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Sampling theorems for Sturm-Liouville problem with moving discontinuity points
Boundary Value Problems volume 2014, Article number: 237 (2014)
Abstract
In this paper, we investigate the sampling analysis for a new Sturm-Liouville problem with symmetrically located discontinuities which are defined depending on a parameter in 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 relations needed for the derivations of the sampling theorems and construct the Green’s function for the problem. Then we derive sampling representations for the solutions and Green’s functions.
MSC: 34B24, 34B27, 94A20.
1 Introduction
We consider the following Sturm-Liouville problem:
with one of the boundary conditions including an eigenparameter:
and transmission conditions at two points of discontinuity, and :
where ; λ is a complex spectral parameter; is a given real valued function which is continuous in , , and and has finite limits , ; (); , ; ; ; , and
In the literature, the Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem and generalization of the WKS sampling theorem (see [1]–[3]) has been investigated extensively (see also [4]–[8]). Sampling theorems associated with Sturm-Liouville problems were investigated in [9]–[13]. Also, [14]–[17] and [18]–[21] are for example works in the direction of sampling analysis for continuous and discontinuous eigenproblems, respectively. The sampling series associated with strings were investigated and one compared them with those associated with Sturm-Liouville problems in [20]. In [21] the author studied the sampling analysis for the discontinuous Sturm-Liouville problem which had transmission conditions at one point of discontinuity and contained an eigenparameter in two boundary conditions. In the present paper, we derive sampling theorems associated with a new Sturm-Liouville problem with moving discontinuity points. The problem studied in this paper was presented in more detail for the first time in [22]. The problem has symmetrically located discontinuities which are defined depending on a parameter in a neighborhood of the midpoint of the interval and with an eigenparameter appearing in a boundary condition. There are many published works on sampling theorems associated with different types of generalized Sturm-Liouville boundary value problems, but the present paper deals with a case that has not been studied before. To derive sampling theorems for the problem (1.1)-(1.7), we establish briefly some spectral properties and construct the Green’s function of the problem (1.1)-(1.7). Then we derive two sampling theorems using solutions and the Green’s function, respectively.
2 An operator formulation and asymptotic formulas
Some properties of the eigenvalues and asymptotic formulas for the eigenvalues and the corresponding eigenfunctions for the same problem were given in [22]. We state the results briefly in this section.
To formulate a theoretic approach to the problem (1.1)-(1.7) we define a Hilbert space with an inner product
where , , , . For convenience we put
Let be the set of all such that f and are absolutely continuous on and , , , . We define an operator by
Thus, the operator is equivalent to the eigenvalue problem (1.1)-(1.7) in the sense that the eigenvalues of A are exactly those of the problem (1.1)-(1.7).
We can prove according to [23], [24] that A is symmetric in H, and all eigenvalues of the problem are real (see [22]).
Let and be two solutions of (1.1) as
satisfying the following conditions, respectively:
and
These functions are entire in λ for all .
Let be the Wronskian of and which is independent of x, since the coefficient of in (1.1) is zero. Let
Now, is an entire function of λ whose zeros are precisely the eigenvalues of the operator A. Using techniques similar to those established by Titchmarsh in [25], see also [22]–[24], the zeros of are real and simple and if , , denote the zeros of , then the two component vectors
are the corresponding eigenvectors of the operator A satisfying the orthogonality relation
Here will be the sequence of eigenfunctions of the problem (1.1)-(1.7) corresponding to the eigenvalues and we denote by the normalized eigenvectors of A, i.e.;
Let be the real constants for which
is the solution determined by (2.3) so, the following integral equations hold for and :
and has the following asymptotic representations for , which holds uniformly for :
if ,
if .
Then we obtain four distinct cases for the asymptotic behavior of for , namely,
Consequently if , are the zeros of , then we have the following asymptotic formulas for sufficiently large n:
3 Green’s 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 the Green’s function of the problem (1.1)-(1.7). We assume without any loss of generality that is not an eigenvalue of A.
