the eigen functions of the following spectral problem with a spectrum in boundary conditions

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

(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.

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

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

and with the normed basis, be a system biorthogonal to it. If the system

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.

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.

where. It is obvious that if for or, then the system is not minimal. Otherwise, excluding x_k in (6), we have:

.

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

At first we consider the system of exponents

where,. For the basicity of system (8) in, the results of the paper [23] will be used. Represent system (8) in the form

(are non-negative integers). Let the condition 2) be fulfilled. Finding from the following inequalities:

assume

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

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 m_p is a constant independent of f, is an ordinary norm in L_p.

2) Let and the sequence of numbers belong to. Then such thatmoreover, where M_p is a constant independent of.

Now, study the basicity of system (3) in. We have

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:

where,. By Lemma 2 operator (12) is Fredholm in L_p. 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 L_p. 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 M_p is independent of f. Thus, system (8) forms a p-basis in L_p. It is easy to see that systems (3) and (8) q-close in L_p. 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 L_p 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 L_p. Denote by

a system biorthogonal to it. Assume. It is clear that

Consider the functions

Thus, it holds

.

Then, as it follows from Theorem KMR, the system forms a basis isomorphic to for L_p. System (3) and the basis differ by a finitely many elements. By denote a biorthogonal system to this basis. Consider

It is obvious that, n,

. Denote by the following determinant

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

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 L_p. Conversely, if system (3) forms a basis for L_p, then as it follows from Lemma 3,. Thus, we established that under accepted conditions system (3) forms a basis for L_p 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 L_p, , 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

forms a basis for. Consider the system

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

Consequently, system (3) is not complete in L_p. 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

forms a basis for L_p. If the conditions 1), 2) are fulfilledthen basis (17) and the system are -close in L_p. Consequently, system (3) is not minimal in L_p. The remaining cases, when, are proved similarly.

Therefore, we obtain the following final result for the basicity of system (3) in L_p.

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.

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 L_p,. Moreover, it either forms a basis for L_p, or it is not complete and not minimal in L_p.

The authors express their deepest gratitude to Professor B. T. Bilalov, for his attention and valuable guidance to this article.