On sampling theories and discontinuous Dirac systems with eigenparameter in the boundary conditions
© Tharwat; licensee Springer. 2013
Received: 8 November 2012
Accepted: 11 March 2013
Published: 29 March 2013
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.
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.
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.
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
where ; the real-valued functions and are continuous in and and have finite limits and ; , ; and .
which are defined on and , respectively.
In the following lemma, we prove that the eigenvalues of problem (2.1)-(2.5) are real.
Lemma 2.1 The eigenvalues of problem (2.1)-(2.5) are real.
This contradicts the conditions and . Consequently, must be real. □
The operator and the eigenvalue problem (2.1)-(2.5) have the same eigenvalues. Therefore they are equivalent in terms of this aspect.
has a unique solution , which is an entire function of parameter λ for each fixed .
has a unique solution for each .
By virtue of equations (2.21) and (2.23), these solutions satisfy both transmission conditions (2.4) and (2.5). These functions are entire in λ for all .
for each .
Corollary 2.3 The zeros of the functions and coincide.
In the following lemma, we show that all eigenvalues of problem (2.1)-(2.5) are simple.
Lemma 2.4 All eigenvalues of problem (2.1)-(2.5) are just zeros of the function . Moreover, every zero of has multiplicity one.
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.
contradicting the assumption . The other case, when , can be treated similarly and the proof is complete. □
where are non-zero constants, since all eigenvalues are simple. Since the eigenvalues are all real, we can take the eigen-vector-functions to be real-valued.
3 Green’s matrix and expansion theorem
and the boundary conditions (2.2), (2.4) and (2.5) with λ is not an eigenvalue of problem (2.1)-(2.5).
Lemma 3.1 The operator is self-adjoint in .
Hence is self-adjoint. □
- (i)For ,(3.15)
- (ii)For ,(3.16)
the series being absolutely and uniformly convergent in the first component for on , and absolutely convergent in the second component.
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.
The series (4.2) converges absolutely on ℂ and uniformly on any compact subset of ℂ, and is the entire function defined in (2.28).
and the fact that , , are entire functions of exponential type, we conclude that is of exponential type. □
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).
where both series converge absolutely on ℂ and uniformly on compact sets of ℂ.
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.
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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