Compressed sensing is a great tool for solving inverse problems and it
has a wide range of applications in signal processing and medical
imaging. However, the current theory covers only problems in finite
dimensions, and thus there are plenty of infinite dimensional inverse
problems that are not covered by the existing results. In this talk I
will show how the finite dimensional theory can be extended to include
problems in infinite dimensions. This allows for recovery of much more
general objects such as analog signals and infinite resolution images.
The tools required come from probability, operator theory, optimization
and geometry of Banach spaces. I\'ll give an introduction to what is
already known (accompanied by numerical examples) and discuss some of
the open questions.