Sufficient conditions for sampling and interpolation on the sphere

TL;DR

This paper establishes sufficient geometric conditions on point arrays on the sphere to serve as sampling and interpolation sets for spherical harmonics, based on mesh norm and separation radius.

Contribution

It provides new criteria involving mesh norm and separation radius for arrays on the sphere to be Marcinkiewicz-Zygmund and interpolating for spherical harmonics.

Findings

Conditions in terms of mesh norm and separation radius are sufficient.

Arrays satisfying these conditions enable stable sampling and interpolation.

Results apply to various dimensions of the sphere.

Abstract

We obtain sufficient conditions for arrays of points, $\mathcal{Z}=\{\mathcal{Z}(L) \}_{L\ge 1},$ on the unit sphere $\mathcal{Z}(L)\subset \mathbb{S}^d,$ to be Marcinkiewicz-Zygmund and interpolating arrays for spaces of spherical harmonics. The conditions are in terms of the mesh norm and the separation radius of $\mathcal{Z}(L)$.