Nonlinear Approximation of Random Functions

Given an orthonormal basis and a certain class X of vectors in a Hilbert space H, consider the following nonlinear approximation process: approach a vector $x\in X$ by keeping only its N largest coordinates, and let N go to infinity. In this paper, we study the accuracy of this process in the case where $H=L^2(I)$, and we use either the trigonometric system or a wavelet basis to expand this space. The class of function that we are interested in is described by a stochastic process. We focus on the case of "piecewise stationary processes" that describe functions which are smooth except at isolated points. We show that the nonlinear wavelet approximation is optimal in terms of mean square error and that this optimality is lost either by using the trigonometric system or by using any type of linear approximation method, i.e., keeping the N first coordinates. The main motivation of this work is the search for a suitable mathematical model to study the compression of images and of certain types of signals.