Ridge functions are a rich class of simple multivariate functions which have found applications in a variety of areas. These include partial differential equations (where they are sometimes termed 'plane waves'), computerised tomography, projection pursuit in the analysis of large multivariate data sets, the MLP model in neural networks, Waring's problem over linear forms, and approximation theory. Ridge Functions is the first book devoted to studying them as entities in and of themselves. The author describes their central properties and provides a solid theoretical foundation for researchers working in areas such as approximation or data science. He also includes an extensive bibliography and discusses some of the unresolved questions that may set the course for future research in the field.
Allan Pinkus has been in the Department of Mathematics at Technion since 1977, and became a full Professor in 1987. He is the author of three research monographs, one textbook, over 100 research articles, and he has edited six proceedings. His main research interests center around approximation theory. Pinkus is a member of various editorial boards and served for ten years as editor-in-chief of the Journal of Approximation Theory. He has held numerous visiting appointments, and has lectured extensively at international conferences.
Preface; Glossary of selected symbols; 1. Introduction; 2. Smoothness; 3. Uniqueness; 4. Identifying functions and directions; 5. Polynomial ridge functions; 6. Density and representation; 7. Closure; 8. Existence and characterization of best approximations; 9. Approximation algorithms; 10. Integral representations; 11. Interpolation at points; 12. Interpolation on lines; References; Supplemental references; Author index; Subject index.