Multivariate Approximation and Splines

Prof. Dr. Guido Walz ist apl. Professor für angewandte Mathematik an der Universität Mannheim, Dozent an verschiedenen Hochschulen und wissenschaftlicher Publizist. Als Projektleiter und Chefredakteur realisierte er u. a. das 'Lexikon der Mathematik in sechs Bänden'.

This book contains the refereed papers which were presented at the interna tional conference on "Multivariate Approximation and Splines" held in Mannheim, Germany, on September 7-10,1996. Fifty experts from Bulgaria, England, France, Israel, Netherlands, Norway, Poland, Switzerland, Ukraine, USA and Germany participated in the symposium. It was the aim of the conference to give an overview of recent developments in multivariate approximation with special emphasis on spline methods. The field is characterized by rapidly developing branches such as approximation, data fit ting, interpolation, splines, radial basis functions, neural networks, computer aided design methods, subdivision algorithms and wavelets. The research has applications in areas like industrial production, visualization, pattern recognition, image and signal processing, cognitive systems and modeling in geology, physics, biology and medicine. In the following, we briefly describe the contents of the papers. Exact inequalities of Kolmogorov type which estimate the derivatives of mul the paper of BABENKO, KOFANovand tivariate periodic functions are derived in PICHUGOV. These inequalities are applied to the approximation of classes of mul tivariate periodic functions and to the approximation by quasi-polynomials. BAINOV, DISHLIEV and HRISTOVA investigate initial value problems for non linear impulse differential-difference equations which have many applications in simulating real processes. By applying iterative techniques, sequences of lower and upper solutions are constructed which converge to a solution of the initial value problem.

Springer Book Archives

Multivariate Inequalities of Kolmogorov Type and Their Applications.- Monotone Iterative Technique for Impulsive Differential-Difference Equations with Variable Impulsive Perturbations.- Multivariate Cosine Wavelets.- On Almost Interpolation by Multivariate Splines.- Locally Linearly Independent Systems and Almost Interpolation.- Exponential-Type Approximation in Multivariate Harmonic Hilbert Spaces.- Interpolation by Continuous Function Spaces.- Discrete Characterization of Besov Spaces and its Applications to Stochastics.- One-Sided Approximation and Interpolation Operators Generating Hyperbolic Sigma-Pi Neural Networks.- Unconstrained Minimization of Quadratic Splines and Applications.- Interpolation by Translates of a Basis function.- On the Sup-Norm Condition Number of the Multivariate Triangular Bernstein Basis.- Integration Methods of Clenshaw-Curtis Type, Based on Four Kinds of Chebyshev Polynomials.- Tensor Products of Convex Cones.- The Curse of Dimension and a Universal Method for Numerical Integration.- Interpolation by Bivariate Splines on Crosscut Partitions.- Necessary and Sufficient Conditions for Orthonormality of Scaling Vectors.- Trigonometric Preconditioners for Block Toeplitz Systems.- The Average Size of Certain Gram-Determinants and Interpolation on Non-Compact Sets.- Radial Basis Functions Viewed From Cubic Splines.- Wavelet Modelling of High Resolution Radar Imaging and Clinical Magnetic Resonance Tomography.- A New Interpretation of the Sampling Theorem and its Extensions.- Gridded Data Interpolation with Restrictions on the First Order Derivatives.- Affine Frames and Multiresolution.- List of Participants.