A Duchon framework for the sphere

Abstract

In his fundamental paper (RAIRO Anal. Numer. 12 (1978) 325) Duchon presented a strategy for analysing the accuracy of surface spline interpolants to sufficiently smooth target functions. In the mid-1990s Duchon's strategy was revisited by Light and Wayne (J. Approx. Theory 92 (1992) 245) and Wendland (in: A. Le Me´haute´, C. Rabut, L.L. Schumaker (Eds.), Surface Fitting and Multiresolution Methods, Vanderbilt Univ. Press, Nashville, 1997, pp. 337-344), who successfully used it to provide useful error estimates for radial basis function interpolation in Euclidean space. A relatively new and closely related area of interest is to investigate how well radial basis functions interpolate data which are restricted to the surface of a unit sphere. In this paper we present a modified version Duchon's strategy for the sphere; this is used in our follow up paper (L p -error estimates for radial basis function interpolation on the sphere, preprint, 2002) to provide new L p error estimates ðpA½1; NÞ for radial basis function interpolation on the sphere. r 2004 Elsevier Inc. All rights reserved.

Inf-sup condition for spherical polynomials and radial basis functions on spheres

“…This can be done using either the so-called power function approach, or using the sampling theorem, which we will discuss in more detail in the next section. The second step consists of a local covering argument due to Duchon (see [3,6]), which then provides the additional convergence order reflecting the fact that the left-hand side is measured in the weaker L 2 -norm. …”

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

“…Here M 1 and M 2 are the lower and upper bounds on M from (30). In the second set of experiments, the spherical basis function is derived from a less smooth radial basis function Ψ 3,2 ∈ C 4 (R 3 ), φ(x · y) = ρ 3,2 ( 2 − 2x · y), ρ 3,2 (r) = (1 − r) 6 + (35r 2 + 18r + 3). The results are shown in Table 3.…”

“…Sobolev spaces on S n can also be defined using local charts (see [12]). Here we use a specific atlas of charts, as in [6]. Let a spherical cap of radius α centered at p ∈ S n be defined by (6) C(p, α) := {x ∈ S n : θ(p, x) ≤ α},…”

Preconditioners for pseudodifferential equations on the sphere with radial basis functions

“…1, Sect. 7.3]) with a specific atlas of charts chosen as in [5]. Let a spherical cap of radius α centered at p ∈ S n be defined by…”