Sufficient Conditions for Sampling and Interpolation on the Sphere

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 \(\mathcal {Z}(L)\) .