The Fourth International Symposium on Multivariate Approximation Theory was held at the Oberwolfach Mathematical Research Insti tute, Black Forest, W.-Germany, during the week of January 20 - 26, 1985. The preceding conferences on this topic were held in 1976, 1979, and 1982 * . We were pleased to have more than 50 mathematicians from 13 countries in attendance. The program in cluded 40 lectures. These Proceedings form a record of most of the papers presented at the Symposium. The topics treated cover different problems on multivariate approximation such as polynomial approximation on simplices, multivariate splines (box-splines, dimension of spline spaces), blending methods, multivariate Hermite interpolation, data smoothing and surface representation, and multivariate summation methods. We would like to thank the director of the Oberwolfach Mathe matical Research Institute, Prof. Dr. M. Barner, and his staff for providing the facilities. Of the people who gave their time to help make this conference a success, we would like to mention in particular Prof. Dr. F.J. Delvos (Siegen), Dr. G. Baszenski (College Station, Texas), and Dipl.-Math. H. Nienhaus (Siegen). Finally, our thanks are due to Carl Einsele of Birkhauser Publishers for his valuable cooperation.

Inhalt
Zur Tagung.- List of Participants.- Program of the Sessions.- L. Bamberger: Interpolation in bivariate spline spaces.- n-th order polynomial spline blending.- The limits of multivariate cardinal splines.- Local smoothing for scattered and noisy data.- Convexity and variation diminishing property of Bernstein polynomials over triangles.- The dimensions of bivariate spline spaces over triangulations.- On bivariate vertex splines.- Biorthogonal system of polynomials on the standard simplex.- Anwendung der Approximationstheorie auf gewisse singulare Randwertaufgaben.- Combinatorial aspects of multivariate splines.- Intermediate blending interpolation.- Hermite-interpolation in N variables and minimal cubature formulae.- Some reflections on multi-dimensional Euler and Poisson summation formulas.- On the dimension of the spline space S 21(D) in special cases.- Degree of best approximation by blending functions.- Shape preserving approximation by polyhedral splines.- Numerical realization of boundary collocation methods.- Interpolation by polynomials and natural splines on normal lattices.- Bivariate approximation procedures.- Construction and applications of Hermite interpolating quadratic spline functions of two and three variables.- Hermite-Fejér and higher Hermite-Fejér interpolation with boundary conditions.- Some uniqueness problems in best Chebyshev and mean multivariate approximation.- Minimal projections in tensor product spaces.- Convergence of Hermite-Fejér interpolation via Korovkin's theorem.- Cubature error bounds using degrees of approximation.- Square functions in the theory of Cesaro summability of double orthogonal series.- Generalized Melkes interpolation.- Approximation theory and analysis of spectral methods.- On multivariate polynomial L1-approximation to zero and related coefficient inequalities.- Abschatzungen von Lagrange-Quadratsummen für die Sphäre mit Hilfe gewisser Eigenwerte.- On Gabor information cells.- Ein Beitrag zur Flachenapproximation über unregelmäßig vertei lten Daten.- The basis and moment problems of some systems of analytic functions.- Normen von Projektionen in mehreren Veranderlichen.- The solution of generalized least squares problems.