Numerical cubature and hyperinterpolation over spherical polygons

Abstract

The purpose of this work is to introduce a strategy for determining the nodes and weights of a low-cardinality positive cubature formula nearly exact for polynomials of a given degree over spherical polygons.

In the numerical section we report the results about numerical cubature over spherical polygons $P_1$, $P_2$, approximating respectively the Australian and African continents. As an application we consider the reconstruction of functions over $P_1$, also affected by perturbations, via hyperinterpolation and some of its variants. The open-source Matlab software used in the numerical tests is available at the author's homepage.