Sampling theories of boundary value problems with several internal points of discontinuity

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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 ).

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