Surface spline approximation on SO(3)

The purpose of this article is to introduce a new class of kernels on SO(3) for approximation and interpolation, and to estimate the approximation power of the associated spaces. The kernels we consider arise as linear combinations of Greenʼs functions of certain differential operators on the rotation group. They are conditionally positive definite and have a simple, closed-form expression, lending themselves to direct implementation via, e.g., interpolation or least-squares approximation. To gauge the approximation power of the underlying spaces, we introduce an approximation scheme providing precise Lp error estimates for linear schemes, namely with Lp approximation order conforming to the Lp smoothness of the target function.