The recovery of a function from a finite set of its values is a common problem in scientific computing. It is required, for instance, in the reconstruction of surfaces from data collected by 3D scanners, and is one of the main underlying problems in the numerical solution of partial differential equations. This talk focuses on the special case of approximating functions from values sampled at equidistant points.

It is known that polynomial interpolants of smooth functions at equally spaced points do not necessarily converge, even if the function is analytic. Instead one may see wild oscillations near the endpoints, an effect known as the Runge phenomenon. Associated with this phenomenon is the exponential growth of the condition number of the approximation process. Several other methods have been proposed for recovering smooth functions from uniform data, such as polynomial least-squares, rational interpolation, and radial basis functions, to name but a few. It is now known that these methods cannot converge at geometric (exponential) rates and remain stable for large data sets. In practice, however, some methods perform remarkably well. This talk will focus on mapped polynomial approximations and how they relate to Fourier continuation and radial basis function approximation.