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.

Preface1 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 FormulaPart 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 ¿¿ 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