Localized bases for kernel spaces on the unit sphere (Q2870616)

For important special cases of radial basis functions, such as thin-plate splines and other surface splines of the Duchon type, and with arguments restricted to the unit sphere, the authors derive stable ``small footprint'' basis functions. Small footprint means that they are constructed with few coefficients, so their \(\ell_0\)-``norm'' is small. These function space bases are quite easily computable and stable with respect to \(L_p\) norms/spaces. They are local -- not usually compactly supported which is normally not possible here, but quickly decaying -- and admit useful approximations by quasi-interpolation using them. The approach is working via Lagrange functions and as mentioned for the two-dimensional unit-sphere (in parts generalizable as remarked by the authors). Numerical examples are included, too, as is a discussion how to use preconditioning of the Gram (interpolation) matrix in this context.