Ebook: Multivariate Approximation and Splines

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.

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Content:
Front Matter....Pages i-x
Multivariate Inequalities of Kolmogorov Type and Their Applications....Pages 1-12
Monotone Iterative Technique for Impulsive Differential-Difference Equations with Variable Impulsive Perturbations....Pages 13-27
Multivariate Cosine Wavelets....Pages 29-43
On Almost Interpolation by Multivariate Splines....Pages 45-58
Locally Linearly Independent Systems and Almost Interpolation....Pages 59-72
Exponential-type Approximation in Multivariate Harmonic Hilbert Spaces....Pages 73-82
Interpolation by Continuous Function Spaces....Pages 83-88
Discrete Characterization of Besov Spaces and Its Applications to Stochastics....Pages 89-98
One-Sided Approximation and Interpolation Operators Generating Hyperbolic Sigma-Pi Neural Networks....Pages 99-112
Unconstrained Minimization of Quadratic Splines and Applications....Pages 113-128
Interpolation by Translates of a Basis Function....Pages 129-140
On the Sup-norm Condition Number of the Multivariate Triangular Bernstein Basis....Pages 141-151
Integration Methods of Clenshaw-Curtis Type, Based on Four Kinds of Chebyshev Polynomials....Pages 153-165
Tensor Products of Convex Cones....Pages 167-176
The Curse of Dimension and a Universal Method For Numerical Integration....Pages 177-187
Interpolation by Bivariate Splines on Crosscut Partitions....Pages 189-203
Necessary and Sufficient Conditions for Orthonormality of Scaling Vectors....Pages 205-218
Trigonometric Preconditioners for Block Toeplitz Systems....Pages 219-234
The Average Size of Certain Gram-Determinants and Interpolation on Non-Compact Sets....Pages 235-243
Radial Basis Functions Viewed From Cubic Splines....Pages 245-258
Wavelet Modelling of High Resolution Radar Imaging and Clinical Magnetic Resonance Tomography....Pages 259-274
A New Interpretation of the Sampling Theorem and Its Extensions....Pages 275-288
Gridded Data Interpolation with Restrictions on the First Order Derivatives....Pages 289-305
Affine Frames and Multiresolution....Pages 307-320
Back Matter....Pages 321-326

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Content:
Front Matter....Pages i-x
Multivariate Inequalities of Kolmogorov Type and Their Applications....Pages 1-12
Monotone Iterative Technique for Impulsive Differential-Difference Equations with Variable Impulsive Perturbations....Pages 13-27
Multivariate Cosine Wavelets....Pages 29-43
On Almost Interpolation by Multivariate Splines....Pages 45-58
Locally Linearly Independent Systems and Almost Interpolation....Pages 59-72
Exponential-type Approximation in Multivariate Harmonic Hilbert Spaces....Pages 73-82
Interpolation by Continuous Function Spaces....Pages 83-88
Discrete Characterization of Besov Spaces and Its Applications to Stochastics....Pages 89-98
One-Sided Approximation and Interpolation Operators Generating Hyperbolic Sigma-Pi Neural Networks....Pages 99-112
Unconstrained Minimization of Quadratic Splines and Applications....Pages 113-128
Interpolation by Translates of a Basis Function....Pages 129-140
On the Sup-norm Condition Number of the Multivariate Triangular Bernstein Basis....Pages 141-151
Integration Methods of Clenshaw-Curtis Type, Based on Four Kinds of Chebyshev Polynomials....Pages 153-165
Tensor Products of Convex Cones....Pages 167-176
The Curse of Dimension and a Universal Method For Numerical Integration....Pages 177-187
Interpolation by Bivariate Splines on Crosscut Partitions....Pages 189-203
Necessary and Sufficient Conditions for Orthonormality of Scaling Vectors....Pages 205-218
Trigonometric Preconditioners for Block Toeplitz Systems....Pages 219-234
The Average Size of Certain Gram-Determinants and Interpolation on Non-Compact Sets....Pages 235-243
Radial Basis Functions Viewed From Cubic Splines....Pages 245-258
Wavelet Modelling of High Resolution Radar Imaging and Clinical Magnetic Resonance Tomography....Pages 259-274
A New Interpretation of the Sampling Theorem and Its Extensions....Pages 275-288
Gridded Data Interpolation with Restrictions on the First Order Derivatives....Pages 289-305
Affine Frames and Multiresolution....Pages 307-320
Back Matter....Pages 321-326
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