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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

### 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

Abstract