Transfinite thin plate spline interpolation

TL;DR

This paper proves that for periodic data on parallel hyperplanes, the polyspline interpolant with Beppo Levi boundary conditions is equivalent to a thin plate spline that minimizes a Duchon type functional, unifying two interpolation methods.

Contribution

It demonstrates that under certain conditions, polyspline interpolation with Beppo Levi boundary conditions coincides with transfinite thin plate spline interpolation.

Findings

Polyspline interpolant with Beppo Levi conditions equals thin plate spline for periodic data.

The equivalence is established for data on finite sets of parallel hyperplanes.

The result unifies transfinite and transfinite spline interpolation methods.

Abstract

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 semi-cardinal 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.