Cost-Reduction in Hyperinterpolation

Abstract

Discretized Newman–Shapiro-operators furnish a generalized hyperinterpolation method on the sphere with valuable mathematical properties. Unfortunately the price is high numerical evaluation cost, which, however, can be reduced significantly, in a first step, by a truncation method. The remaining, relevant terms, now small in number, are values of a (zonal) kernel function with arguments near the pole. Here, and with respect to the degree, the kernel function satisfies an asymptotic formula. It is based on a generalized Mehler–Heine-type formula which concerns certain ‘divided’ Gegenbauer-polynomials and Bessel-functions. This formula is proved and used in order to reduce, in a second step, the evaluation cost once more, such that the discretized Newman–Shapiro-operators become a competitive direct numerical polynomial approximation method on the sphere. For example, the graph of a degree 160 approximation to a rather complicated spherical function has been calculated with a time (cost) reduction, in total, by a factor about 10−4.