constructive approximation

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  • Constructive Approximationon the SphereWith Applicationsto Geomathematics

    W. FREEDENGeomathematics Group, University of Kaiserslautern

    T. GERVENSUniversity ofOsnabriick (FH)

    M. SCHREINERGeomathematics Group, University of Kaiserslautern

    CLARENDON PRESS OXFORD1998

  • CONTENTS

    List of Symbols xiii

    Introduction 1

    1 Notation 71.1 Basic Settings 71.2 Spherical Nomenclature 8

    SCALARS

    2 Polynomials 212.1 Homogeneous Polynomials 212.2 Homogeneous Harmonic Polynomials 252.3 Addition Theorem 29

    3 Scalar Spherical Harmonics 353.1 Definition 353.2 Legendre Polynomials 383.3 Fundamental Systems 463.4" Closure and Completeness 523.5 Eigenfunctions of the Beltrami Operator 593.6 Funk-Hecke Formula 60

    4 Green's Functions 674.1 'Definition 674.2 Integral Formulae 724.3 Beltrami Differential Equation 78

    5 Spherical Radial Basis Functions 815.1 Sobolev Spaces and Pseudodifferential Operators 815.2 Reproducing Kernel Sobolev Spaces 88

  • viii CONTENTS

    5.3 Integral Formulae 925.4 Peano's Theorem 975.5 Localization 995.6 Examples of Radial Basis Functions 105

    5.6.1 Green's Kernels Corresponding to Iterated Bel-trami Operators 105

    5.6.2 Kernels Corresponding to Invertible Pseudo-differential Operators 107

    5.6.3 Abel-Poisson Kernel and Related Functions 1085.6.4 Gaufi-WeierstraB Kernel and Related Func-

    tions 1125.7 Positive Definiteness 1155.8 Further Examples of Radial Basis Functions 120

    5.8.1 Kernels Related to Radial Basis Functions ofEuclidean Spaces 120

    5.8.2 Locally Supported Kernels 1235.9 Strict Positive Definiteness 135

    6 Spherical Splines 1396.1 Variational Characterization of Spline Interpolation 139

    6.1.1 Spherical Spline Interpolation, m = 1 1396.1.2 Spherical Spline Interpolation, m > 0 143

    6.2 Error Estimates 1496.3 Convergence 1536.4 Complete Spline Bases 1556.5 Combined Interpolation and Smoothing 157

    7 Approximate Integration 1657.1 Low Discrepancy Method 1667.2 Interpolatory Polynomial Integration 1777.3 Best Approximations 1857.4 Interpolatory Spline Integration 188

    8 Spherical Singular Integrals 1938.1 Approximate Identities 1938.2 Examples of Spherical Singular Integrals 197

    8.2.1 Abel-Poisson Singular Integral 197

  • CONTENTS

    8.2.2 Gaufi-Weierstrafi Singular Integral 1998.2.3 __ Singular Integrals Corresponding to Locally

    Supported Kernel Functions 2018.2.4 Spherical Up Function 207

    9 Gabor and Toeplitz Transforms 2099.1 Gabor Transform 2109.2 Combined Fourier and Gabor Transform 2159.3 Toeplitz Transform 2169.4 Combined Fourier and Toeplitz Transform 2209.5 Combined Gabor and Toeplitz Transform 223

    10 Continuous Wavelet Transform 22710.1 Continuous Wavelet Transform (Linear Theory) 22910.2 Continuous Wavelet Transform (Bilinear Theory) 234

    10.2.1 Reconstruction Formula 23410.2.2 Scaling Function 23710.2.3 Examples 24210.2.4 Least Squares Property 243

    10.3 Scale Discretized Wavelet Transform 24610.3.1 P-Scale Discretized Wavelets 24610.3.2 M-scale Discretized Wavelets. 25210.3.3 Multiresolution Analysis by Means of Scale

    Discretized Wavelets. 25510.3.4 >-scale Discretized Wavelet Transform 26210.3.5 Least Squares Property 269

    10.4 Fully Discretized Wavelet Transform 27010.5 Combined Spherical Harmonic and Wavelet Expan-

    sion 272

    11 Discrete Wavelet Transform 27511.1 Scale Discrete Scaling Function 276

    11.1.1 Admissibility Condition 27611.1.2 Scale Discrete Scaling Function 27911.1.3 Discrete Approximate Identity 280

    11.2 Scale Discrete Wavelet-Transform 28211.3 Non-band-limited Wavelets 287

    11.3.1 Rational Wavelets 28711.3.2 Modified Rational Wavelet 288

  • x CONTENTS

    11.3.3 Exponential Wavelets 28911.4 Band-limited Wavelets 290

    11.4.1 Shannon Wavelets 29111.4.2 De la Valid Poussin Wavelets 29411.4.3 Cubic Polynomial Wavelets 29511.4.4 Exact Fully Discrete Wavelet Transform 29611.4.5 A Pyramid Scheme 301

    11.5 Scale Discrete Wavelet Transform of Positive Order 31111.5.1 Fixed Order 31211.5.2 Increasing Order 315

    VECTORS

    12 Vector Spherical Harmonics 32112.1 Notation 32212.2 Definition of Vector Spherical Harmonics 32312.3 Homogeneous Harmonic Vector Polynomials 32412.4 Decomposition of Spherical Vector Fields 32912.5 Vectorial Beltrami Operator 33112.6 Vectorial Addition Theorem 33312.7 Vectorial Funk-Hecke Formulae 340

    13 Vectorial Approximation Methods 34513.1 Sobolev Spaces and Pseudodifferential Operators 34513.2 Vectorial Splines 34813.3 Vectorial Wavelets 35013.4 Scale Continuous Spherical Wavelets 35213.5 Scale Discrete Spherical Wavelets 352

    TENSORS

    14 Tensor Spherical Harmonics 35714.1 Notation 35814.2 Definition of Tensor Spherical Harmonics 37014.3 Homogeneous Harmonic Tensor Polynomials 371

  • CONTENTS xi

    14.4 Decomposition Theorem 37614.5 Tensorial Beltrami Operator 38014.6 Tensorial Addition Theorem 38314.7 Terisorial Funk-Hecke Formula 392

    15 Tensorial Approximation Methods 399

    References 403

    Index 421