Nonlinear approximation has recently found computational applications such as data compression, statistical estimation or adaptive schemes for partial differential or integral equations, especially through the development of wavelet-based methods. The goal of this paper is to provide with a short survey of nonlinear wavelet approximation in the perspective of these applications, as well as to stress some remaining open questions. 1. Introduction Numerous problems of approximation theory have in common the following general setting: we are given a family of subspaces (SN ) N0 of a normed space X, and for f 2 X, we consider the best approximation error oe N (f) := inf g2SN kf \Gamma gkX : (1) Typically, N represents the number of parameters needed to describe an element in SN , and in most cases of interest, oe N (f) goes to zero as this number tends to infinity. For a given f , we can then study the rate of approximation, i.e. the range of r 0 for which there exists C ? 0 such th...

The purpose of this paper is to explain the interest and importance of (approximate) models and model selection in Statistics. Starting from the very elementary example of histograms we present a general notion of finite dimensional model for statistical estimation and we explain what type of risk bounds can be expected from the use of one such model. We then give the performance of suitable model selection procedures from a family of such models. We illustrate our point of view by two main examples: the choice of a partition for designing a histogram from an n-sample and the problem of variable selection in the context of Gaussian regression. 1 Introduction: a story of histograms 1.1 Histograms as graphical tools Assume we are given a (large) set of real valued measurements or data x1,...,xn, corresponding to lifetimes of some human beings in a specific area, or lifetimes of some manufactured goods, or to the annual income of families in some country,....

We propose adaptive multiscale refinement algorithms for approximating and encoding curves by polygonal curves. We establish rates of approximation of these algorithms in the Hausdorff metric. For example, we show that under the mere assumption that the original curve has finite length then the first of these algorithms gives a rate of convergence O(1/n) where n is the number of vertices of the approximating polygonal curve. Similar results giving second order approximation are proven under weak assumptions on the curvature such as Lp integrability, p> 1. Note that for nonadaptive algorithms, to obtain the same order of approximation would require that the curvature is bounded. Key Words: Polygonal approximation of planar curves, adaptive multiscale refinement, maximal functions, convergence rates, encoding

Approximation of piecewise smooth

Approximation using scattered shifts of a multivariate function ∗