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Frames from Cosines with the Degenerate Coefficients

Sadigova Sabina Rahib*, Mamedova Zahira Vahid

Institute of Mathematics and Mechanics of NAS of Azerbaijan, B.Vahabzade 9, AZ1141,Baku, Azerbaijan

Abstract: The system of cosines with a degenerate coefficient in exponential form is considered. A necessary and sufficient condition on the degree of degeneration is found that makes the considered system a frame in Lebesgue spaces. It is proved that if the degenerate coefficient satisfies the Muckenhoupt condition, then the basicity holds. If the Muckenhoupt condition does not hold, then the system has a finite defect, and does not form a frame.

Keywords: systems of cosines, degeneration, frames

Received December 23, 2012; Revised May 09, 2013; Accepted May 10, 2013

Citation: Sadigova Sabina Rahib, Mamedova Zahira Vahid, Frames from Cosines with the Degenerate Coefficients. American Journal of Applied Mathematics and Statistics, 1 (3). 36-40. January 2013. doi:10.12691/ajams-1-3-1

1. Introduction

Basis properties of classical system of exponents (is the set of all integers) in Lebesgue spaces , , are well studied in the literature (see [6,7,17,18,19]). N.K.Bari in her fundamental work [2] 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 [1]. 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 [8]. In [13], the condition on the weight was found which make the system form a basis for the weight space with a norm . 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, [9]). It should be noted that the above-stated is true for the systems of sines and cosines, too.

Basis properties of the system of exponents and sines with the linear phase in weighted Lebesgue spaces have been studied in [14,15,16]. Those of thpe systems of exponents with degenerate coefficients have been studied in [3,4].

In this work, we study the frame properties of the system of cosines with degenerate coefficients in Lebesgue spaces. Similar problems have previously been considered in paper [11,12].

2. Needful Information

To obtain our main results, we will use some concepts and facts from the theory of bases.

We will use the standard notation. 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.

Let be some Banach space with a norm . Then will denote its conjugate with a norm . By we denote the linear span of the set , and will stand for the closure of .

System is said to be complete in if . It is called minimal in if .

System is said to be uniformly minimal in if

.

The following criteria of completeness and minimality are available:

Criterion 1. System is complete in if .

Criterion 2. System is minimal in it has a biorthogonal system i.e. is the Kronecker symbol.

Criterion 3. Complete system is uniformly minimal in , where is a system biorthogonal to it.

System is said to be a basis for if , .

System is said to be a frame if , .

If system forms a basis for , then it is uniformly minimal.

More details about these facts can be found in [5,10,12].

3. Completeness and Minimality

We consider a system of cosines

() with a degenerate coefficient :

where for . The symbol , means that in sufficiently small neighborhood of the point there holds the inequality .

Theorem 1. Let the conditions

[picture] | (1)

be satisfied. Then the system is complete in . If the following relation holds

[picture] | (2)

then for it forms a basis for , but in a case this system is complete and minimal in , but does not form a basis for it.

Proof. Thus, it is clear that the system belongs to the space , if and only if the relation (1) holds.

Let us consider the completeness of this system. Let , cancels the system out, that is

[picture] | (3)

where is a complex conjugate. By we denote the Banach space of functions which are continuous on with a -norm. It is absolutely clear that . As the system of cosines is complete in , we obtain from the relations (2) that . This proves the completeness of system in . Now consider the minimality of this system in . It is clear that the system belongs to if and only if

The theorem is proved.

It should be noted that these facts can be directly obtained from the classical results.

4. Defective Case

Here we consider the defective system of cosines , where , is some number.

The following theorem is true.

Theorem 2. Let the necessary condition , be satisfied. Then the system is a frame (basis) in if and only if . Moreover, for , , it has a defect equal to , where .

Assume and . Suppose that cancels system out, that is, .

Consequently,

[picture] | (4)

As and system is complete and minimal in , from (4) we get , where is some constant. Then from expression follows that if and only if . Thus, system is complete in .

Now we consider the minimality of this system. Let

We have

On the other hand , and, consequently, . From these relations we immediately find that . Consequently, system is complete and minimal in . Then, it is clear that the system has a defect equal to 1 in this case.

Consider the basicity of system in . By we denote the ordinary norm in . It is obvious that . We have

( is some constant, independent of ) where the interval () does not contain the points and . Assume

We have

It is clear that , and as a result

[picture] | (5)

It is obvious that

Taking the previous relations into account, from (5) we have

where . Hence

Consequently, it directly follows that

Thus

Concerning biorthogonal system, we have

where ( and in sequel also) by we’ll denote positive constants, which may be different in different places. Consequently

As , then it is clear that

Consider

where . At first consider the case , i.e. . We have

As, , and , then it is clear that

and consequently, , . Let , i.e. . It is obvious that

On the other hand

where . Thus

It is absolutely clear that

Taking into account this relation we have

where . Consequently, , . As a result, we obtain . Then, it is clear that . As a result, we get that the system is not uniformly minimal in [see, for example 16] and at the same time does not form a basis for it. Let us show that in this case the system is not a frame in . Assume that it is not true. Then, it is clear that zero has a non-trivial decomposition, i.e.

It is evident that . It follows directly that the arbitrary element can be expanded with respect to the system . We came upon a contradiction which proves that this system does not form a basis for .

Let us consider the case . We look at the system , where are some numbers. Let cancel this system out, that is

Using the previous reasoning, we find that for some constants the following is true

Using representations

we obtain

( is some constant ), where

and is some function. Thus, if and only if . Assume , . As and , it is clear that , . Suppose . We have

It follows directly that for sufficiently small we have

where is some constant. As a result . Consequently, , and it is clear that . Thus, we obtain the following algebraic system with respect to the constants

It is clear that . And, consequently . As a result, , which, in turn, implies that the system is complete in . Let us show that it is also minimal in . Assume . Consider the system

We obtain directly from the following representation , , that . On the other hand

where , . From these relations follows the minimality of system in . Thus, if , then the system is complete and minimal in , and, as a result, the system has a defect equal to 2. Continuing this way, we obtain that if , where , then the system is complete and minimal in , where , , with .

Proceeding in an absolutely similar way as we did in the previous case, we can prove that absolutely similar to the case we establish that for the system is not uniformly minimal in , and consequently, it does not form a basis for it. Let us show that for , the system is not frame in . Let . Assume that the system is a frame in . Then zero has a non-trivial decomposition: . It is clear that , and let . It follows directly that the system is a frame in . The further reasoning is absolutely similar to the case of . This scheme is applicable for .

The theorem is proved.

Acknowledgement

This paper was performed under financial support of Science Foundation of the State Oil Company of Azerbaijan Republic.

The authors would like to express their profound gratitude to Prof. Bilal Bilalov, for his valuable guidance to this article.

References

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