Error estimates for Hermite interpolation on spheres

Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels

“…This is half the approximation order that we prove, although we believe that their result could be doubled by an analogous trick to the one that we employ. There are other notable differences between our results and those in [15]; we give our final error bounds for ||u − s u || t rather than…”

“…After this work was close to completion, we learned of the work by Levesley and Luo [15]. Using a different approach, their results show an approximation order of h…”

“…Let L be a pseudodifferential operator of order t ¥ R, for which K 0 (L) is finite, let o be a zonal kernel satisfying (5) and (15), and let…”

“…It will be instructive to compare Theorem 16 to related work on Hermite interpolation [15], and on Lagrange interpolation [12,14]. Some of these cited papers assume certain continuity of the shape function k o , rather than conditions on the decay of the b k (k o ).…”

“…, the dependence on f is explicit, we cover the more general case of non-invertible operators, and we do not require integer continuity of k o p cos. On the other hand, the paper [15] does not require that the data points become dense on the entire sphere, just in a neighborhood of the evaluation point p.…”