Local Radial Basis Function Approximation on the Sphere

Abstract

In this paper we derive local error estimates for radial basis function interpolation on the unit sphere S 2 ⊂ R 3 . More precisely, we consider radial basis function interpolation based on data on a (global or local) point set X ⊂ S 2 for functions in the Sobolev space H s (S 2 ) with norm · s , where s > 1. The zonal positive definite continuous kernel φ, which defines the radial basis function, is chosen such that its native space can be identified with H s (S 2 ). Under these assumptions we derive a local estimate for the uniform error on a spherical cap S(z; r ): the radial basis function interpolant X f of f ∈ H s (S 2 ) satisfies sup x∈S(z;r ) | f (x) − X f (x)| ≤ ch (s−1)/2 f s , where h = h X,S(z;r ) is the local mesh norm of the point set X with respect to the spherical cap S(z; r ). Our proof is intrinsic to the sphere, and makes use of the Videnskii inequality. A numerical test illustrates the theoretical result.2000 Mathematics subject classification: 41A05, 41A15, 41A25, 65D05, 65D07.

Local uniform error estimates for spherical basis functions interpolation

“…In particular, the extreme case is of interest, where we have only distinct points on a spherical cap, then we have h X ≈ π ∕ 2, and so the global error estimates give no information, which inspires us to study the local error estimates on the spherical cap. After a statement of some notations and preliminaries about the spherical cap, we refer to a result about local estimates that apply to target functions within the native space, which is obtained in (see Theorem 4.1 in later text). Then, we are concerned with local estimates for target functions that are not smooth enough to be within the native space, which is the main result of this section (see Theorem 4.7 in later text).…”

“…The following theorem is quoted from , which presents a local estimate that apply to target functions within the native space . Theorem Let be the spherical cap with center z and radius .…”

“…The following theorem is quoted from [13], which presents a local estimate that apply to target functions within the native space N .…”

“…Theorem 4.1 (see [13,Theorem 1]) Let D.z, r/ S 2 be the spherical cap with center z and radius 0 < r Ä 2 . For a given point set X D fx 1 , x 2 , : : : , x N g D.z, r/, let f˚.…”

“…If there are points only on a spherical cap, where the global mesh norm of the point set is close to π ∕ 2, then the global error estimates give no information. In , Hesse and Le Gia first shifted attention to target functions restricted on a spherical cap and studied the local error estimates in an appropriate native space of continuous functions on , which is defined by an underlying strictly positive definite kernel Φ . They obtained a local error estimate for target functions within the native space , where the error estimate was expressed in terms of the local mesh norm of the given point set.…”