6. Wavelets and adaptability

We have already observed, in the case of the Haar system, that wavelet representations possess properties of local adaptativity that are reflected in the approximations obtained by thresholding the coefficients. These properties play a central role in the most relevant applications of wavelet bases: in image processing for compression and denoising, and in adaptive numerical simulation. A common feature of these applications is the central role played by nonlinear approximation theory in the mathematical analysis and optimization of wavelet transformation methods. We'll start with an overview of this theory, before turning to the applications.