Greedy Approximation with Regard to Bases and General Minimal Systems

By Sergei V. Konyagin, Vladimir N. Temlyakov. 2002

54 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f8bbeeb00a2b4a0bf62e943996e9df79a0297285

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Gustavo Garrigós
2. Eugenio Hernández
3. Vladimir N. Temlyakov
4. Timur Oikhberg
5. Pablo M. Berná

In Computer Vision and Pattern Recognition

1. Eugenio Hernández
2. Vladimir N. Temlyakov
3. David L. Donoho

In Statistics and Probability

1. Eugenio Hernández
2. Gustavo Garrigós
3. Vladimir N. Temlyakov
4. Timur Oikhberg

Citations in the corpus

• Lebesgue-Type Inequalities for Quasi-greedy Bases (2011, background): “3 can also be found in [13]; namely, given N and k there exists a set A of cardinality not exceeding N + k such that ∥x− SA(x) ∥∥ ≤ c ( 1 + hr(N) hl(k) ) σN(x), ∀ x ∈ X.”
• Greedy approximation (2008, methodology): “There are several surveys that discuss greedy approximation: see DeVore (1998), Temlyakov (2003a), Konyagin and Temlyakov (2002), Wojtaszczyk (2002a) and Temlyakov (2006b).”