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.
V. Temlyakov is Carolina Distinguished Professor in the Department of Mathematics at the University of South Carolina. He has written several books on approximation theory, and has received numerous honours and awards. His research interests include greedy approximation, compressed sensing, learning theory and numerical integration.
1. Approximation of univariate functions; 2. Optimality and other properties of the trigonometric system; 3. Approximation of functions from anisotropic Sobolev and Nikol'skii classes; 4. Hyperbolic cross approximation; 5. The widths of classes of functions with mixed smoothness; 6. Numerical integration and approximate recovery; 7. Entropy; 8. Greedy approximation; 9. Sparse approximation; Appendix. Classical inequalities; References; Index.