Radially symmetric thin plate splines interpolating a circular contour map

TL;DR

This paper investigates radially symmetric thin plate spline surfaces that minimize Beppo Levi energy over an infinite domain, introducing new solutions, analyzing their properties, and providing numerical representations.

Contribution

It extends previous work by constructing and analyzing minimizers over the full semi-axis, including singular and non-singular profiles, with new approximation and representation results.

Findings

Identified two types of energy-minimizing profiles: non-singular and singular.

Established $L^{p}$-approximation order $3/2+1/p$ for both profiles.

Developed a novel basis function representation for the minimizers.

Abstract

Profiles of radially symmetric thin plate spline surfaces minimizing the Beppo Levi energy over a compact annulus $R_{1}\leq r\leq R_{2}$ have been studied by Rabut via reproducing kernel methods. Motivated by our recent construction of Beppo Levi polyspline surfaces, we focus here on minimizing the radial energy over the full semi-axis $0<r<\infty$. Using a $L$-spline approach, we find two types of minimizing profiles: one is the limit of Rabut's solution as $R_{1}\rightarrow0$ and $R_{2}\rightarrow\infty$ (identified as a `non-singular' $L$-spline), the other has a second-derivative singularity and matches an extra data value at $0$. For both profiles and $p\in\left[ 2,\infty\right] $, we establish the $L^{p}$-approximation order $3/2+1/p$ in the radial energy space. We also include numerical examples and obtain a novel representation of the minimizers in terms of dilates of a basis function.