Greedy Approximation.
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| Author / Creator: | Temlyakov, Vladimir. |
|---|---|
| Imprint: | Cambridge : Cambridge University Press, 2011. |
| Description: | 1 online resource (434 pages) |
| Language: | English |
| Series: | Cambridge Monographs on Applied and Computational Mathematics ; v. 20 Cambridge monographs on applied and computational mathematics. |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/11827554 |
Table of Contents:
- Cover; CAMBRIDGE MONOGRAPHS ON APPLIED AND COMPUTATIONAL MATHEMATICS; 20 Greedy Approximation; Title; Copyright; Contents; Preface; 1 Greedy approximation with regard to bases; 1.1 Introduction; 1.2 Schauder bases in Banach spaces; 1.3 Greedy bases; 1.4 Quasi-greedy and almost greedy bases; 1.5 Weak Greedy Algorithms with respect to bases; 1.6 Thresholding and minimal systems; 1.7 Greedy approximation with respect to the trigonometric system; 1.8 Greedy-type bases; direct and inverse theorems; 1.9 Some further results; 1.10 Systems Lp-equivalent to the Haar basis; 1.11 Open problems.
- 2 Greedy approximation with respect to dictionaries: Hilbert spaces2.1 Introduction; 2.2 Convergence; 2.3 Rate of convergence; 2.3.1 Upper bounds for approximation by general dictionaries; 2.3.2 Upper estimates for weak-type greedy algorithms; 2.4 Greedy algorithms for systems that are not dictionaries; 2.5 Greedy approximation with respect to?-quasi-orthogonal dictionaries; 2.6 Lebesgue-type inequalities for greedy approximation; 2.6.1 Introduction; 2.6.2 Proofs; 2.7 Saturation property of greedy-type algorithms; 2.7.1 Saturation of the Pure Greedy Algorithm.
- 2.7.2 A generalization of the Pure Greedy Algorithm2.7.3 Performance of the n-Greedy Algorithm with regard to an incoherent dictionary; 2.8 Some further remarks; 2.9 Open problems; 3 Entropy; 3.1 Introduction: definitions and some simple properties; 3.2 Finite dimensional spaces; 3.3 Trigonometric polynomials and volume estimates; 3.3.1 Univariate trigonometric polynomials; 3.3.2 Multivariate trigonometric polynomials; The Dirichlet kernels.; The Fejér kernels.; The de la Vallée Poussin kernels.; The Rudin-Shapiro polynomials.; 3.3.3 Volume estimates; generalized Rudin-Shapiro polynomials.
- 3.4 The function classes3.5 General inequalities; 3.6 Some further remarks; 3.7 Open problems; 4 Approximation in learning theory; 4.1 Introduction; 4.1.1 Approximation theory; recovery of functions; 4.1.2 Statistics; regression theory; 4.1.3 Learning theory; 4.2 Some basic concepts of probability theory; 4.2.1 The measure theory and integration; 4.2.2 The concentration of measure inequalities; 4.2.3 The Kullback-Leibler information and the Hellinger distance; 4.3 Improper function learning; upper estimates; 4.3.1 Introduction; 4.3.2 First estimates for classes from Sr.
- 4.3.3 Further estimates for classes from Sr chaining technique; 4.3.4 Least squares estimators for convex hypothesis spaces; 4.3.5 Least squares estimators for non-convex hypothesis spaces; 4.3.6 Estimates for classes from Sr2; 4.3.7 Estimates for classes from Sr1; 4.4 Proper function learning; upper estimates; 4.4.1 Introduction; 4.4.2 The least squares estimators; 4.4.3 Some examples; 4.5 The lower estimates; 4.5.1 Introduction; 4.5.2 The projection learning; 4.5.3 Lower estimates for the Bernoulli scheme; 4.5.4 The proper function learning.