Tri-cubic polynomial natural spline interpolation for scattered data

In this paper an interpolation problem for 3D scattered data defined on a rectangular parallelepiped with natural boundary conditions is considered. By using spline function theory in Hilbert space, we discuss the existence, uniqueness and characterization of the solution of the interpolation problem as well as its convergence. We show that the solution can be constructed in a simple way without using reproducing kernel semi-Hilbert space theory. Moreover, the solution can be written as the sum of tri-linear polynomials and piecewise tri-cubic polynomials and its coefficients can be determined by solving a positive semi-definite linear system. Numerical examples are presented to illustrate the proposed approach.