Title:
Multivariate polysplines : applications to numerical and wavelet analysis
Personal Author:
Publication Information:
California : Academic Press, 2001
ISBN:
9780124224902
Subject Term:
Available:*
Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Summary
Summary
Multivariate polysplines are a new mathematical technique that has arisen from a synthesis of approximation theory and the theory of partial differential equations. It is an invaluable means to interpolate practical data with smooth functions.
Multivariate polysplines have applications in the design of surfaces and "smoothing" that are essential in computer aided geometric design (CAGD and CAD/CAM systems), geophysics, magnetism, geodesy, geography, wavelet analysis and signal and image processing. In many cases involving practical data in these areas, polysplines are proving more effective than well-established methods, such as kKriging, radial basis functions, thin plate splines and minimum curvature.
Author Notes
Ognyan Kounchev received his M.S. in partial differential equations from Sofia University, Bulgaria and his Ph.D. in optimal control of partial differential equations and numerical methods from the University of Belarus, Minsk. He was awarded a grant from the Volkswagen Foundation (1996-1999) for studying the applications of partial differential equations in approximation and spline theory. Currently, Dr Kounchev is a Fulbright Scholar at the University of Wisconsin-Madison where he works in the Wavelet Ideal Data Representation Center in the Department of Computer Sciences.
Table of Contents
| Preface |
| 1 Introduction |
| 1.1 Organization of Material |
| 1.1.1 Part I: Introduction of Polysplines |
| 1.1.2 Part II: Cardinal Polysplines |
| 1.1.3 Part III: Wavelet Analysis Using Polysplines |
| 1.1.4 Part IV: Polysplines on General Interfaces |
| 1.2 Audience |
| 1.3 Statements |
| 1.4 Acknowledgements |
| 1.5 The Polyharmonic Paradigm |
| 1.5.1 The Operator, Object and Data Concepts of the Polyharmonic Paradigm |
| 1.5.2 The Taylor Formula |
| Part I Introduction to Polysplines |
| 2 One-Dimensional Linear and Cubic Splines |
| 2.1 Cubic Splines |
| 2.2 Linear Splines |
| 2.3 Variational (Holladay) Property of the Odd-Degree Splines |
| 2.4 Existence and Uniqueness of Odd-Degree Splines |
| 2.5 The Holladay Theorem |
| 3 The Two-Dimensional Case: Data and Smoothness Concepts |
| 3.1 The Data Concept in Two Dimensions According to the Polyharmonic Paradigm |
| 3.2 The Smoothness Concept According to the Polyharmonic Paradigm |
| 4 The Objects Concept: Harmonic and Polyharmonic Functions in Rectangular Domains in ?2 |
| 4.1 Harmonic Functions in Strips or Rectangles |
| 4.2 "Parametrization" of the Space of Periodic Harmonic Functions in the Strip: the Dirichlet Problem |
| 4.3 "Parametrization" of the Space of Periodic Polyharmonic Functions in the Strip: the Dirichlet Problem |
| 4.4 Nonperiodicity in y |
| 5 Polysplines on Strips in ?2 |
| 5.1 Periodic Harmonic Polysplines on Strips, p = |
| 5.2 Periodic Biharmonic Polysplines on Strips, p = |
| 5.3 Computing the Biharmonic Polysplines on Strips |
| 5.4 Uniqueness of the Interpolation Polysplines |
| 6 Application of Polysplines to Magnetism and CAGD |
| 6.1 Smoothing Airborne Magnetic Field Data |
| 6.2 Applications to Computer-Aided Geometric Design |
| 6.3 Conclusions |
| 7 The Objects Concept: Harmonic and Polyharmonic Functions in Annuli in ?2 |
| 7.1 Harmonic Functions in Spherical (Circular) Domains |
| 7.2 Biharmonic and Polyharmonic Functions |
| 7.3 "Parametrization" of the Space of Polyharmonic Functions in the Annulus and Ball: the Dirichlet Problem |
