Many of the different types of geometries we are concentrating on in this book have representations as n-unisolvent sets, that is, sets of continuous functions that solve one of the Lagrange interpolation problems. For example, the Euclidean plane corresponds, in the obvious way, to the set of all linear functions over the reals that solves the Lagrange interpolation problem of order 2. Also, the tubular circle planes of rank n correspond to sets of continuous periodic or half-periodic functions that solve the Lagrange interpolation problem of order n.

In this chapter we summarize many important results about interpolating sets of functions and their corresponding geometries following the exposition in Polster [1998d], [1998f], and Polster–Steinke [20XXd]. We find that many of the results that we encountered in the previous chapters have counterparts in this very general setting. However, many more of these counterparts are still waiting to be proved.

The results in this chapter form part of the topological foundation of the theory of approximation and interpolation. There are two properties that make n-unisolvent sets important for classical interpolation and approximation theory.