On the Frame Properties of System of Exponents with Piecewise Continuous Phase

A double system of exponents with piecewise continuous complex-valued coefficients are considered. Under definite conditions on the coefficients the frame property of this system in Lebesgue spaces of functions is investigated. Such systems arise in the spectral problems for discontinuous differential operators.


Introduction
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 holds a.e. on all the segment    [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][8][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 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 , a a (wave velocity in medium) and 1 2 , 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]. 2

L
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 prop- 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  

Necessary Information and Main Assumptions
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"; !  K R  or K C  will stand for the set of real or complex numbers, respectively; nk  is The Banach space will be called a B-space. X  is a space conjugate to space X. By   L M we denote the linear span of the set M X  , and M will stand for the closure of M.
is said to be a ba- It holds the following In the case of basicity, such a system will be called a p-basis.
The following lemma is also valid.
While ob the fol easily provable lemma.
taining the basic result, we will use lowing th the basis x  i.e.
Let the condition

Basic Results
At first we consider the syste ponents m of ex Based on Theorem 1 of the paper [23] we ca conclude the following Statement 1. Let the conditions 1), 2) be fu where p e L  is a function. Let the conditions fulfilled for system (3) and Then, it is easy to see that system (16) and bas in   certain conditions on the functions defining the phase, we prove that this system ma 1 p    . Moreover, it either forms a basis for L p , or it is not complete and not minimal in L p .