A practical guide the using kernels and distance functions for approximations

Resumen: Two- or higher dimensional approximations are useful for employing numerical methods such as quadrature rules or solving partial differential equations in physics. Kernels methods are known to be extremely versatile with respect to dimension and which approximands are required. At the same time we must choose the right distance functions or approaches to approximations when modelling data even in simple instances like two dimensions. We shall show some methods beginning with least-squares approximations on circles and spheres to illustrate these points before we proceed to the general and new cases of many types of kernel functions in any dimensions. The least squares approximations on circles and spheres are chosen as examples in order to honour Carl Friedrich Gauß from Göttingen whose 250th birthday is soon and who used least squares approximations on circles and ellipses to approximate the positions of the planetoid Ceres to extreme accuracy without employing a computer.