Now let not be an eigenvalue of A and consider the inhomogeneous problem for , ;
where I is the identity operator. Since
we have
Now we can represent the general solution of the homogeneous differential equation (1.1), appropriate to (3.1) in the following form:
in which () are arbitrary constants. By applying the method of variation of the constants, we shall search the general solution of the non-homogeneous linear differential (3.1) in the following form:
where the functions () satisfy the following linear system of equations:
Since λ is not an eigenvalue and , , , each of the linear systems in (3.4) has a unique solution, which leads to
where () are arbitrary constants. Substituting (3.5) into (3.3), we obtain the solution of (3.1) as
Then, from the boundary conditions (3.2), (1.2), and the transmission conditions (1.4)-(1.7), we get
Substituting (3.7) and (2.5) into (3.6), (3.6) can be written as
Hence we have
where
4 The sampling theorems
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. Letand
Then is an entire function of exponential type that can be reconstructed from its values at the points via the sampling formula
The series (4.1) converges absolutely on ℂ and uniformly on compact subset of ℂ. Hereis the entire function defined in (2.5).
Proof
The relation (4.1) can be rewritten as an inner product of H as follows:
where
Since both and are in H, then they have the Fourier expansions
where
Applying Parseval’s identity to (4.3) and using (4.4), we obtain
Now we calculate and . To prove formula (4.2), we need to show that
By the definition of the inner product of H, we have
From the Green’s identity [26], we have
then from (2.3) and (2.4), the equality (4.11) becomes
Thus
From (2.2), (2.4), and (2.5), we have
Equations (2.5), (2.6), and yield
Substituting from (4.8), (4.9), and (4.10) into (4.6), we get
Letting in (4.11), since the zeros of are simple, we get
Therefore from (4.11) and (4.12), we get (4.5). Hence (4.2) is proved with a pointwise converge on ℂ. Now we investigate the convergence of (4.2). First we prove that it is absolutely convergent on ℂ. Using the Cauchy-Schwarz’s inequality for ,
Since , both series in the right-hand side of (4.13) converge. Thus the series (4.2) converges absolutely on ℂ. For uniform convergence let be compact. Let and . Define to be
Using the same method developed above
Therefore
Since is compact, then we can find a positive constant such that
Therefore,
uniformly on M. In view of Parseval’s equality,
Thus uniformly on M. Hence (4.2) converges uniformly on M. As a result is analytic on compact subsets of ℂ and hence it is entire function. From the relation
and the fact that , and are entire functions of exponential type , we conclude that is also exponential type . □
Remark 1
To see that the expansion (4.2) is a Lagrange type interpolation, we may replace by the canonical product
From Hadamard’s factorization theorem (see [27]), , where is an entire function with no zeros. Thus,
and (4.1), (4.2) remain valid for the function . Hence
We may redefine (4.1) by taking the kernel to get
The next theorem is devoted to giving interpolation sampling expansions associated with the problem (1.1)-(1.7) defined in terms of the Green’s function (these steps were introduced for the first time in [4], [10] and recently in [21], [28]). As we see in (3.9), the Green’s function of the problem (1.1)-(1.7) has simple poles at . Let the function to be , where is a fixed point and is the function defined in (2.5) or the canonical product (4.14).
Theorem 2
Let and be the integral transform
Then is an entire function of exponential type which admits the sampling representation
The series (4.16) converges absolutely on ℂ and uniformly on a compact subset of ℂ.
Proof
The integral transform (4.15) can be rewritten as
where
Applying Parseval’s identity to (4.17) with respect to , we obtain
Let . Since each is an eigenvector of A,
Thus
From (3.8) and (4.9), we obtain
Using , (4.20) becomes
Hence (4.21) can be rewritten as
From the definition of , we have
From (3.8), we have
Combining (4.24), and (2.6), together with (4.23), yields
Substituting from (4.22) and (4.25), we get
As an element of H, has the eigenvectors expansion
Taking the limit when in (4.17), we get
Making use of (4.27), we may rewrite (4.28) as
The interchange of the limit and summation is justified by the asymptotic behavior of and . If , then (4.29) gives
Combining (4.26), (4.27), and (4.18) under the assumption that for all n. If , for some n, the same expansion holds with . The convergence properties as well as the analytic and growth properties can be established as in Theorem 2. □
Now we give an example to illustrate the sampling transform.
Example
Consider the boundary value problem:
is a special case of the problem (1.1)-(1.7) when and as . The eigenvalues of the problem (4.31) are the zeros of the function
By Theorem 1, the transform
has the following expansion:
where are the zeros of (4.32).
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All authors are very grateful to the anonymous referees for their valuable suggestions.
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Hıra, F., Altınışık, N. Sampling theorems for Sturm-Liouville problem with moving discontinuity points. Bound Value Probl 2014, 237 (2014). https://doi.org/10.1186/s13661-014-0237-9
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DOI: https://doi.org/10.1186/s13661-014-0237-9