Abstract
In this paper, our boundary value problem is a Dirac system with transmission conditions at several points of discontinuity. The main purpose of this paper is to derive the sampling theorems of this boundary value problem. To derive the sampling theorems including the construction of the Green’s matrix as well as the vector-valued eigenfunction expansion theorem, we briefly study the spectral analysis of the problem as in Levitan and Sargsjan (Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Translations of Mathematical Monographs, vol. 39, 1975; Sturm-Liouville and Dirac Operators, 1991) in a way similar to that of Fulton (Proc. R. Soc. Edinb., Sect. A 77:293-308, 1977). We derive sampling representations for transforms whose kernels are either solutions or the Green’s matrices of the problem. In the special case when our problem has one point of discontinuity, the obtained results coincide with the corresponding results in Tharwat et al. (Numer. Funct. Anal. Optim. 34:323-348, 2013).
Highlights
Let H(D) be a class of complex-valued functions defined on D, where D is a subset of C, which may coincide with C
To derive the sampling theorems including the construction of the Green’s matrix as well as the vector-valued eigenfunction expansion theorem, we briefly study the spectral analysis of the problem as in Levitan and Sargsjan (Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Translations of Mathematical Monographs, vol 39, 1975; Sturm-Liouville and Dirac Operators, 1991) in a way similar to that of Fulton
We say that a sampling theorem holds for the class H(D) if there are two sequences {λk}∞ k= ⊂ D and {Sk(λ)}∞ k= ⊂ H(D), such that
Summary
Let H(D) be a class of complex-valued functions defined on D, where D is a subset of C, which may coincide with C. It is shown in many articles that Kramer’s expansions are nothing more than Lagrange-type interpolation ones when the kernels of the sampled integral transforms are solutions of certain self adjoint eigenvalue problems or are to be expressed in terms of the Green’s functions of these problems; see [ – ]. In the case of boundary value problems with one point of discontinuity, Tharwat et al [ ], discussed sampling theorems of Dirac systems; see [ , ]. {χ (·, λn)}∞ n=–∞ is another set of vector-valued eigenfunctions which is related by {φ(·, λn)}∞ n=–∞ with χ (x, λn) = τnφ(x, λn), x ∈ [a, c ) ∪ (c , c ) ∪ (c , c ) ∪ · · · ∪ Since the eigenvalues are all real, we can take the vector-valued eigenfunctions to be real valued
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