Mathematical signal processing (aka applied and computational harmonic analysis) deals with the theory of data acquisition, representation, reconstruction (and more). Viewing a signal as a vector in an appropriate space, one seeks sampling theorems that prescribe how to measure the signal, and subsequently recover it from the measurements. A classical example is the standard sampling theorem for band-limited functions. On the other hand, compressing (transform coding) a class of signals entails representing its members sparsely in an appropriate system. So one is interested both in finding such a system and obtaining theoretical guarantees on the error resulting from representing functions using a few its elements. Also of interest are practical methods of digitizing the measurements, and here we seek theoretical guarantees to quantify tradeoffs between the number of measurements and bits, and the reconstruction accuracy. In recent years, this area has seen many exciting developments based on the following simple observation. Many real-world signals can be modeled as vectors having only a few degrees of freedom, and more specifically as linear combinations of relatively few basis elements. Such vectors are said to be sparse. Great strides in sparse approximation theory and its application have been made, spurred by the rapidly growing area of compressed sensing. This is a sampling paradigm that entails efficiently recovering estimates of sparse N-dimensional vectors from m linear measurements where $m<