# Hermite-Birkhoff interpolation on scattered data on the sphere and other manifolds

## TL;DR

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

## Contribution

It proposes a new interpolation approach that constructs basis functions with specific derivative properties, avoiding linear system solutions and applicable to arbitrary manifolds.

## Key findings

- Effective interpolation on spheres demonstrated through numerical tests.
- Basis functions depend on geodesic distance and have zero derivatives at data points.
- Method belongs to partition of unity class, ensuring flexibility and efficiency.

## Abstract

The Hermite-Birkhoff interpolation problem of a function given on arbitrarily distributed points on the sphere and other manifolds is considered. Each proposed interpolant is expressed 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 depend on the geodesic distance, are orthonormal with respect to the point-evaluation functionals, and have all derivatives equal zero up to a certain order at the interpolation points. A remarkable feature of such interpolants, which belong to the class of partition of unity methods, is that their construction does not require solving linear systems. Numerical tests are given to show the interpolation performance.