Transfinite Thin Plate Spline Interpolation

Duchon's method of thin plate splines defines a polyharmonic interpolant to scattered data values as the minimizer of a certain integral functional. For transfinite interpolation, i.e., interpolation of continuous data prescribed on curves or hypersurfaces, Kounchev has developed the method of polysplines, which are piecewise polyharmonic functions of fixed smoothness across the given hypersurfaces and satisfy some boundary conditions. Recently, Bejancu has introduced boundary conditions of Beppo-Levi type to construct a semicardinal model for polyspline interpolation to data on an infinite set of parallel hyperplanes. The present paper proves that, for periodic data on a finite set of parallel hyperplanes, the polyspline interpolant satisfying Beppo-Levi boundary conditions is in fact a thin plate spline, i.e., it minimizes a Duchon type functional. The construction and variational characterization of the Beppo-Levi polysplines are based on the analysis of a new class of univariate exponential a"'-splines satisfying adjoint natural end conditions.