The sampling theory says that a function may be determined by its sampled values at some certain points provided the function satisfies some certain conditions. In this paper we consider a Dirac system which contains an eigenparameter appearing linearly in one condition in addition to an internal point of discontinuity. We closely follow the analysis derived by Annaby and Tharwat (J. Appl. Math. Comput. 2010, doi:10.1007/s12190-010-0404-9) to establish the needed relations for the derivations of the sampling theorems including the construction of Green’s matrix as well as the eigen-vector-function expansion theorem. We derive sampling representations for transforms whose kernels are either solutions or Green’s matrix of the problem. In the special case, when our problem is continuous, the obtained results coincide with the corresponding results in Annaby and Tharwat (J. Appl. Math. Comput. 2010, doi:10.1007/s12190-010-0404-9).
MSC: 34L16, 94A20, 65L15.
Keywords:Dirac systems; transmission conditions; eigenvalue parameter in the boundary conditions; discontinuous boundary value problems
Sampling theory is one of the most powerful results in signal analysis. It is of great need in signal processing to reconstruct (recover) a signal (function) from its values at a discrete sequence of points (samples). If this aim is achieved, then an analog (continuous) signal can be transformed into a digital (discrete) one and then it can be recovered by the receiver. If the signal is band-limited, the sampling process can be done via the celebrated Whittaker, Shannon and Kotel’nikov (WSK) sampling theorem [1-3]. By a band-limited signal with band width σ, , i.e., the signal contains no frequencies higher than cycles per second (cps), we mean a function in the Paley-Wiener space of entire functions of exponential type at most σ which are -functions when restricted to ℝ. In other words, if there exists such that, cf.[4,5],
The sampling series (1.2) is absolutely and uniformly convergent on compact subsets of ℂ.
The WSK sampling theorem has been generalized in many different ways. Here we are interested in two extensions. The first is concerned with replacing the equidistant sampling points by more general ones, which is very important from the practical point of view. The following theorem which is known in some literature as the Paley-Wiener theorem  gives a sampling theorem with a more general class of sampling points. Although the theorem in its final form may be attributed to Levinson  and Kadec , it could be named after Paley and Wiener who first derived the theorem in a more restrictive form; see [6,7,10] for more details.
and G is the entire function defined by
then, for any function of the form (1.1), we have
The series (1.6) converges uniformly on compact subsets of ℂ.
The sampling series (1.6) can be regarded as an extension of the classical Lagrange interpolation formula to ℝ for functions of exponential type. Therefore, (1.6) is called a Lagrange-type interpolation expansion.
The second extension of WSK sampling theorem is the theorem of Kramer,  which states that if I is a finite closed interval, is a function continuous in t such that for all . Let be a sequence of real numbers such that is a complete orthogonal set in . Suppose that
Series (1.7) converges uniformly wherever , as a function of t, is bounded. In this theorem, sampling representations were given for integral transforms whose kernels are more general than . Also Kramer’s theorem is a generalization of WSK theorem. If we take , , , then (1.7) is (1.2).
The relationship between both extensions of WSK sampling theorem has been investigated extensively. Starting from a function theory approach, cf., it was proved in  that if , , , satisfies some analyticity conditions, then Kramer’s sampling formula (1.7) turns out to be a Lagrange interpolation one; see also [14-16]. In another direction, it was shown that Kramer’s expansion (1.7) could be written as a Lagrange-type interpolation formula if and were extracted from ordinary differential operators; see the survey  and the references cited therein. The present work is a continuation of the second direction mentioned above. We prove that integral transforms associated with Dirac systems with an internal point of discontinuity can also be reconstructed in a sampling form of Lagrange interpolation type. We would like to mention that works in direction of sampling associated with eigenproblems with an eigenparameter in the boundary conditions are few; see, e.g., [18-20]. Also, papers in sampling with discontinuous eigenproblems are few; see [21-24]. However, sampling theories associated with Dirac systems which contain eigenparameter in the boundary conditions and have at the same time discontinuity conditions, do not exist as far as we know. Our investigation is be the first in that direction, introducing a good example. To achieve our aim we briefly study the spectral analysis of the problem. Then we derive two sampling theorems using solutions and Green’s matrix respectively.
2 The eigenvalue problem
In this section, we define our boundary value problem and state some of its properties. We consider the Dirac system
and transmission conditions
In  the authors discussed problem (2.1)-(2.5) but with the condition instead of (2.3). To formulate a theoretic approach to problem (2.1)-(2.5), we define the Hilbert space with an inner product, see [19,20],
where ⊤ denotes the matrix transpose,
Equation (2.1) can be written as
In the following lemma, we prove that the eigenvalues of problem (2.1)-(2.5) are real.
Lemma 2.1The eigenvalues of problem (2.1)-(2.5) are real.
