ABSTRACT
Spline theory is strongly related to zonal kernel exact approximation, where no polynomial precision is of interest. It is canonically based on a variational approach under certain polynomial accuracy (cf. W. Freeden [1978a, 1981a], G. Wahba [1981], W. Freeden [1990], W. Freeden et al. [1998], K.Hesse, I.H. Sloan [2005, 2006]) that minimizes a weighted Sobolev norm of the interpolant, with a large class of spline manifestations provided by pseudodifferential operators being at the disposal of the user. In fact, the Sobolev space framework involving rotation-invariant pseudodifferential operators as realized in Chapter 12 yields some important benefits of spline integration as preparatory tool for spherical sampling. Global spherical spline approximation has its roots in physically motivated problems. It was independently developed by W. Freeden [1978a, 1981a], G. Wahba [1981]. A spline survey is given e.g., by W. Freeden et al. [1996, 1998], G. Whaba [1990]. Generalizations to multi-dimensional hyperspheres are due to W. Freeden, P. Hermann [1985], W. Freeden, M. Gutting [2013] and many others, while generalizations to invariant vectorial and tensorial problems of geosciences and satellite-based disciplines have been discussed by J. Cui, W. Freeden [1997], W. Freeden [1982b, 1987, 1990], R. Reuter [1982], L. Shure et al. [1982], and many others. Related work in multivariate periodic theory can be found in Chapter 21 (see also the references therein).