| 8 Polysplines on annuli in ?2 |
| 8.1 The Biharmonic Polysplines, p = 2 |
| 8.2 Radially Symmetric Interpolation Polysplines |
| 8.3 Computing the Polysplines for General (Nonconstant) Data |
| 8.4 The Uniqueness of Interpolation Polysplines on Annuli |
| 8.5 The change v = log r and the Operators Mk,p |
| 8.6 The Fundamental Set of Solutions for the Operator Mk,p(d/dv) |
| 9 Polysplines on Strips and Annuli in ?n |
| 9.1 Polysplines on Strips in ?n |
| 9.2 Polysplines on Annuli in ?n |
| 10 Compendium on Spherical Harmonics and Polyharmonic Functions |
| 10.1 Introduction |
| 10.2 Notations |
| 10.3 Spherical Coordinates and the Laplace Operator |
| 10.4 Fourier Series and Basic Properties |
| 10.5 Finding the Point of View |
| 10.6 Homogeneous Polynomials in ?n |
| 10.7 Gauss Pepresentation of Homogeneous Polynomials |
| 10.8 Gauss Representation: Analog to the Taylor Series, the Polyharmonic Paradigm |
| 10.9 The Sets ?k are Eigenspaces for the Operator ?? |
| 10.10 Completeness of the Spherical Harmonics in L2(??n-1) |
| 10.11 Solutions of ?w(x) = 0 with Separated Variables |
| 10.12 Zonal Harmonics : the Functional Approach |
| 10.13 The Classical Approach to Zonal Harmonics |
| 10.14 The Representation of Polyharmonic Functions Using Spherical Harmonics |
| 10.15 The Operator is Formally Self-Adjoint |
| 10.16 The Almansi Theorem |
| 10.17 Bibliographical Notes |
| 11 Appendix on Chebyshev Splines |
| 11.1 Differential Operators and Extended Complete Chebyshev Systems |
| 11.2 Divided Differences for Extended Complete Chebyshev Systems |
| 11.3 Dual Operator and ECT-System |
| 11.4 Chebyshev Splines and One-Sided Basis |
| 11.5 Natural Chebyshev Splines |
| 12 Appendix on Fourier Series and Fourier Transform |
| 12.1 Bibliographical Notes |
| Bibliography to Part I |
| Part II Cardinal Polysplines in ?n |
| 13 Cardinal L-Splines According to Micchelli |
| 13.1 Cardinal L-Splines and the Interpolation Problem |
| 13.2 Differential Operators and their Solution Sets UZ+1 |
| 13.3 Variation of the Set UZ+1[?] with ? and Other Properties |
| 13.4 The Green Function (x) of the Operator ?Z+1 |
| 13.5 The Dictionary: L-Polynomial Case |
| 13.6 The Generalized Euler Polynomials AZ(x; ?) |
| 13.7 Generalized Divided Difference Operator |
| 13.8 Zeros of the Euler-Frobenius Polynomial ?Z(?) |
| 13.9 The Cardinal Interpolation Problem for L-Splines |
| 13.10 The Cardinal Compactly Supported L-Splines QZ+1 |
| 13.11 Laplace and Fourier Transform of the Cardinal TB-Spline QZ+1 |
| 13.12 Convolution Formula for Cardinal TB-Splines |
| 13.13 Differentiation of Cardinal TB-Splines |
| 13.14 Hermite-Gennocchi-Type Formula |
| 13.15 Recurrence Relation for the TB-Spline |
| 13.16 The Adjoint Operator ?*Z+1 and the TB-Spline Q*Z+1(x) |
| 13.17 The Euler Polynomial AZ(x; ?) and the TB-Spline QZ+1(x) |
| 13.18 The Leading Coefficient of the Euler-Frobenius Polynomial ?Z(?) |
| 13.19 Schoenberg's "Exponential" Euler L-Spline ?Z(x; ?) and AZ(x; ?) |
| 13.20 Marsden's Identity for Cardinal L-Splines |
| 13.21 Peano Kernel and the Divided Difference Operator in the Cardinal Case |
| 13.22 Two-Scale Relation (Refinement Equation) for the TB-Splines QZ+1[?; h] |
| 13.23 Symmetry of the Zeros of the Euler-Frobenius Polynomial ?Z(?) |
| 13.24 Estimates of the Functions AZ(x; ?) and QZ+1(x) |
| 14 Riesz Bounds for the Cardinal L-Splines QZ+1 |
| 14.1 Summary of Necessary Results for Cardinal L-Splines |
| 14.2 Riesz Bounds |
| 14.3 The Asymptotic of AZ(0; ?) in k |
| 14.4 Asymptotic of the Riesz Bounds A, B |
| 14.5 Synthesis of Compactly Supported Polysplines on Annuli |
| 15 Cardinal interpolation Polysplines on annuli 287 |
| 15.1 Introduction |
