Vector field multiplier operators and matrix-valued kernel quasi-interpolation

TL;DR

This paper introduces a new class of matrix-valued spherical convolution kernels that improve vector field approximation on the sphere, offering optimal error estimates, robustness to noise, and computational efficiency.

Contribution

It develops a novel vector-valued quasi-interpolation method based on these kernels, with advantages over existing kernel-based interpolation techniques.

Findings

Optimal Sobolev error estimates achieved.

Robustness of the algorithm against noisy data.

Enhanced computational efficiency and robustness.

Abstract

We develop and analyze a class of matrix-valued spherical-convolution kernels stemming from scaled zonal functions on $\mathbb{S}^2,$ the unit sphere embedded in $\mathbb{R}^3$. The construct of these kernels utilizes the Legendre differential equation and requires less stringent regularity conditions on the original zonal kernels. The induced integral operators are simple Fourier-Legendre multipliers that not only deliver optimal Sobolev error estimates (in terms of the scaling parameter) but also yield natural Helmholtz-Hodge decompositions on the $L_2$-tangential vector fields on $\mathbb{S}^2$. Via discretization of the underlying convolution integrals, we harvest a family of vector-valued quasi-interpolants that accomplish our approximation goal in the divergence/curl-free vector field. The quasi-interpolation algorithm is robust against noisy data. The implementation process is adaptive to human-improvision, involving neither evaluating the convolution integrals nor solving systems of linear equations. The computational efficiency and executory robustness of the quasi-interpolation algorithm stand in sharp contrast to the existing kernel-based vector field interpolation method.