Hyperbolic Cross Approximation

Description

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. Front Matter ....Pages i-xi Introduction (Dinh Dũng, Vladimir Temlyakov, Tino Ullrich)....Pages 1-13 Trigonometric Polynomials (Dinh Dũng, Vladimir Temlyakov, Tino Ullrich)....Pages 15-33 Function Spaces on \( \mathbb{T}^d\) (Dinh Dũng, Vladimir Temlyakov, Tino Ullrich)....Pages 35-43 Linear Approximation (Dinh Dũng, Vladimir Temlyakov, Tino Ullrich)....Pages 45-70 Sampling Recovery (Dinh Dũng, Vladimir Temlyakov, Tino Ullrich)....Pages 71-94 Entropy Numbers (Dinh Dũng, Vladimir Temlyakov, Tino Ullrich)....Pages 95-105 Best m-Term Approximation (Dinh Dũng, Vladimir Temlyakov, Tino Ullrich)....Pages 107-126 Numerical Integration (Dinh Dũng, Vladimir Temlyakov, Tino Ullrich)....Pages 127-152 Related Problems (Dinh Dũng, Vladimir Temlyakov, Tino Ullrich)....Pages 153-170 High-Dimensional Approximation (Dinh Dũng, Vladimir Temlyakov, Tino Ullrich)....Pages 171-179 Appendix (Dinh Dũng, Vladimir Temlyakov, Tino Ullrich)....Pages 181-185 Back Matter ....Pages 187-218