Multivariate Polysplines

  • ID: 1768512
  • Book
  • 498 Pages
  • Elsevier Science and Technology
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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
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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


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Kounchev, Ognyan
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.
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