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2018 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Hyperbolic Cross Approximation

verfasst von: Dinh Dũng, Prof. Vladimir Temlyakov, Prof. Dr. Tino Ullrich

herausgegeben von: Prof. Sergey Tikhonov

Verlag: Springer International Publishing

Buchreihe : Advanced Courses in Mathematics - CRM Barcelona

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book provides a systematic survey of classical and recent results on hyperbolic cross approximation.
Motivated by numerous applications, the last two decades have seen great success in studying multivariate approximation. Multivariate problems have proven to be considerably more difficult than their univariate counterparts, and recent findings have established that multivariate mixed smoothness classes play a fundamental role in high-dimensional approximation. The book presents essential findings on and discussions of linear and nonlinear approximations of the mixed smoothness classes. Many of the important open problems explored here will provide both students and professionals with inspirations for further research.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
This book is a survey on multivariate approximation. The 20th century was a period of transition from univariate problems to multivariate problems in a number of areas of mathematics. For instance, it is a step from Gaussian sums to Weil’s sums in number theory, a step from ordinary differential equations to PDEs, a step from univariate trigonometric series to multivariate trigonometric series in harmonic analysis, a step from quadrature formulas to cubature formulas in numerical integration, a step from univariate function classes to multivariate function classes in approximation theory. In many cases this step brought not only new phenomena but also required new techniques to handle the corresponding multivariate problems. In some cases even a formulation of a multivariate problem requires a nontrivial modification of a univariate problem.
Dinh Dũng, Vladimir Temlyakov, Tino Ullrich
Chapter 2. Trigonometric Polynomials
Abstract
Functions of the form
$$ t(x) = \sum_{|k|\leq n} c_{k} e^{ikx} = a_{0} / 2 + \sum^{n}_{k=1} (a_k \,\, \text{cos}\,\, kx + b_{k} \,\, \text{sin}\,\, kx), $$
(2.1.1)
where \( c_{k}, \,\,a_{k}, \,\,b_{k} \) are complex numbers, will be called trigonometric polynomials of order n. The set of such polynomials we shall denote by \( \mathcal{T} (n),\,\text{and by} \, \mathcal{RT} (n)\,\text{the subset of} \,\, \mathcal{T}(n) \) the subset of T (n) of real polynomials.
We first consider a number of concrete polynomials which play an important role in approximation theory.
Dinh Dũng, Vladimir Temlyakov, Tino Ullrich
Chapter 3. Function Spaces on
Abstract
We begin with the univariate case in order to illustrate the action of the differential operator on periodic functions. For a trigonometric polynomial \( f \in \mathcal(T)(n) \) we have
$$ (D_x f)(x) = f^\prime (x) = \sum_{|k|\leq n} ik {\hat{f}}(k) e^{ikx}. $$
Dinh Dũng, Vladimir Temlyakov, Tino Ullrich
Chapter 4. Linear Approximation
Abstract
By linear approximation we understand approximation from a fixed finite-dimensional subspace. In the study of approximation of the univariate periodic functions the idea of representing a function by its Fourier series is very natural and traditional.
Dinh Dũng, Vladimir Temlyakov, Tino Ullrich
Chapter 5. Sampling Recovery
Abstract
In Chapter 4 we discussed approximation of functions with mixed smoothness by elements of finite-dimensional subspaces.
Dinh Dũng, Vladimir Temlyakov, Tino Ullrich
Chapter 6. Entropy Numbers
Abstract
The concept of entropy is also known as Kolmogorov entropy and metric entropy. This concept allows us to measure how big is a compact set.
Dinh Dũng, Vladimir Temlyakov, Tino Ullrich
Chapter 7. Best m-Term Approximation
Abstract
The last two decades have seen great successes in studying nonlinear approximation which was motivated by numerous applications.
Dinh Dũng, Vladimir Temlyakov, Tino Ullrich
Chapter 8. Numerical Integration
Abstract
A cubature rule Λm(f, ξ) approximates the integral I(f) = f[0,1]d f(x) dx by computing a weighted sum of finitely many function values at Xm = {x1, . . . , xm},
Dinh Dũng, Vladimir Temlyakov, Tino Ullrich
Chapter 9. Related Problems
Abstract
In this section we briefly discuss the development of the hyperbolic cross approximation theory with emphasis put on the development of methods and connections to other areas of research.
Dinh Dũng, Vladimir Temlyakov, Tino Ullrich
Chapter 10. High-Dimensional Approximation
Abstract
We explained in Sections 9.1 and 9.2 that classes with mixed smoothness play a central role among the classes of functions with finite smoothness
Dinh Dũng, Vladimir Temlyakov, Tino Ullrich
Chapter 11. Appendix
Abstract
Let us start with introducing some notations
Dinh Dũng, Vladimir Temlyakov, Tino Ullrich
Backmatter
Metadaten
Titel
Hyperbolic Cross Approximation
verfasst von
Dinh Dũng
Prof. Vladimir Temlyakov
Prof. Dr. Tino Ullrich
herausgegeben von
Prof. Sergey Tikhonov
Copyright-Jahr
2018
Electronic ISBN
978-3-319-92240-9
Print ISBN
978-3-319-92239-3
DOI
https://doi.org/10.1007/978-3-319-92240-9

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