This self-contained, systematic treatment of multivariate approximation begins with classical linear approximation, and moves on to contemporary nonlinear approximation. It covers substantial new developments in the linear approximation theory of classes with mixed smoothness, and shows how it is directly related to deep problems in other areas of mathematics. For example, numerical integration of these classes is closely related to discrepancy theory and to nonlinear approximation with respect to special redundant dictionaries, and estimates of the entropy numbers of classes with mixed smoothness are closely related to (in some cases equivalent to) the Small Ball Problem from probability theory. The useful background material included in the book makes it accessible to graduate students. Researchers will find that the many open problems in the theory outlined in the book provide helpful directions and guidance for their own research in this exciting and active area.

'This excellent book covers a variety of topics in univariate and multivariate approximation as well as their connection to computational mathematics … The exposition is designed in such a way that the reader is familiarized with the univariate results first and then the transition to the multivariate case is performed, highlighting the challenges and the new methods. The book is self-contained and is accessible to readers with graduate and advanced undergraduate background.'

Andriy V. Prymak Source: MathSciNet