Greedy Approximation with Regard to Bases and General Minimal Systems

Abstract

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 non- linear approximation. This was motivated by numerous applications. Non- linear 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 approx- imation 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 ap- proximated. 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.