1. Introduction

The study of approximative properties of function systems in Lebesgue spaces represents special scientific interest for applications in various areas of mathematics. In particular, these matters are important in the spectral theory of differential operators and in the theory of wavelet analysis. Obvious examples are the classical systems of exponentials, sines, cosines, and their perturbation. Approximative properties of these systems in various functional spaces are well studied, and there are extensive bibliographies devoted to them (see, e.g.[1-5]). Relationship between the basis properties of these systems are known, and it is not difficult to establish it. In general, systems of sines and cosines can be written as follows

;

,

where , N is the set of all positive integers, and

.

It is easy to see that the set of values of the functions and fill up the whole segment,

where the basis properties of the system of exponentials (are integers ) are examined. Considering the generalization of this case, we obtain a system of the following form

(1)

It turns out the value of ( i.e. the number of exponential summands) plays an important role in studying of the basis properties of systems of sines and cosines. We will establish some relations between the basis properties of systems and considered in various Banach spaces. We assume that the system is defined on the segment.

It is interesting that, under natural conditions on functions and, , if the system forms a basis for, then the system is non-minimal in for. This phenomenon does not happen in the case of the system of exponentials, , since in this case we deal with (or if a system of cosines is considered), the half of the basis . Apparently similar problem is considered for the first time. These and other approximative properties of systems are closely related to the matters of continuation of the basis on a wide interval which have previously been considered in[5-7].

2. Main Assumptions and Auxiliary Facts

Let [a,b] be a segment on the real axis. As usual, by, we mean a Lebesgue space of functions whose absolute value raised to the p^th power is summable in. The norm in this space is defined as

It is known that is isometrically isomorphic to, where is a number conjugated to. In other words

.

Let us state some ideas from the theory of bases. Let be some Banach space and be its conjugate. We denote by the linear span of the set, and will be the closure of in . We will assume that all the considered spaces are complex.

Definition 1. System is called complete in if.

Definition 2. System is called minimal in X if,.

The following criteria of these properties are well known

Statement 1. System is complete in if and only if from

, it follows that.

Statement 2. System is minimal in if and only if, , where is the Kronecker symbol.

Recall also the definition of the basis

Definition 3. System forms a basis for if for :

, where is a field of complex numbers.

More details of these and other facts from the theory of bases can be found in the monographs [8-11]. Thus, the completeness of the system is equivalent to the fact that, , implies. Minimality of the system means that :

.

We make the following basic assumptions concerning functions , are piecewise smooth, monotonous functions on; moreover, ,, and, where denotes the image of the set, i.e. .

are measurable functions on and the inequality holds, where enotes the derivative of in t.

Throughout this paper we will use the notation.

2.1. Continuation of the Basis

Let systems and form bases for spaces, respectively. By and we denote the corresponding biorthogonal systems. Let’s consider arbitrary functions and:

.

We introduce the following functions

where

.

Consider the double system

(2)

This system is minimal in, and the system biorthogonal to it has the following form

In fact

Similarly we can show that

.

Now let us prove the completeness of the system (2) in. Let the following relations be true for some:

.

We have

From it follows that

.

Consequently, , and this proves the completeness of the system (2) in.

Let us consider and the partial sum

,

where

.

Denote

.

Then

.

Let . In this case we have

Similarly, for we have

From these relations it follows that

,

and thus, the double system (2) forms a basis for if summation is made symmetrically, i.e. this system forms a symmetrical basis for.

2.2. Some Approximative Properties of Function Systems in Lebesgue Spaces

By e mean the Lebesgue measure of the set. All the subsets of real axis we consider are assumed to be Lebesgue measurable. It is easily seen that if the system is minimal in , then it is also minimal in; and if it is complete in, then it is also complete in. An interesting fact should be noted that the system can be complete and minimal at the same time in and in for . Relevant nontrivial example can be found e.g. in [6]. In the case of basis we have the following

Lemma 1. If the system forms a basis for, , then it is nonminimal in for:.

In fact, let this system be minimal in, and let be a corresponding biorthogonal system. Assume

Evidently, is a system biorthogonal to in. Taking the function

Moreover, a basis in can be complete also in for (it is minimal in, of course). Let us give an appropriate example. Let us take an arbitrary orthonormal basis in . Let be some complete system in (in case when are intervals, we can easily make such a choice). Assume, and construct a new system in the following way

Let us consider the system defined by the expression

(3)

It is obvious that forms a basis for. Show that it is complete in . Let

for some . It follows from construction of and (3) that for :

.

Evidently, , and, as a result, . It follows from completeness of the system that . The further reasoning is obvious.

3. Main Results

3.1. Single Case

We proceed to the main results. Let us consider the system (1). The following theorem is true.

