Some Problem Research About Rational Interpolants On The Hypersphere

As long as interpolation problem on the hypersphere is important tointerpolation theory, and rational interpolation method have some crucial valuesin the studying of computational mathematics. So it is important in both theoreticand practical sense to operate an overall research of interpolation problem on thehypersphere. This article tackles in some important problems in interpolationproblem on the hypersphere, such as constructing the properly posed set of nodes forinterpolation, constructing a Thiele type rational interpolation and so on.As we all know, constructing the properly posed set of nodes for interpolation is themost important part of interpolation theory. Many works have been done infinding ways to construct the properly posed set of nodes for interpolation in plain,such as Parallel-Lines Process and so on. Researchers have achieved someeffective ways in constructing the properly posed set of nodes for interpolation on thesphere, such as Circumference-Superposition Process, Weft-Superposition Process andso on. This article combining Weft-Superposition Process and CircumferenceSuperposition Process gives a new algorithm, Regular Polyhedrons-SuperpositionProcess, which enjoys the property of low complexity, thus avoiding the problemof singular point in computer coding.Another part of this article probes into the problem that how to construct abivariate Thiele type rational function. In order to construct a bivariate Thiele typerational function, the construction process of it is based on the Samelson inverse forvectors. We generalize Thiele type rational interpolants on the hypersphere of Thierry G.and give the numerical example to show validity of interpolant.