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El. knyga: Sparse Approximation with Bases

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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.

Recenzijos

This book mainly deals with a kind of interesting and current nonlinear approximation called m-term approximation, or sparse approximation, with bases. The book is based on numerous research papers of the author and is recommended to researchers working in approximation theory, numerical analysis, harmonic analysis and functional analysis. Also, it could be used for various graduate courses in the above mentioned topics. (Sorin Gheorghe Gal, zbMATH 1317.41019, 2015)

Preface ix
1 Introduction
1.1 General setting of approximation problems
1(4)
1.2 Existence and uniqueness of best approximation
5(5)
1.3 Schauder bases in Banach spaces
10(6)
1.4 Unconditional bases
16(5)
2 Lebesgue-type Inequalities for Greedy Approximation with Respect to Some Classical Bases
2.1 Introduction
21(4)
2.2 The trigonometric system
25(7)
2.3 Wavelet bases
32(8)
2.4 Greedy bases
40(2)
2.5 Some examples
42(4)
2.5.1 Unconditionality does not imply democracy
42(1)
2.5.2 Democracy does not imply unconditionality
43(1)
2.5.3 Superdemocracy does not imply unconditionality
43(1)
2.5.4 A quasi-greedy basis is not necessarily an unconditional basis
44(2)
2.6 Further results
46(8)
2.6.1 Direct and inverse theorems
46(5)
2.6.2 Greedy approximation in L1 and L∞
51(3)
2.7 Some inequalities for the tensor product of greedy bases
54(7)
3 Quasi-greedy Bases and Lebesgue-type Inequalities
3.1 Introduction
61(3)
3.2 Properties of quasi-greedy bases
64(13)
3.3 Construction of quasi-greedy bases
77(7)
3.4 Uniformly bounded quasi-greedy systems
84(6)
3.5 Lebesgue-type inequalities for quasi-greedy bases
90(4)
3.6 Lebesgue-type inequalities for uniformly bounded quasi-greedy bases
94(5)
3.7 Lebesgue-type inequalities for uniformly bounded orthonormal quasi-greedy bases
99(4)
4 Almost Greedy Bases and Duality
4.1 Introduction
103(2)
4.2 Greedy conditions for bases
105(3)
4.3 Democratic and conservative bases
108(4)
4.4 Bidemocratic bases
112(4)
4.5 Duality of almost greedy bases
116(5)
5 Greedy Approximation with Respect to the Trigonometric System
5.1 Introduction
121(6)
5.2 Convergence. Conditions on Fourier coefficients
127(23)
5.2.1 Introduction
127(3)
5.2.2 Sufficient conditions in terms of Fourier coefficients. Proof of Theorem 5.2.1
130(7)
5.2.3 Sufficient conditions in terms of the decreasing rearrangement of Fourier coefficients. Proof of Theorem 5.2.2
137(3)
5.2.4 Convergence in the uniform norm. Proof of Theorems 5.2.3--5.2.5
140(10)
5.3 Convergence. Conditions on greedy approximants
150(24)
5.3.1 Introduction
150(2)
5.3.2 Some inequalities
152(6)
5.3.3 Sufficient conditions in the case p (2, ∞)
158(3)
5.3.4 Necessary conditions in the case p (2, ∞)
161(8)
5.3.5 Necessary and sufficient conditions in the case p = ∞
169(5)
5.4 An application of WCGA
174(5)
5.4.1 Convergence
175(1)
5.4.2 Rate of approximation
175(2)
5.4.3 Constructive approximation of function classes
177(2)
5.5 Constructive nonlinear trigonometric m-term approximation
179(8)
6 Greedy Approximation with Respect to Dictionaries
6.1 Introduction
187(6)
6.2 The Weak Chebyshev Greedy Algorithm
193(7)
6.3 Relaxation. Co-convex approximation
200(2)
6.4 Free relaxation
202(4)
6.5 Fixed relaxation
206(6)
6.6 Relaxation. X-greedy algorithms
212(2)
6.7 Greedy expansions
214(15)
6.7.1 Introduction
214(3)
6.7.2 Convergence of the Dual-Based Expansion
217(4)
6.7.3 A modification of the Weak Dual Greedy Algorithm
221(5)
6.7.4 Convergence of WDGA
226(3)
7 Appendix
7.1 Lp-spaces and some inequalities
229(7)
7.1.1 Modulus of continuity
229(2)
7.1.2 Some inequalities
231(5)
7.2 Duality in Lp spaces
236(3)
7.3 Fourier series of functions in Lp
239(4)
7.4 Trigonometric polynomials
243(6)
7.5 Bernstein--Nikol'skii Inequalities. The Marcinkiewicz Theorem
249(8)
Bibliography 257
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