BibTeX
@MISC{Temlyakov02nonlinearmethods,
author = {V.N. Temlyakov},
title = {Nonlinear Methods of Approximation},
year = {2002}
}
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Abstract
Our main interest in this paper is nonlinear approximation. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a xed linear space but are allowed to depend on the function being approximated. While the scope of this paper is mostly theoretical, we should note that this form of approximation appears in many numerical applications such as adaptive PDE solvers, compression of images and signals, statistical classication, and so on. The standard problem in this regard is the problem of m-term approximation where one xes a basis and looks to approximate a target function by a linear combination of m terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is the starting point for compression algorithms. We are interested in the quantitative aspects of this type of approximation. Namely, we want to understand the properties (usually smoothness) of the function which govern its rate of approximation in some given norm (or metric). We are also interested in stable algorithms for nding good or near best approximations using m terms. Some of our earlier work has introduced and analyzed such algorithms. More recently, there has emerged another more complicated form of nonlinear approximation which we call highly nonlinear approximation. It takes many forms but has the basic ingredient that a basis is replaced by a larger system of functions that is usually redundant. Some types of approximation that fall into this general category are mathematical frames, adaptive pursuit (or greedy algorithms) and adaptive basis selection. Redundancy on the one hand oers much promise for greater eciency in terms of approximation rate, but on the other hand gives rise to h...
