Multivariate Polysplines

Key Features

· Part 1 assumes no special knowledge of partial differential equations and is intended as a graduate level introduction to the topic
· Part 2 develops the theory of cardinal Polysplines, which is a natural generalization of Schoenberg's beautiful one-dimensional theory of cardinal splines
· Part 3 constructs a wavelet analysis using cardinal Polysplines. The results parallel those found by Chui for the one-dimensional case
· Part 4 considers the ultimate generalization of Polysplines - on manifolds, for a wide class of higher-order elliptic operators and satisfying a Holladay variational property.

Multivariate Polysplines is aimed principally at specialists in approximation and spline theory, wavelet analysis and signal and image processing. It will also prove a valuable text for people using computer aided geometric design (CAGD and CAD/CAM) systems or smoothing and spline methods in geophysics, geodesy, geology, magnetism etc. as it offers a flexible alternative to traditional tools such as Kriging, Radial Basis Functions and Minimum Curvature.

The book is also suitable as a text for graduate courses on these topics.

Contents

Preface

• Introduction

• One-dimensional linera and cubic splines
• The twxo-dimensional case: data and smoothness concepts
• The objects concepts: harmonic and polyharmonic functions in rectangular domains in R²
• Polysplines on strips in R²
• Application of polysplines to magnetism and CAGD
• The object concpet: harmonic and polyharmonic functions in annuli in R²
• Polysplines on annuli R²
• Polysplines on strips and annuli in R²
• Compendium on spherical harmonics and polyharmonic functions
• Appendix on Chebyshev splines
• Appendix on Fourier series and Fourier transfomr
• Bibliography to part I

• Cardinal L-splines according to Micchelli
• Risz bound for the cardinal L-splines Q_z+1
• Cardinal interpolation polysplines on annuli
• Bibliography to part II

• Chui's cardinal spline wavelet analysis
• Cardinal L-spline wavelet analysis
• Polyharmonic wavelet analysis: scalling and rotationally invariant spaces
• Bibliography to part III

• Heuristic arguments
• Definition of polysplines and uniqueness for general interfaces
• A priori estimates and Fredholm operators
• Existence and convergence of polysplines
• Appendix on elliptic boundary value problems in Sobolev and Hölder spaces
• Afterwords
• Bibliography to part IV
• Index