| 15.2 Formulation of the Cardinal Interpolation Problem for Polysplines |
| 15.3 ? = 0 is good for all L-Splines with L = Mk,p |
| 15.4 Explaining the Problem |
| 15.5 Schoenberg's Results on the Fundamental Spline L(X) in the Polynomial Case |
| 15.6 Asymptotic of the Zeros of ?Z(?; 0) |
| 15.7 The Fundamental Spline Function L(X) for the Spherical Operators Mk,p |
| 15.8 Synthesis of the Interpolation Cardinal Polyspline |
| 15.9 Bibliographical Notes |
| Bibliography to Part II |
| Part III Wavelet Analysis |
| 16 Chui's Cardinal Spline Wavelet Analysis |
| 16.1 Cardinal Splines and the Sets Vj |
| 16.2 The Wavelet Spaces Wj |
| 16.3 The Mother Wavelet ? |
| 16.4 The Dual Mother Wavelet ? |
| 16.5 The Dual Scaling Function ? |
| 16.6 Decomposition Relations |
| 16.7 Decomposition and Reconstruction Algorithms |
| 16.8 Zero Moments |
| 16.9 Symmetry and Asymmetry |
| 17 Cardinal L-Spline Wavelet Analysis |
| 17.1 Introduction: the Spaces Vj and Wj |
| 17.2 Multiresolution Analysis Using L-Splines |
| 17.3 The Two-Scale Relation for the TB-Splines QZ+1(x) |
| 17.4 Construction of the Mother Wavelet ?h |
| 17.5 Some Algebra of Laurent Polynomials and the Mother Wavelet ?h |
| 17.6 Some Algebraic Identities |
| 17.7 The Function ?h Generates a Riesz Basis of W0 |
| 17.8 Riesz Basis from all Wavelet Functions ? (x) |
| 17.9 The Decomposition Relations for the Scaling Function QZ+1 |
| 17.10 The Dual Scaling Function ? and the Dual Wavelet ? |
| 17.11 Decomposition and Reconstruction by L-Spline Wavelets and MRA |
| 17.12 Discussion of the Standard Scheme of MRA |
| 18 Polyharmonic Wavelet Analysis: Scaling and Rotationally Invariant Spaces |
| 18.1 The Refinement Equation for the Normed TB-Spline QZ+1 |
| 18.2 Finding the Way: some Heuristics |
| 18.3 The Sets PVj and Isomorphisms |
| 18.4 Spherical Riesz Basis and Father Wavelet |
| 18.5 Polyharmonic MRA |
| 18.6 Decomposition and Reconstruction for Polyharmonic Wavelets and the Mother Wavelet |
| 18.7 Zero Moments of Polyharmonic Wavelets |
| 18.8 Bibliographical Notes |
| Bibliography to Part III |
| Part IV Polysplines for General Interfaces |
| 19 Heuristic Arguments |
| 19.1 Introduction |
| 19.2 The Setting of the Variational Problem |
| 19.3 Polysplines of Arbitrary Order p |
| 19.4 Counting the Parameters |
| 19.5 Main Results and Techniques |
| 19.6 Open Problems |
| 20 Definition of Polysplines and Uniqueness for General Interfaces |
| 20.1 Introduction |
| 20.2 Definition of Polysplines |
| 20.3 Basic Identity for Polysplines of even Order p = 2q |
| 20.4 Uniqueness of Interpolation Polysplines and Extremal Holladay-Type Property |
| 21 A Priori Estimates and Fredholm Operators |
| 21.1 Basic Proposition for Interface on the Real Line |
| 21.2 A Priori Estimates in a Bounded Domain with Interfaces |
| 21.3 Fredholm Operator in the Space H2p+r(D\ST ) for r ? 0 |
| 22 Existence and Convergence of Polysplines |
| 22.1 Polysplines of Order 2q for Operator L = L |
| 22.2 The Case of a General Operator L |
| 22.3 Existence of Polysplines on Strips with Compact Data |
| 22.4 Classical Smoothness of the Interpolation Data gj |
| 22.5 Sobolev Embedding in Ck,? |
| 22.6 Existence for an Interface which is not C? |
| 22.7 Convergence Properties of the Polysplines |
| 22.8 Bibliographical Notes and Remarks |
| 23 Appendix on Elliptic Boundary Value Problems in Sobolev and Hölder Spaces |
| 23.1 Sobolev and Hölder Spaces |
| 23.2 Regular Elliptic Boundary Value Problems |
| 23.3 Boundary Operators, Adjoint Problem and Green Formula |
| 23.4 Elliptic Boundary Value Problems |
| 23.5 Bibliographical Notes |
| 24 Afterword |
| Bibliography to Part IV |
| Index |
