Nonlinear Methods of Approximation |
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Authors: | Temlyakov |
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Institution: | (1) Department of Mathematics University of South Carolina Columbia, SC 29208, USA temlyak@math.sc.edu, US |
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Abstract: | 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 fixed 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 classification, and so on. The standard
problem in this regard is the problem of m -term approximation where one fixes 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 finding 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 offers much promise for greater efficiency in terms of approximation rate, but on the
other hand gives rise to highly nontrivial theoretical and practical problems. With this motivation, our recent work and the
current activity focuses on nonlinear approximation both in the classical form of m -term approximation (where several important problems remain unsolved) and in the form of highly nonlinear approximation
where a theory is only now emerging. |
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Keywords: | AMS Classification 41A17 41A25 41A46 41A65 42A10 42C10 42C15 46A35 46C99 46E35 46N40 65D15 65J05 |
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