An adaptive hierarchical approximation method on the sphere using axisymmetric locally supported basis functions
The authors approximate scattered data on the sphere by singular integrals based on non-orthogonal sets of kernel functions \(\widetilde B_\varrho\) derived from functions \(B_\varrho^{(k)}\) which are \(0\) on \(-1\leq t<\varrho\) and are the \(k\)th Bernstein polynomial on \([\varrho,1]\). They derive a quadrature formula for functions defined on the sphere of the type \[ \int F(\eta)\widetilde B_\varrho(\eta\xi) d\omega(\eta)= d_\Gamma\sum_{\zeta\in \Gamma} F(\zeta)\widetilde B_\varrho(\zeta\xi), \] where \(\Gamma\) is a given pointset, \(d\) an appropriately defined constant, and multiplication is scalar multiplication of unit vectors. They derive error estimates necessary for an adaptive use of the formula for the evaluation of iterated convolutions. Since for \(k\to\infty\) and \(\varrho\to 1\) the basis functions tend to be concentrated at 1, they develop an adaptive algorithm that starts with a relatively small value of \(\varrho\) and then computes error estimates at the points of the grid and refines the computations only in those domains where the errors exceed a given threshold. An example given is a topographic map of Australia that shows generally good results except for one corner where because of the inaccessibility of the terrain the number of data is small and, therefore, no good approximation is to be expected.
axisymmetric locally supported basis functions
scientific article; zbMATH DE number 935213