TABLE OF CONTENTS Wavelets have often been promoted as being the correct tool for processing non-stationary signals having strong transients. In contrast, Fourier analysis is the appropriate tool for studying stationary Gaussian processes. However, as Patrick Flandrin pointed out [110], being nonstationary is a negatively defined concept, and it is too broad to be mathematically useful; it is a jungle, a terra incognita waiting for proper exploration and clarification. Does this mean that our advertisement about wavelets and nonstationary signals belongs to the collection of unfulfilled claims made by the pioneers of the wavelet saga? Should we be pessimistic and conclude that wavelets have nothing to do with nonstationary signals? Not at all. Thanks to the work of a group at the University of South Carolina, this debate was settled when they found the following result: There exist well-defined classes of signals that are characterized by the fact that their wavelet expansions are sparse. If the wavelet expansion of a signal is sparse, then an efficient approximation of the signal requires only a few terms of its wavelet expansion. This paves the way to an efficient compression and transmission. Moreover, these classes are also characterized by optimal nonlinear rational approximation. Ronald DeVore, Bjorn Jawerth, P. Petrushev, and V. Popov delimited some precisely defined territories inside the jungle of nonstationary signals. These territories are new function spaces, and they happen to be nicely related to certain Besov spaces. (See Appendix D for the definition of Besov spaces and their characterization in terms...
