A Duchon framework for the sphere
The paper is motivated by finding error bounds for radial basis functions interpolation in the case when the data points lie on the surface \(S^{d-1}\) of the unit ball of the Euclidean space \({\mathbb R}^d\). For a given \(z \in S^{d-1}\) and \(\theta \in (0,2\pi]\), let \(G(z, \theta)\) denote the set of all points \(\zeta \in S^{d-1}\) whose geodesic distance from \(z\) is \(<\theta\). The authors study relations, in terms of appropriate Sobolev spaces \(W_2^\beta\), between the local and global smoothness of functions \(f \in S^{d-1}\to {\mathbb R}\). They show that for every \(\theta\) it is possible to cover \(S^{d-1}\) with a finite collection of geodesic balls \(G_i=G(z_i, \theta)\), each of the same radius \(\theta\), so that for every \(f \in W_2^\beta(S^{d-1})\) one has \[ \sum_{G_i} \| f | _{G_i}\| ^2_{W_2^\beta (G_i)} \leq Q \| f\| ^2_{W_2^\beta (S^{d-1})}\;, \] where the constant \(Q\) is independent of \(\theta\). The second purpose of the paper is to construct a bounded linear extension operator from \(W_2^\beta (G(z, \theta))\) to \(W_2^\beta (S^{d-1})\). It is shown that the norm of such an operator necessarily depends on \(\theta\). However the dependence on \(\theta\) can be eliminated if one takes only those functions \(f \in W_2^\beta (G(z, \theta))\) that vanish on a certain finite subset of \(G(z, \theta)\).
scientific article; zbMATH DE number 2093758