Multivariate Approximation Theory II

The Third International Symposium on Hultivariate Approximation Theory was held at the Oberwolfach !1athematical Research Insti tute, Black Forest, February 8-12, 1982. The preceding conferen ces on this topic were held in 1976 and 1979 . The conference brought together 50 mathematicians from 14 coun tries. These Proceedings form arecord of most of the papers pre sented at the Symposium. The topics treated cover different problems on multivariate approximation theory such as new results concerning approxima tion by polynomials in Sobolev spaces, biorthogonal systems and orthogonal series of functions in several variables, multivariate spline functions, group theoretic and functional analytic methods, positive linear operators, error estimates for approximation procedures and cubature formulae, Boolean methods in multivari ate interpolation and the numerical application of summation procedures. Special emphasis was posed on the application of multivariate approximation in various fields of science. One mathematician was sorely missed at the Symposium. Professor Arthur Sard who had actively taken part in the earlier conferen ces passed away in August of 1980. Since he was a friend of many of the participants, the editors wish to dedicate these Procee dings to the memory of this distinguished mathematician. Abrief appreciation of his life and mathematical work appears as well "Constructive Theory of Functions of Several Variables". Edited by w. Schempp and Karl Zeller. Lecture Notes in 1-1athematics, Vol.

Some remarks on product approximations.- Approximation theory on a snowflake.- Estimates and bounds for the doubly perturbed eigenvalue problem.- Remarks on reduced Hermite interpolation.- Ein Beispiel zur Theorie der besten Approximation.- On spaces of piecewise polynomials with boundary conditions. I. Rectangles.- Some remarks on multivariate B-splines.- On discrete trivariate blending interpolation.- On precision sets of interpolation projectors.- Which closed convex subsets of an inner product space are Chebyshev?.- Interpolation methods for numerical analytic continuation.- Blending-Splines auf Dreiecksnetzen.- Remainder terms in numerical integration formulas of the sphere.- A generalized Sylvester's identity on determinants and its applications to interpolation problems.- Biorthogonality in approximation.- Cubature remainder and biorthogonal systems.- Product approximation: Error estimates.- A Lagrangian method for multivariate continuous Chebyshev approximation problems.- Construction of surfaces of class &k on a domain ? C ?2, after triangulation.- Minimal projections and near-best approximations by multivariate polynomial expansion and interpolation.- Sharp "a priori" error bounds for polynomial approximation in Sobolev spaces.- An immediate construction of numerical integration and differentiation formulae.- On the approximation by multiple orthogonal series.- Quantitative theorems on approximation processes of positive linear operators.- Some relationships between surface splines and Kriging.- A view of mathematics.- Drei statt einer reellen Variablen?.- Eine Methode zur Konstruktion von c1-Flächen zur Interpolation unregelmässig verteilter Daten.- Approximation theory and "domain of dependence" for P.D.E. of hyperbolic type.- Properties ofspline projections.- A survey of recent results on approximation theory in China.- Vector splines on the sphere, with application to the estimation of vorticity and divergence from discrete, noisy data.