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# Multivariate Polysplines: Applications to Numerical and Wavelet Analysis / Edition 1

Multivariate Polysplines: Applications to Numerical and Wavelet Analysis / Edition 1 available in Hardcover, NOOK Book

## Overview

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.

- 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

## Product Details

ISBN-13: | 9780124224902 |
---|---|

Publisher: | Elsevier Science |

Publication date: | 07/04/2001 |

Pages: | 498 |

Product dimensions: | 1.19(w) x 6.14(h) x 9.21(d) |

## Read an Excerpt

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.

- 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

## First Chapter

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.

- 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

## Table of Contents

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 ℝ2

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 ℝ2

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 ℝ2

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 ℝ2

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 ℝn

9.1 Polysplines on Strips in ℝn

9.2 Polysplines on Annuli in ℝn

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 ℝn

10.7 Gauss Pepresentation of Homogeneous Polynomials

10.8 Gauss Representation: Analog to the Taylor Series, the Polyharmonic Paradigm

10.9 The Sets ℋk are Eigenspaces for the Operator ∆θ

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 ℝn

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[Λ] with Λ and Other Properties

13.4 The Green Function (x) of the Operator ℒZ+1

13.5 The Dictionary: L-Polynomial Case

13.6 The Generalized Euler Polynomials AZ(x; λ)

13.7 Generalized Divided Difference Operator

13.8 Zeros of the Euler–Frobenius Polynomial ΠZ(λ)

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 ℒ*Z+1 and the TB-Spline Q*Z+1(x)

13.17 The Euler Polynomial AZ(x; λ) and the TB-Spline QZ+1(x)

13.18 The Leading Coefficient of the Euler–Frobenius Polynomial ΠZ(λ)

13.19 Schoenberg’s “Exponential” Euler L-Spline ΦZ(x; λ) and AZ(x; λ)

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[Λ; h]

13.23 Symmetry of the Zeros of the Euler–Frobenius Polynomial ΠZ(λ)

13.24 Estimates of the Functions AZ(x; λ) 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; λ) 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 α = 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 ΠZ(λ; 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 ψ

16.4 The Dual Mother Wavelet ψ

16.5 The Dual Scaling Function φ

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 ψh

17.5 Some Algebra of Laurent Polynomials and the Mother Wavelet ψh

17.6 Some Algebraic Identities

17.7 The Function ψh Generates a Riesz Basis of W0

17.8 Riesz Basis from all Wavelet Functions ψ(x)

17.9 The Decomposition Relations for the Scaling Function QZ+1

17.10 The Dual Scaling Function φ and the Dual Wavelet ψ

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,α

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

## Reading Group Guide

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 ℝ2

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 ℝ2

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 ℝ2

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 ℝ2

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 ℝn

9.1 Polysplines on Strips in ℝn

9.2 Polysplines on Annuli in ℝn

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 ℝn

10.7 Gauss Pepresentation of Homogeneous Polynomials

10.8 Gauss Representation: Analog to the Taylor Series, the Polyharmonic Paradigm

10.9 The Sets ℋk are Eigenspaces for the Operator ∆θ

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 ℝn

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[Λ] with Λ and Other Properties

13.4 The Green Function (x) of the Operator ℒZ+1

13.5 The Dictionary: L-Polynomial Case

13.6 The Generalized Euler Polynomials AZ(x; λ)

13.7 Generalized Divided Difference Operator

13.8 Zeros of the Euler–Frobenius Polynomial ΠZ(λ)

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 ℒ*Z+1 and the TB-Spline Q*Z+1(x)

13.17 The Euler Polynomial AZ(x; λ) and the TB-Spline QZ+1(x)

13.18 The Leading Coefficient of the Euler–Frobenius Polynomial ΠZ(λ)

13.19 Schoenberg’s “Exponential” Euler L-Spline ΦZ(x; λ) and AZ(x; λ)

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[Λ; h]

13.23 Symmetry of the Zeros of the Euler–Frobenius Polynomial ΠZ(λ)

13.24 Estimates of the Functions AZ(x; λ) 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; λ) 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 α = 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 ΠZ(λ; 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 ψ

