Adaptive wavelet methods

Wavelet bases, initially introduced as a tool for signal and image processing, have rapidly obtained recognition in many different application fields. In this lecture notes we will describe some of the interesting properties that such functions display and we will illustrate how such properties (and in particular the simultaneous good localization of the basis functions in both space and frequency) allow to devise several adaptive solution strategies for partial differential equations.While some of such strategies are based mostly on heuristic arguments, for some other a complete rigorous justification and analysis of convergence and computational complexity is available.