Discrete polynomial interpolation, Green's functions, maximum principles, error bounds and boundary value problems

Abstract

We construct discrete interpolating polynomials, provide explicit representations of discrete Green's functions, give several identities and inequalities for these Green's functions, use the explicit forms of the interpolating polynomials and that of Green's functions to establish several maximum principles. Further, we obtain error bounds in discrete polynomial interpolation and use them to study existence and uniqueness of the discrete boundary value problems. These bounds are also used to provide sufficient conditions for the convergence of the Picard's method, the approximate Picard's method, quasilinearization and the approximate quasilinearization. The monotone convergence of the Picard's iterative method is also analysed.