Hermite-Birkhoff Interpolation on Arbitrarily Distributed Data on the Sphere and Other Manifolds

TL;DR

This paper introduces a new Hermite-Birkhoff interpolation method for scattered data on spheres and manifolds, using basis functions based on geodesic distances that do not require solving linear systems.

Contribution

It develops a novel interpolation approach on manifolds that relies on basis functions with specific properties, avoiding linear system solutions.

Findings

Interpolants depend on geodesic distances.

Basis functions are orthonormal and have zero derivatives at data points.

Method is a partition of unity approach that simplifies computation.

Abstract

We consider the problem of interpolating a function given on scattered points using Hermite-Birkhoff formulas on the sphere and other manifolds. We express each proposed interpolant as a linear combination of basis functions, the combination coefficients being incomplete Taylor expansions of the interpolated function at the interpolation points. The basis functions have the following features: (i) depend on the geodesic distance; (ii) are orthonormal with respect to the point-evaluation functionals; and (iii) have all derivatives equal zero up to a certain order at the interpolation points. Moreover, the construction of such interpolants, which belong to the class of partition of unity methods, takes advantage of not requiring any solution of linear systems.