On Basis Properties of Function Systems in Lebesgue Spaces

Some approximative issues related to function systems in Lebesgue spaces are treated in this work, such as the continuation of basis, the non-minimality of basis in subinterval, the relationship between completeness and minimality of sine and cosine type systems. It is proved that the basis properties of sines and cosines type systems in Lebesgue space of functions depend on the number of exponential summands in expressions of these systems.


Introduction
The study of approximat ive properties of function systems in Lebesgue spaces represents special scientific interest for applications in various areas of mathematics. In particu lar, these matters are important in the spectral theory of differential operators and in the theory of wavelet analysis. Obvious examp les are the classical systems of exponentials, sines, cosines, and their perturbation. Approximat ive properties of these systems in various functional spaces are well studied, and there are extensive b ibliographies devoted to them (see, e.g. [1][2][3][4][5]). Relationship between the basis properties of these systems are known, and it is not difficu lt to establish it. In general, systems of sines and cosines can be written as follows It turns out the value of 2 = r ( 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 int { } n Z e ∈ . Apparently similar problem is considered for the first time. These and other approximat ive 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][6][7].

Main Assumptions and Auxiliary Facts
Let us state some ideas fro m 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.
where denotes the derivative of in t. Throughout this paper we will use the notation .

Continuati on of the Basis
We introduce the following functions where . Consider the double system (2) This system is min imal in , and the system biorthogonal to it has the following form 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

Let
. In this case we have Fro m these relations it fo llows that as , and thus, the double system (2) forms a basis for if summation is made symmetrically, i.e. this system forms a symmetrical basis for .

Some Approxi mati ve Properties of Function Systems in Lebesgue Spaces
By we 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 min imal in , then it is also minimal in ; and if it is co mplete in , then it is , . An interesting fact should be noted that the system can be complete and minimal at the same time in and in for , . Relevant nontrivial examp le 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 Ev idently, is a system biorthogonal to in . Taking the function

Single Case
We proceed to the main results. Let us consider the system (1). The following theorem is true.
The validity of the theorem follows directly fro m (4) and 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 valid ity of the theorem fo llo ws direct ly fro m Lemma 1 and (6). The following result was absolutely unexpected for the authors. Proof. Let be a system biorthogonal to . It is ev ident that the system forms a basis for . Consequently, this system is non min imal in , where , is any interval. Assume that the system is min imal 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 fro m to , we have where .
It is obvious that Taking into account this expression in (7), we obtain , where .
It is easy to see that , and, consequently, is min imal in .
Ev idently, for . So we have a contradiction by virtue of Lemma 1.
The theorem is proved.
In fact, the apparent equality , and Theorem 1 imp ly that the system is comp lete in .

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 follo wing conditions be satisfied:  ( 8 2 ) It is easy to prove the follo wing  We have , .
Similarly we establish , .
We derive fro m the relat ions (9) and (10) We have .
Similarly we obtain .
The following theorem is valid. Theorem 5. Let the conditions be fulfilled. If the system (8 1 ) ( (8 2 )) forms a basis for , and This theorem is an analogue of Theorem 2 for double systems. The below theorem wh ich is an analogue of Theorem 3 is valid as well.
Theorem 6. Let the condition be fulfilled and . If the system (8 1 ) ((8 2 )) forms a basis for , then at least one of the systems ( ; ) is non-minimal in .
In fact, let the systems be min imal in . If , then we have a contradiction by virtue of Lemma 1. Therefo re we will assume that . It is evident that the system biorthogonal to (8 1 ) is defined by the relation (11). Denote by the system biorthogonal to (8 1 ). It is easily seen that . Let us assume . Fro m relat ions (11) we derive .
Taking into account the latter relat ion, 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 co me to the following Corollary 2. Let be arbitrary non-triv ial complex nu mbers. 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][13][14].