Multivariate Approximation Theory forms a rapidly evolving field in Applied Mathematics. The reason for its particular current interest lies in its impact on Computer Aided Geometric Design (CAGD), Image Processing, Pattern Recogni tion, and Mult idimensional Signal Processing. Mul ti var iate Bernstein polynomials and box splines, for example, play an important role in CAGD. Conversely, the highly important filter bank design problem of signal processing, for instance, gives rise to a new family of multivariate approximating functions, the Gabor wavelets, with interesting technological and biological applications. The conferences on Multivariate Approximation Theory held at the Mathematical Research Institute at Oberwolfach, Black Forest, in 1976, 1979, 1982, 1985 and 1989 ref lect the progress made in this area and related fie Ids. The present volume which is a continuation of the preceding volumes Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics 571 (1977) Multivariate Approximation Theory, ISNM 51 (1979) Multivariate Approximation Theory II, ISNM 61 (1982) Multivariate Approximation Theory III, ISNM 75 (1985) is based on the conference held on February 12-18, 1989. It includes most of the lectures presented at the Oberwolfach meeting and reveals the wide spectrum of activities in the field of multivariate approximation. The organizers are grateful to the Director of the Oberwolfach Mathematical Research Institute, Professor Dr. M. Barner, and his staff for providing the facili ties, and to Dr. G. Baszenski, Professor Dr. F. J. Delvos, Dr. H.

Inhalt
A Recursion Formula for the Dimension of Super Spline Spaces of Smoothness r and Degree d ** r2k.- Approximation with Barycentric Coordinates: The Hilbertian Case.- A Discrete Fourier Transform Scheme for Boolean Sums of Trigonometric Operators.- A Local Basis for Certain Smooth Bivariate PP Spaces.- Polynomial Ideals and Multivariate Splines.- Cardinal Interpolation with Radial Basis Functions: An Integral Transform Approach.- On the Evaluation of Multivariate Lagrange Formulae.- Asymptotically Optimal Sampling Schemes for Periodic Functions II: The Multivariate Case.- Generalized Bochner-Riesz Means of Fourier Integrals.- An Algorithm for Best Approximating Algebraic Polynomials in Lp over a Simplex.- Rational and Algebraic Approximation for Initial- and Boundary-Value-Problems.- R-th Order Blending Rectangle Rules.- A Trivariate Boolean Cubature Scheme.- Multivariate Band-Limited Functions: Sampling Representation and Approximation.- Multidimensional Irregular Sampling of Band-Limited Functions in Lp-Spaces.- L2-Synthesis by Ambiguity Functions.- Vector Spherical Spline Interpolation.- Simultaneous Approximation by Generalized n-th Order Blending Operators.- Some Results on Quadratic Splines of Three (and more) Variables.- On Iterates of Linear Variation Diminishing Operators and Characterization of Bernstein Type Polynomials.- Methoden der Fourier-Transformation bei der kardinalen Interpolation periodischer Daten.- Dual Bases Associated with Box Splines.- Some Pointwise Negative Results in Multivariate Approximation by Convolution Processes of Fejér's Type.- On Multidimensional Lebesgue-Stieltjes Convolution Operators.- The Singularity of Distance Matrices.- Cardinal Interpolation with Polyharmonic Splines.- Multivariate Rational Interpolation.- On the Role of the Exponential Eigen Splines in Translation Invariant Periodic Spline Spaces.- Uniform Closure of Tensor Product of Linear Subspaces.- On Periodic Hermite-Birkhoff Interpolation by Translation.- Problems and Results in the Calculation of Extremal Fundamental Systems for Sphere and Ball.- Bernstein Quasi-Interpolants on [0,1].- Elementary Holograms, Artificial Neural Networks, and Theta-Null Values.- Some Recent Results on Complex Interpolatory Approximation.- Index Transforms for Multidimensional Discrete Fourier Transforms.- Minimal Properties of Periodic Box-Spline Interpolation on a Three Direction Mesh.- Strong Uniform Approximation by Bochner-Riesz Means.