Local radial basis function approximation on the sphere

In this paper we derive local error estimates for radial basis function interpolation on the unit sphere $\\mathbb{S}^2\\subset\\mathbb{R}^3$. More precisely, we consider radial basis function interpolation based on data on a (global or local) point set $X\\subset\\mathbb{S}^2$ for functions in the Sobolev space $H^s(\\mathbb{S}^2)$ with norm $\\|\\cdot\\|_s$, where $s>1$. The zonal positive definite continuous kernel $\\phi$, which defines the radial basis function, is chosen such that its native space can be identified with $H^s(\\mathbb{S}^2)$. Under these assumptions we derive a local estimate for the uniform error on a spherical cap $S(\\mathbf{z};r)$: the radial basis function interpolant $\\Lambda_X f$ of $f\\in H^s(\\mathbb{S}^2)$ satisfies $\\sup_{\\mathbf{x}\\in S(\\mathbf{z};r)} |f(\\mathbf{x}) - \\Lambda_X f(\\mathbf{x})| \\leq c h^{(s-1)/2} \\|f\\|_{s}$, where $h=h_{X,S(\\mathbf{z};r)}$ is the local mesh norm of the point set $X$ with respect to the spherical cap $S(\\mathbf{z};r)$. Our proof is intrinsic to the sphere, and makes use of the Videnskii inequality. A numerical test illustrates the theoretical result.