ABSTRACT
Polynomial integration on the sphere is an intensively studied research area in mathematics (see, e.g., F. Filbir, W. Themistaclakis [2008], G.B. Folland [2001], W. Freeden et al. [1998], K. Hesse et al. [2010], M. Reimer [2003] and the references therein for points of departure). In our work we start with spherical counterparts to Lagrangian polynomial integration in terms of spherical harmonics. The difficulty in contrast to the one-dimensional theory is that pointwise interpolation cannot always be carried out automatically for any prescribed nodal system of pairwise different points because of Haar’s theorem (see, e.g., P.J. Davis [1963]). As a consequence, the concept of unisolvence for point sets has to be taken int account (cf. W. Freeden et al. [1998]). The chapter ends with an overview on extremal point systems and designs. It culminates in the statement that Gaussian integration cannot be realized adequately on the unit sphere S 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315195674/823ed023-dd6c-41bb-a456-22174b64a780/content/imath10_1.tif"/> .
