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## Über dieses Buch

This book systematically presents recent fundamental results on greedy approximation with respect to bases.

Motivated by numerous applications, the last decade has seen great successes in studying nonlinear sparse approximation. Recent findings have established that greedy-type algorithms are suitable methods of nonlinear approximation in both sparse approximation with respect to bases and sparse approximation with respect to redundant systems. These insights, combined with some previous fundamental results, form the basis for constructing the theory of greedy approximation. Taking into account the theoretical and practical demand for this kind of theory, the book systematically elaborates a theoretical framework for greedy approximation and its applications.

The book addresses the needs of researchers working in numerical mathematics, harmonic analysis, and functional analysis. It quickly takes the reader from classical results to the latest frontier, but is written at the level of a graduate course and does not require a broad background in the field.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
We will always consider approximation problems in a Banach space. We briefly recall the definition of a Banach space. Let X be a linear (vector) space.

### Chapter 2. Lebesgue-type Inequalities for Greedy Approximation with Respect to Some Classical Bases

Abstract
Let a Banach space X, with a basis $$\Psi\;=\;\mathop{\left\{\psi_k\right\}}\nolimits^\infty_{k=1}$$, be given. We assume that $$\|\psi_k\|\geq\;C\;>\;0,\;k\;=\;1,2,\ldots,$$ , and consider the following theoretical greedy algorithm.

### Chapter 3. Quasi-greedy Bases and Lebesgue-type Inequalities

Abstract
Our primary interest in this chapter is in approximation in L p with respect to quasi-greedy bases. The presentation of this chapter is based on the recent paper [21].

### Chapter 4. Almost Greedy Bases and Duality

Abstract
Let X be a Banach space with a semi-normalized basis $$\Psi\;=\;\mathop{\left\{\psi_n\right\}}\nolimits^\infty_{n=1}$$.

### Chapter 5. Greedy Approximation with Respect to the Trigonometric System

Abstract
The trigonometric system is a classical system that inspired the creation of wonderful deep theories and proofs of a myriad of beautiful difficult theorems. In this chapter we present some results on greedy approximation with respect to the trigonometric system.

### Chapter 6. Greedy Approximation with Respect to Dictionaries

Abstract
In this chapter we consider greedy algorithms with respect to general systems in Banach spaces. We already pointed out in Chapter 5 that greedy algorithms designed for general systems turn out to be good for the trigonometric system.

### Chapter 7. Appendix

Abstract
This chapter contains well-known results in analysis. For the sake of completeness, some of these are proved.