Hyperinterpolations on the Sphere

Abstract

Hyperinterpolation on the unit sphere of the Euclidean space was proposed by Sloan in 1995. But pointwise convergence in the uniform form can not be achieved by hyperinterpolation. Reimer later proposed the generalized hyperinterpolation, which has the advantages that uniform convergence for all continuous functions can be achieved and it requires positive cubature formulas of less precision and hence significantly reduces the cost of computations. However, Reimers technique only allows him to obtain best approximation order result for C1 functions. The main purpose of this thesis is to consider generalized hyperinterpolation of higher order and a best approximation order is obtained. The smoothness of functions is measured in terms of a new K-functional we introduced. Finally, the thesis also collects several useful positive cubature formulas and discusses briefly the construction methods.