Greedy Approximation with Regard to Bases and General Minimal Systems

*This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003

This paper is a survey which also contains some new results on the nonlinear approximation with regard to a basis or, more generally, with regard to a minimal system. Approximation takes place in a Banach or in a quasi-Banach space. The last decade was very successful in studying nonlinear approximation. This was motivated by numerous applications. Nonlinear approximation is important in applications because of its increased efficiency. Two types of nonlinear approximation are employed frequently in applications. Adaptive methods are used in PDE solvers. The m-term approximation considered here is used in image and signal processing as well as the design of neural networks. 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. The fundamental question of nonlinear approximation is how to construct good methods (algorithms) of nonlinear approximation. In this paper we discuss greedy type and thresholding type algorithms.