16.4 The Dual Mother Wavelet ψ

16.5 The Dual Scaling Function φ

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 ψh

17.5 Some Algebra of Laurent Polynomials and the Mother Wavelet ψh

17.6 Some Algebraic Identities

17.7 The Function ψh Generates a Riesz Basis of W0

17.8 Riesz Basis from all Wavelet Functions ψ(x)

17.9 The Decomposition Relations for the Scaling Function QZ+1

17.10 The Dual Scaling Function φ and the Dual Wavelet ψ

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,α

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

## Interviews

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 ℝ2

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 ℝ2

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 ℝ2

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 ℝ2

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 ℝn

9.1 Polysplines on Strips in ℝn

9.2 Polysplines on Annuli in ℝn

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 ℝn

10.7 Gauss Pepresentation of Homogeneous Polynomials

10.8 Gauss Representation: Analog to the Taylor Series, the Polyharmonic Paradigm

10.9 The Sets ℋk are Eigenspaces for the Operator ∆θ

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 ℝn

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[Λ] with Λ and Other Properties

13.4 The Green Function (x) of the Operator ℒZ+1

13.5 The Dictionary: L-Polynomial Case

13.6 The Generalized Euler Polynomials AZ(x; λ)

13.7 Generalized Divided Difference Operator

13.8 Zeros of the Euler–Frobenius Polynomial ΠZ(λ)

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 ℒ*Z+1 and the TB-Spline Q*Z+1(x)

13.17 The Euler Polynomial AZ(x; λ) and the TB-Spline QZ+1(x)

13.18 The Leading Coefficient of the Euler–Frobenius Polynomial ΠZ(λ)

13.19 Schoenberg’s “Exponential” Euler L-Spline ΦZ(x; λ) and AZ(x; λ)

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[Λ; h]

13.23 Symmetry of the Zeros of the Euler–Frobenius Polynomial ΠZ(λ)

13.24 Estimates of the Functions AZ(x; λ) 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; λ) 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 α = 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 ΠZ(λ; 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 ψ

16.4 The Dual Mother Wavelet ψ

16.5 The Dual Scaling Function φ

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 ψh

17.5 Some Algebra of Laurent Polynomials and the Mother Wavelet ψh

17.6 Some Algebraic Identities

17.7 The Function ψh Generates a Riesz Basis of W0

17.8 Riesz Basis from all Wavelet Functions ψ(x)

17.9 The Decomposition Relations for the Scaling Function QZ+1

17.10 The Dual Scaling Function φ and the Dual Wavelet ψ

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,α

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

## Recipe

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 ℝ2

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 ℝ2

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 ℝ2

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 ℝ2

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 ℝn

9.1 Polysplines on Strips in ℝn

9.2 Polysplines on Annuli in ℝn

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 ℝn

10.7 Gauss Pepresentation of Homogeneous Polynomials

10.8 Gauss Representation: Analog to the Taylor Series, the Polyharmonic Paradigm

10.9 The Sets ℋk are Eigenspaces for the Operator ∆θ

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 ℝn

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[Λ] with Λ and Other Properties

13.4 The Green Function (x) of the Operator ℒZ+1

13.5 The Dictionary: L-Polynomial Case

13.6 The Generalized Euler Polynomials AZ(x; λ)

13.7 Generalized Divided Difference Operator

13.8 Zeros of the Euler–Frobenius Polynomial ΠZ(λ)

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 ℒ*Z+1 and the TB-Spline Q*Z+1(x)

13.17 The Euler Polynomial AZ(x; λ) and the TB-Spline QZ+1(x)

13.18 The Leading Coefficient of the Euler–Frobenius Polynomial ΠZ(λ)

13.19 Schoenberg’s “Exponential” Euler L-Spline ΦZ(x; λ) and AZ(x; λ)

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[Λ; h]

13.23 Symmetry of the Zeros of the Euler–Frobenius Polynomial ΠZ(λ)

13.24 Estimates of the Functions AZ(x; λ) 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; λ) 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 α = 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 ΠZ(λ; 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 ψ

16.4 The Dual Mother Wavelet ψ

16.5 The Dual Scaling Function φ

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 ψh

17.5 Some Algebra of Laurent Polynomials and the Mother Wavelet ψh

17.6 Some Algebraic Identities

17.7 The Function ψh Generates a Riesz Basis of W0

17.8 Riesz Basis from all Wavelet Functions ψ(x)

17.9 The Decomposition Relations for the Scaling Function QZ+1

17.10 The Dual Scaling Function φ and the Dual Wavelet ψ

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,α

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