1. Introduction

The study of bases in different linear structures plays an important scientific and practical interest in many areas mathematics and natural science. There are numerous monographs as Singer I.[1;2], Day М.М.[3], Young R.[4], Heil Ch.[5], Christensen O.[6;7], Charles K. Chui[8] and others, and even review articles ( see e.g.[9]) devoted to them. From the point of view of applications recently interest in the study of various generalizations of bases (frames and their modifications) is increased. More details about related problems can be found in[4-8]. In this theory, the special role played the Banach space of sequences of scalars, including the space of coefficients having a canonical basis.

In this paper in the term of the Banach space of coefficients generated by the non-degenerate system of some Banach space is considered. In the case of completeness and minimality of this system in above stated space (even if doesn’t form a basis), it is shown that it forms a basis for the obtained space. Some concrete examples are given. We also consider the example of an exponential bases in the weighted space on the real line.

2. Needful concepts and facts

We will use the usual notations: will be a set of all positive integers; is the set of all integers; is the set of all real numbers; will stand for the field of complex numbers; Banach space will be referred to as -space; will stand for a space conjugated to ; is a linear

span of the set ; is a closure of the set in the corresponding topology; will stand for a space conjugated to ; a domain of definition (a range of values) of the operator ; is the Kronecker symbol; is canonical system, where . We will need some concepts and facts from the theory of basis.

Definition 1. System is called complete in if .

Definition 2. System is called minimal in if , .

The following criteria of completeness and minimality are available.

Statement 1. System is complete in if and only if : , , implies .

Statement 2. System is minimal in if and only if : , .

Also recall the definition of a basis.

Definition 3. System forms a basis for if :.

Basicity criteria. The following basicity criteria of systems in -spaces is true.

Theorem 1. System forms a basis for -space the following conditions are fulfilled:

1) is complete in ;

2) is minimal in ;

3) Projectors are uniformly bounded , i.e. :

, ,

where is an appropriate biorthogonal system to , and is a norm in .

Bases are called a monotone basis in -space , if the following inequality holds

.

Let be some -space and be minimal system with conjugate system . Let be some -space of sequences of scalars.

If , we will said that the system has -property.

3. Space

Let be -space, be complete and minimal with the conjugate system in it. Assume

.

It is easy to see that is a normed space with a norm

,

.

The completion of with respect to the norm will be denoted by . We have

.

Hence it directly follows that the functional is bounded on , for and its extension by continuity on denote by . Thus, . From , follows that the system is minimal in . Consider the projectors

.

We have

Consequently, the family is uniformly bounded in . Completeness of the system in is obvious. Then from the basicity criteria we obtain the validity of the following theorem.

Theorem 2. Let be complete and minimal system in -space , be -space with a norm generated by , . Then this system forms a monotone basis for .

Indeed, the fact that the system forms a basis for , is proved. It is easy to see that it holds

.

Consequently, the system forms a monotone basis for .

Consider the operator . It is clear that . is an invertible operator, since . Let be bounded on , i.e.

(1)

So, is a dense in , continuing the operator of the continuity from (1) we obtain

(2)

Similarly, we obtain that if the operator is bounded, then holds

(3)

Inequalities (2) and (3) is called the direct and inverse inequalities of Hausdorff-Young type.

Consider the operator , defined by the expression . Consequently, , where is a canonical system. We have

.

It is clear that if the system is complete in , then the operator provides an isometric isomorphism between and . Consequently, for , the spaces and are isomorphic. Assume that the spaces and are isomorphic and the inequalities (2), (3) hold. Then it is easy to see that the operator provides an isomorphism between and , moreover, . Consequently, in this case the system forms a basis for and its space of coefficients coincides with the space . Isomorphism between the spaces and will be denoted as . So, let , i.e. and the inequality (2) holds. Hence, . Then by the results of[10] we obtain that the system has -property. Conversely, if the inequality (3) holds, then according to the results of[10], the system is -system in . Thus, if the inequality (2) holds, then , if the inequality (3), then conversely, . As a result, we obtain the validity of the following theorem.

Theorem 3. Let be complete and minimal system in -space , be - space generated by , . Then , and . If , then it is clear that and forms a basis for .

4. Space

Let , be an ordinary Lebesgue space of functions. We denote by , the Fourier transform of the function

.

Let . Put , where is some number and accept the norm in :

,

where . It is clear that is the normalized, linear space. We show that it is Banach space too. Let be some fundamental sequence: , be Fourier coefficient of functions . From the completeness of space follows that . On the other hand, from the evaluation of , it directly follows that . It is easy to see that and in . Take and consider the functional

.

It is easy to see that , as a result, and the function in we will identify with the corresponding functionals. It is clear that, the system is a system biorthogonal to in .

Consider the completeness of the system in . First, consider the case . In this case the system forms a basis for . Take . Let be an arbitrary number. It is obvious that , where . On the other hand . Put and assume . We have .

This immediately implies the completeness of the system in . Consider the projectors

.

Consequently, the system forms a basis for , .

Consider the case . It is obvious that is continuously embedded in , i.e. , . As a result . Let the functional cancels out the system . Since and forms a basis for , then it is clear that . Thus, the following theorem is true.

Theorem 4. System forms a basis for if ; is complete and minimal in it if , .

Separately, we consider the case . Let . By Statement 1 implies the system is complete and minimal in . Take . Consequently, . Hence, , and . Consider the partial sums.

We have

,

where is an absolute constant. So the following theorem is true.

Theorem 5. The system of exponents forms a basis for , for .

It is absolutely clear that .

5. On exponential Bases in

5.1. Abstract Case

Let -space with a norm , . Assume , . Let us define linear operations of addition and multiplication by scalars coordinate-wise. Define

.

We denote the obtained -space by . It is absolutely clear that the subspace of the elements of the form is isometrically isomorphic to . Therefore, accurate to within an isometry, the direct expansion holds. Assume that the system forms a basis for . Consider the system , where , . It is obvious that , . Denote by , , the system biorthogonal to . Before proceeding with further considerations, we define the following space. Let and . Define the norm as follows.

.

We define the linear operations in a set of such elements coordinate-wise. Denote the obtained -space by . Let us show that . Take and define

.

We have

.

It is clear that is a linear continuous functional on . Consequently, . Thus, . Consider the system , where , . It is clear that , . We have

.

Consequently, is a system biorthogonal to . Take and consider partial sums

.

We have

Taking into account an expression for the , we have

Hence

Since, forms a basis for , passing to the limit as yields . In fact

.

As a result, we obtain

It is clear that , . Thus, the following theorem is true.

Theorem 6. Let the system forms a basis for ,. Then the system forms a basis for , where , .

5.2. Realization

Here we consider the realization of this approach on the example of the weighted Lebesgue space with the norm

,

where is some weight function. Let , and be a characteristic function on half-interval . Suppose

.

Let us assume that the weight function satisfies the Muckenhoupt condition[11]

(4)

where is a Lebesgue measure of the set . Then the system of exponents forms a basis for (see, e.g.[12-14]). The system biorthogonal to it has the form . Since the conjugate of is identified with the space and the every functional is related to the element through

.

Consequently, is a system biorthogonal to in , where . Take . Denote by the biorthogonal coefficients of the function . Consider the partial sums

.

Taking into account the expression for we obtain .

We have

.

Moreover

On the other hand

,.

As a result

,and, consequently

.

Thus, the following theorem is true.

Theorem 7. System forms a basis for , , if the function satisfies the Muckenhoupt condition (4).

From this theorem we immediately obtain

Corollary 1. System forms an orthonormal basis for .