Then from (2.2), (2.3) and transmission conditions, we have, respectively,
Proof Equation (2.15) follows immediately from the orthogonality of the corresponding eigenelements:
Now, we construct a special fundamental system of solutions of equation (2.1) for λ not being an eigenvalue. Let us consider the next initial value problem:
By virtue of Theorem 1.1 in , this problem has a unique solution , which is an entire function of for each fixed . Similarly, employing the same method as in the proof of Theorem 1.1 in , we see that the problem
Let us construct two basic solutions of equation (2.1) as follows:
Let denote the Wronskian of and defined in [, p.194], i.e.,
It is obvious that the Wronskian
In the following lemma, we show that all eigenvalues of problem (2.1)-(2.5) are simple.
Proof Since the functions and satisfy the boundary condition (2.2) and both transmission conditions (2.4) and (2.5), to find the eigenvalues of the (2.1)-(2.5), we have to insert the functions and in the boundary condition (2.3) and find the roots of this equation.
We show that equation
Now we derive the asymptotic formulae of the eigenvalues and the eigen-vector-functions . We transform equations (2.1), (2.17), (2.21) and (2.24) into the integral equations, see , as follows:
For the following estimates hold uniformly with respect to x, , cf. [, p.55],
Now we find an asymptotic formula of the eigenvalues. Since the eigenvalues of the boundary value problem (2.1)-(2.5) coincide with the roots of the equation
then from the estimates (2.47), (2.48) and (2.49), we get
which can be written as
Then, from (2.45) and (2.46), equation (2.50) has the form
3 Green’s matrix and expansion theorem
Let , where , be a continuous vector-valued function. To study the completeness of the eigenvectors of , and hence the completeness of the eigen-vector-functions of (2.1)-(2.5), we derive Green’s function of problem (2.1)-(2.5) as well as the resolvent of . Indeed, let λ be not an eigenvalue of and consider the inhomogeneous problem
where I is the identity operator. Since
then we have
and the boundary conditions (2.2), (2.4) and (2.5) with λ is not an eigenvalue of problem (2.1)-(2.5).
Now, we can represent the general solution of (3.1) in the following form:
Substituting equations (3.6) and (3.7) into (3.3), we obtain the solution of (3.1)
Then from (2.2), (3.2) and the transmission conditions (2.4) and (2.5), we get
Then (3.8) can be written as
which can be written as
Expanding (3.12) we obtain the concrete form
The matrix is called Green’s matrix of problem (2.1)-(2.5). Obviously, is a meromorphic function of λ, for every , which has simple poles only at the eigenvalues. Although Green’s matrix looks as simple as that of Dirac systems, cf., e.g., [25,26], it is rather complicated because of the transmission conditions (see the example at the end of this paper). Therefore
4 The sampling theorems
The first sampling theorem of this section associated with the boundary value problem (2.1)-(2.5) is the following theorem.
Proof The relation (4.1) can be rewritten in the form
Applying Parseval’s identity to (4.3), we obtain
From Green’s identity, see [, p.51], we have
Then (4.9) and the initial conditions (2.21) imply
From (2.40), (2.19) and (2.7), we have
Also, from (2.40) we have
Then from (2.26) and (4.12) we obtain
Substituting from (4.10), (4.11) and (4.13) into (4.8), we get
Since and are arbitrary, then (4.14) and (4.15) hold for all and all . Therefore, from (4.14) and (4.15), we get (4.7). Hence (4.2) is proved with a pointwise convergence on ℂ. Now we investigate the convergence of (4.2). First we prove that it is absolutely convergent on ℂ. Using Cauchy-Schwarz’ inequality for ,
Using the same method developed above, we get
uniformly on M. In view of Parseval’s equality,
From Hadamard’s factorization theorem, see , , where is an entire function with no zeros. Thus,
The next theorem is devoted to giving vector-type interpolation sampling expansions associated with problem (2.1)-(2.5) for integral transforms whose kernels are defined in terms of Green’s matrix. As we see in (3.12), Green’s matrix of problem (2.1)-(2.5) has simple poles at . Define the function to be , where is a fixed point and is the function defined in (2.28) or it is the canonical product (4.22).
The vector-valued series (4.26) converges absolutely on ℂ and uniformly on compact subsets of ℂ. Here (4.26) means
where both series converge absolutely on ℂ and uniformly on compact sets of ℂ.
Proof The integral transform (4.25) can be written as
From (3.14) and (4.30) we obtain
Then from (2.26) and (2.40) in (4.31), we get
Hence equation (4.32) can be rewritten as
Moreover, from (3.12) we have
Then from (4.35), (2.26) and (2.40) in (4.34), we obtain
Combining (4.36) and (4.33), yields
Combining (4.37), (4.40) and (4.29), we get (4.28) under the assumption that and for all n. If , for some or 2, the same expansions hold with . The convergence properties as well as the analytic and growth properties can be established as in Theorem 4.1 above. □
Now we derive an example illustrating the previous results.
Consider the system
and transmission conditions
The eigenvalues are the solutions of the equation
which can be rewritten as
Green’s function of problem (4.41)-(4.45) is given by
By Theorem 4.1, the transform
has the following expansion:
has the following vector-valued expansion:
The author declares that he has no competing interests.
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.
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