Sampling Theorems for Sturm Liouville Problem with Moving Discontinuity Points

In this paper, we investigate the sampling analysis for a new Sturm-Liouville problem with symmetrically located discontinuities which are defined to depending on a neighborhood of a midpoint of the interval. Also the problem has transmission conditions at these points of discontinuity and includes an eigenparameter in a boundary condition. We establish briefly the needed relations for the derivations of the sampling theorems and construct Green's function for the problem. Then we derive sampling representations for transforms whose kernels are either solutions or Green's functions.


An Operator Formulation and Asymptotic Formulas
To formulate a theoretic approach to the problem (1.1)-(1.7) we define the Hilbert space H = L 2 (a, b) ⊕ C with an inner product (2.1) For convenience we put We can prove in a manner similar to that of [23,25,26,28] that A is symmetric in H, all eigenvalues of the problem are real.
Let φ λ (.) and χ λ (.) be two solutions of (1.1) as satisfying the following conditions, respectively; (2.11) Then ω (λ) is an entire function of λ whose zeros are precisely the eigenvalues of the operator A. Using techniques similar of those established by Titchmarsh in [22] , see also [25,26,28] the zeros of ω (λ) are real and simple and if λ n , n = 0, 1, 2, ... denote the zeros of ω (λ) , then the two component vectors are the corresponding eigenvectors of the operator A satisfying the orthogonality relation Here φ λn (.) ∞ n=0 will be the sequence of eigenfunctions of the problem (1.1)-(1.7) corresponding to the eigenvalues {λ n } ∞ n=0 . We denote by Let k n = 0 be the real constants for which The asymptotics of the eigenvalues and eigenfunctions can be derived similar to the classical techniques of [23,25,26,28] . We state the results briefly. φ λ (.) is the solution determined by equations (2.5)-(2.7) above then the following integral equations hold for k = 0 and k = 1 : sufficiently large λ and φ λ (.) have the following asymptotic representations for |λ| → ∞, which hold uniformly for x ∈ I: Then we obtain four distinct cases for the asymptotic behaviour of ω (λ) as |λ| → ∞, namely; (2.25) .., are the zeros of ω (λ) , then we have for sufficiently large n the following asymptotic formulas

Green Function
To study the completeness of the eigenvectors of A, and hence the completeness of the eigenfunctions of the problem (1.1)-(1.7), we construct the resolvent of A as well as Green's function of the problem (1.1)-(1.7). We assume without any loss of generality that λ = 0 is not an eigenvalue of A. Now let λ ∈ C not be an eigenvalue of A and consider the inhomogenous problem for f( and I is the identity operator. Since Now we can represent the general solution of homogeneous differential equation (1.1), appropriate to equation (3.3) in the following form: in which c i i = 1, 6 are arbitrary constants. By applying the method of variation of the constants, we shall search the general solution of the non-homogeneous linear differential equation (3.3) in the following form: where the functions c i (x, λ) i = 1, 6 satisfy the linear system of equation Since λ is not an eigenvalue and ω −ε (λ) = 0, ω ε (λ) = 0, ω +ε (λ) = 0, each of the linear systems in (3.6)-(3.8) have a unique solution which leads x θ+ε φ +ε,λ (y) f (y) dy + c 6 (λ) , where c i (λ) i = 1, 6 are arbitrary constants. Substituting (3.9)-(3.11) into (3.5), we obtain the solution of the equation (3.3) (3.12) Then from the boundary conditions Substituting (3.13) and (2.11) into (3.12), then (3.12) can be written as (3.14) Hence, we have is Green's function of the problem (1.1)-(1.7).

The Sampling Theorem
In this section we derive two sampling theorems associated with the problem (1.1)-(1.7). For convenience we may assume that the eigenvectors of A are real valued.
Theorem 1. Consider the problem (1.1)-(1.7), and let be the solution defined above. Let g (.) ∈ L 2 (a, b) and Then F (λ) is an entire function of exponential type (b − a) that can be reconstructed from its values at the points {λ n } ∞ n=0 via the sampling formula The series (4.2) converges absolutely on C and uniformly on compact subset of C. Here ω (λ) is the entire function defined in (2.11).
Theorem 2. Let g(.) ∈ L 2 (a, b) and F (λ) the integral transform Then F (λ) is an entire function of exponential type (b − a) which admits the sampling representation .
Series (4.31) converges absolutely on C and uniformly on compact subset of C.