Redundant function approximation in theory and in practice

Functions are usually approximated numerically in a basis, a non-redundant and complete set of functions that span a certain space. In this talk we highlight a number of benefits of using overcomplete sets, in particular using the more general notion of a "frame". The main 

benefit is that frames are easily constructed even for functions of several variables on domains with irregular shapes. On the other hand, allowing for possible linear depencies naturally leads to ill-conditioning of approximation algorithms. The ill-conditioning is 

potentially severe. We give some useful examples of frames and we first address the numerical stability of best approximations in a frame. Next, we briefly describe special point sets in which interpolation turns out to be stable. Finally, we review so-called Fourier extensions and an efficient algorithm to approximate functions with spectral accuracy on domains without structure.