On the occasion of his 60th birthday in October 2009, friends, collaborators, and admirers of Wolfgang Dahmen have organized this volume which touches on va- ous of his research interests. This volume will provide an easy to read excursion into many important topics in applied and computational mathematics. These include nonlinear and adaptive approximation, multivariate splines, subdivision schemes, multiscale and wavelet methods, numerical schemes for partial differential and boundary integral equations, learning theory, and high-dimensional integrals. College Station, Texas, USA Ronald A. DeVore Paderborn, Germany Angela Kunoth June 2009 vii Acknowledgements We are deeply grateful to Dr. Martin Peters and Thanh-Ha Le Thi from Springer for realizing this book project and to Frank Holzwarth for technical support. ix Contents Introduction: Wolfgang Dahmen's mathematical work...1 Ronald A. DeVore and Angela Kunoth 1 Introduction ...1 2 The early years: Classical approximation theory...2 3 Bonn, Bielefeld, Berlin, and multivariate splines ...2 3. 1 Computer aided geometric design ...3 3. 2 Subdivision and wavelets ...4 4 Wavelet and multiscale methods for operator equations...5 4.1 Multilevel preconditioning ...5 4. 2 Compression of operators...5 5 Adaptive solvers ...6 6 Constructionandimplementation...7 7 Hyperbolic partial differential equations and conservation laws ...8 8 Engineering collaborations ...9 9 Thepresent ...9 10 Finalremarks...10 Publications by Wolfgang Dahmen (as of summer 2009)...10 The way things were in multivariate splines: A personal view...19 Carl de Boor 1 Tensor product spline interpolation...19 2 Quasiinterpolation ...20 3 MultivariateB-splines ...21 4 Kergininterpolation ...