Multivariate Interpolation and Approximation by Translates of a Basis Function

This contribution will touch the following topics: Short introduction into the theory of multivariate interpolation and approximation by nitely many (irregular) translates of a (not necessarily radial) basis function, motivated by optimal recovery of functions from discrete samples. Native spaces of functions associated to conditionally positive definite functions, and relations between such spaces. Error bounds and condition numbers for interpolation of functions from native spaces. Uncertainty Relation: Why are good error bounds always tied to bad condition numbers? Shift and Scale: How to cope with the Uncertainty Relation? x1. Introduction and Overview This contribution contains the author's view of a certain area of multivari-ate interpolation and approximation. It is not intended to be a complete survey of a larger area of research, and it will not account for the history of the theory it deals with. Section 2 will motivate why we are mainly interested in multivariate functions that are linear combinations