Theorem 1. Let the conditions),) be fulfilled. Then : 1) it follows from the completeness of the system in that the system is complete in; 2) from minimality of the system in it follows that the system is minimal in,.

Proof. Let us take any function and consider

(4)

Without loss of generality, we will assume that all functions are increasing. We denote by an inverse of the function ,. We set,. Let us assume , , where, and introduce the function:

Under these notations, the integrals can be written as follows

(5)

The validity of the theorem follows directly from (4) and (5). In case of minimality, the systems and, are biorthogonal to the systems and, respectively, are related by the following formula

(6)

The theorem is proved.

Theorem 2. Let the conditions),) be fulfilled. Then, if the system forms a basis for, , and contains a nontrivial interval ( i.e. ), then the system is nonminimal in.

The validity of the theorem follows directly from Lemma 1 and (6). The following result was absolutely unexpected for the authors.

Theorem 3. Let the conditions),) be fulfilled and. Then if forms a basis for , then the system is nonminimal in,.

Proof. Let be a system biorthogonal to. It is evident that the system forms a basis for. Consequently, this system is nonminimal in, where is any interval. Assume that the system is minimal in and is a system biorthogonal to it. The uniqueness of a system biotrhogonal to the complete one and the relation (6) imply that

where

Without loss of generality, we will assume that . Since doing otherwise we would have, by virtue of Theorem 2, that the system is nonminimal in. Let us introduce the function

.

We have

.

On the other hand

.

As a result we obtain

.

Summing this equality over from to, we have

(7)

where

It is obvious that

Taking into account this expression in (7), we obtain

where

It is easy to see that, and, consequently, is minimal in. Evidently, . So we have a contradiction by virtue of Lemma 1.

The theorem is proved.

Corollary 1. System , is complete i .

In fact, the apparent equality

,

and Theorem 1 imply that the system is complete in.

3.2. Double Case

The similar conclusions can be made for the following function systems

In this case we assume that the functions , are defined on the segment . Let the following conditions be satisfied:

, are piecewise smooth, monotonous functions on with ;, where , are measurable functions on with

Assume. Introduce

, and consider the following double systems

It is easy to prove the following

Theorem 4. Let the conditions be fulfilled. Then: 1) if the system (8₁), ((8₂)) is complete in , then the systems are complete in it follows from the minimality of the systems that the systems (8₁) ((8₂)) are minimal in .

In fact, let us take and suppos .

We have

Thus

(9)

Similarly we establish

(10)

We derive from the relations (9) and (10) that the completeness of the system (8₁) ((8₂)) in implies the completeness of each of the systems .

Let us assume that the systems are minimal in and are the corresponding biorthogonal systems. Assume

and consider the system

(11)

We have

Similarly we obtain

.

From these relations we obtain the minimality of system (8₁) ((8₂)) in .

The following theorem is valid.

Theorem 5. Let the conditions be fulfilled. If the system (8₁) ((8₂)) forms a basis for, and contains a non-trivial interval, then at least one of the systems is non-minimal in.

This theorem is an analogue of Theorem 2 for double systems. The below theorem which is an analogue of Theorem 3 is valid as well.

Theorem 6. Let the condition be fulfilled and. If the system (8₁) ((8₂)) forms a basis for, then at least one of the systems is non-minimal in.

In fact, let the systems be minimal in . If , then we have a contradiction by virtue of Lemma 1. Therefore we will assume that . It is evident that the system biorthogonal to (8₁) is defined by the relation (11). Denote by the system biorthogonal to (8₁). It is easily seen that. Let us assume. From relations (11) we derive

.

Taking into account the latter relation, we have

.

Hence

In the same manner we get

Consequently

Continuing in the same way as we did when proving Theorem 3, we finish the proof of Theorem 6. This theorem is an analogue of Theorem 3 for double systems. Using these two theorems, we come to the following

Corollary 2. Let be arbitrary non-trivial complex numbers. Then each of the systems is complete in,. However, at least one of them is nonminimal in it with

.

In fact, denoting and, we can apply Theorem 6 to this system.

It should be noted that some relationship between the unitary and double power systems are considered in [12-14].

4. Conclusions

Summing up, we arrive at the following conclusions:

1) a method for constructing a basis in the direct product of Lebesgue spaces is suggested;

2) it is shown that if the system of functions forms a basis for Lebesgue space , then it is not minimal in for ;

3) let us reduce an example of the basis in , which is complete in ;

4) the unitary system of the form (1) is considered and some relations between the basis properties of the systems and is established;

5) it is proved that for , from the basicity of the system follows the nonminimality of the system ;

6) analogous results is obtained with respect to the double system of functions.

ACKNOWLEDGEMENTS

The authors are thankful to the reviewers for their valuable comments.