Greedy Approximation with Regard to Bases and General Minimal Systems

Abstract

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 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.

Greedy Type Bases in Banach Spaces

“…Thus from (10), (11) and (12) using the fact that is decreasing, we obtain that (13) ■ Theorems 3 and 4 are quoted from [40] but the almost the same arguments were used earlier in [11] and [27].…”

“…The first result was first proved in [33] but we present argument given in [22] and [40] which is a bit easier. (27) For each t 1 we apply the inductive hypothesis (note that the number of different B's is at most J) and we continue the estimates (28) Now we apply the estimate (24) for d = 1 and we continue as (29) Due to Proposition 4 we can complete the proof of (24). The inequality (25) follows by duality from (24) …”

Greedy wavelet projections are bounded on BV

“…It is known that the Haar basis is not quasi-greedy for L 1 (R d ) (see [13]). On the other hand, it follows from what we have proved in this paper, that the wavelet bases are quasi-greedy in W 1 (L 1 (R d )).…”

Lebesgue-Type Inequalities for Greedy Approximation

“…In the case of approximation with regard to bases (or minimal systems), the Lebesgue-type inequalities are known both in linear and in nonlinear settings (see surveys [4,7]). It would be very interesting to prove the Lebesgue-type inequalities for redundant systems (dictionaries).…”