An Approximate Tour in Approximation Theory

Applying mathematics to real life problems often requires different layers of approximation. For example, to study very complex phenomena we rely on simplified models (e.g. differential equations) that usually do not have explicit solutions. To estimate such solutions we pass trough discretizations and algorithms whose results depend on the amount of invested resources (e.g. time, computational power). In this context, approximation theory takes care of how functions can be approximated using simpler ones and how the approximation error behaves with respect to the properties of the functions involved, exploiting knowledges from different areas of mathemat- ics. In this talk we review fundamental notions and results of constructive approximation and use them to introduce wavelets, frames and subdivision schemes, while showing examples linking also to other topics.