Phase space localization appears in different fields, under different names. It may be most familiar from classical or quantum mechanics, where it takes the form of a simultaneous localization in position and momentum. In analysis, it is fundamental to the study of pseudodifferential operators, which are represented by symbols that are functions of two dual variables. Under the name time-frequency localization it plays an important role in the study of non-stationary signals in electrical engineering.
In the eighties, a synthesis took place of phase space ideas and approaches developed independently in these different fields. This gave rise to the development of tools that are not only elegant and powerful mathematically, but also easy and fast to implement numerically. The most prominent examples of such tools are wavelets and localized sine bases, together with many other constructions derived from them. In some cases, mathematical understandng led to new algorithms that would not have been considered as natural extensions of the existing algorithms without the intervening stage of abstaction. On the other hand, it was also striking that some of the deepest mathematical insights came from applications. The talk will sketch these developments and what drove them, and discuss some recent applications.