Discontinuous Boundary-value Problems: Expansion and Sampling Theorems

This paper is devoted to the derivation of expansion and sampling theorems associated with nth order discontinuous eigenvalue problems defined on [−1, 1], illustrated with detailed examples. The problem consists of nth order differential expressions and n boundary and n compatibility conditions at x = 0. The differential expressions are defined, in general, in two different ways throughout [−1, 1]. We derive an eigenfunction expansion theorem for the Green’s function of the problem as well as a theorem of uniform convergence of the Birkhoff series of a certain class of functions. Then we derive a sampling theorem for integral transforms whose kernels are the product of the Green’s function and the characteristic determinant of the problem.