Spherical Functions of Mathematical Geosciences

by Freeden, Willi; Schreiner, Michael

• ISBN13: 9783540851110
• ISBN10: 3540851119
• Format: Hardcover
• Copyright: 2009-10-27
• Publisher: Springer Verlag

Summary

This book collects all material developed by the Geomathematics Group, TU Kaiserslautern, during the few last years to set up a theory of spherical functions of mathematical (geo-)physics. The work shows a twofold transition: First, the natural transition from the scalar to the vectorial and tensorial theory of spherical harmonics is given in coordinate-free representation, based on new variants of the addition theorem and the Funk-Hecke formulas. Second, the canonical transition from spherical harmonics via zonal (kernel) functions to the Dirac kernel is presented in close orientation to an uncertainty principle classifying the space/frequency (momentum) behavior of the functions for purposes of constructive approximation and data analysis. In doing so, the whole palette of spherical (trial) functions is provided for modeling and simulating phenomena and processes of the Earth system.

Author Biography

Willi Freeden born in 1948 in Kaldenkirchen/Germany, Studies in Mathematics, Geography, and Philosophy at the RWTH Aachen, 1971 GÇÿDiplom' in Mathematics, 1972 GÇÿStaatsexamen' in Mathematics and Geography, 1975 PhD in Mathematics, 1979 GÇÿHabilitation' in Mathematics, 1981/1982 Visiting Research Professor at the Ohio State University, Columbus (Department of Geodetic Sciences and Surveying), 1984 Professor of Mathematics at the RWTH Aachen (Institute of Pure and Applied Mathematics), 1989 Professor of Technomathematics, 1994 Head of the Geomathematics Group, 2002-2006 Vice-president for Research and Technology at the University of Kaiserslautern.Michael Schreiner born in 1966 in Mertesheim/Germany, Studies in Industrial Mathematics, Mechanical Engineering, and Computer Science at the University of Kaiserslautern, 1991 GÇÿDiplom' in Industrial Mathematics, 1994 PhD in Mathematics, 2004 GÇÿHabilitation' in Mathematics. 1997-2001 researcher and project leader at the Hilti Corp. Schaan, Liechtenstein, 2002 Professor for Industrial Mathematics at the University of Buchs NTB, Buchs, Switzerland. 2004 Head of the Department of Mathematics of the University of Buchs, 2004 also Lecturer at the University of Kaiserslautern.

Table of Contents

Preface	p. xiii
Introduction	p. 1
Motivation	p. 3
Layout	p. 14
Basic Settings and Spherical Nomenclature	p. 19
Scalars, Vectors, and Tensors	p. 19
Differential Operators	p. 24
Spherical Notation	p. 30
Function Spaces	p. 32
Differential Calculus	p. 35
Integral Calculus	p. 39
Orthogonal Invariance	p. 48
Scalar Spherical Harmonics	p. 57
Homogeneous Harmonic Polynomials	p. 58
Addition Theorem	p. 65
Exact Computation of Basis Systems	p. 71
Definition of Scalar Spherical Harmonics	p. 81
Legendre Polynomials	p. 87
Orthogonal (Fourier) Expansions	p. 97
Legendre (Spherical) Harmonics	p. 110
Funk-Hecke Formula	p. 115
Eigenfunctions of the Beltrami Operator	p. 117
Irreducibility of Scalar Harmonics	p. 119
Degree and Order Variances	p. 122
Associated Legendre Polynomials	p. 129
Associated Legendre (Spherical) Harmonics	p. 138
Exact Computation of Legendre Basis Systems	p. 153
Bibliographical Notes	p. 158
Green's Functions and Integral Formulas	p. 159
Green's Function with Respect to the Beltrami Operator	p. 159
Space Regularized Green Function with Respect to the Beltrami Operator	p. 162
Frequency Regularized Green Function with Respect to the Beltrami Operator	p. 170
Modified Green Functions	p. 173
Integral Formulas	p. 176
Differential Equations	p. 181
Approximate Integration and Spline Interpolation	p. 183
Integral Formulas with Respect to Iterated Beltrami Operators	p. 189
Differential Equations Respect to Iterated Beltrami Operators	p. 198
Bibliographical Notes	p. 200
Vector Spherical Harmonics	p. 201
Normal and Tangential Fields	p. 202
Definition of Vector Spherical Harmonics	p. 203
Helmholtz Decomposition Theorem for Spherical Vector Fields	p. 208
Orthogonal (Fourier) Expansions	p. 212
Homogeneous Harmonic Vector Polynomials	p. 220
Exact Computation of Orthonormal Systems	p. 223
Orthogonal Invariance	p. 228
Vectorial Beltrami Operator	p. 236
Vectorial Addition Theorem	p. 238
Vectorial Funk-Hecke Formulas	p. 244
Counterparts of the Legendre Polynomial	p. 248
Degree and Order Variances	p. 252
Vector Homogeneous Harmonic Polynomials	p. 257
Alternative Systems of Vector Spherical Harmonics	p. 260
Vector Legendre Kernels	p. 266
Bibliographical Notes	p. 271
Tensor Spherical Harmonics	p. 273
Some Nomenclature	p. 274
Normal and Tangential Fields	p. 275
Integral Theorems	p. 278
Definition of Tensor Spherical Harmonics	p. 283
Helmholtz Decomposition Theorem	p. 289
Orthogonal (Fourier) Expansions	p. 293
Homogeneous Harmonic Tensor Polynomials	p. 301
Tensorial Beltrami Operator	p. 306
Tensorial Addition Theorem	p. 309
Tensorial Funk-Hecke Formulas	p. 318
Counterparts to the Legendre Polynomials	p. 323
Tensor Homogeneous Harmonic Polynomials	p. 325
Alternative Systems of Tensor Spherical Harmonics	p. 328
Tensor Legendre Kernels	p. 334
Bibliographical Notes	p. 337
Scalar Zonal Kernel Functions	p. 339
Zonal Kernel Functions in Scalar Context	p. 339
Convolutions Involving Scalar Zonal Kernel Functions	p. 341
Classification of Zonal Kernel Functions	p. 343
Dirac Families of Zonal Scalar Kernel Functions	p. 357
Examples of Dirac Families	p. 366
Bibliographical Notes	p. 386
Vector Zonal Kernel Functions	p. 389
Preparatory Material	p. 390
Tensor Zonal Kernel Functions of Rank Two in Vectorial Context	p. 391
Vector Zonal Kernel Functions in Vectorial Context	p. 396
Convolutions Involving Vector Zonal Kernel Functions	p. 399
Dirac Families of Zonal Kernel Functions	p. 401
Bibliographical Notes	p. 403
Tensorial Zonal Kernel Functions	p. 405
Preparatory Material	p. 406
Tensor Zonal Kernel Functions of Rank Four in Tensorial Context	p. 406
Convolutions Involving Zonal Tensor Kernel Functions	p. 408
Tensor Zonal Kernel Functions of Rank Two in Tensorial Context	p. 410
Dirac Families of Zonal Tensor Kernel Functions	p. 414
Bibliographical Notes	p. 415
Zonal Function Modeling of Earth's Mass Distribution	p. 417
Key Observables	p. 418
Gravity Potential	p. 428
Inner/Outer Harmonics	p. 435
Limit Formulas and Jump Relations	p. 454
Gravity Anomalies and Deflections of the Vertical	p. 458
Geostrophic Ocean Flow and Dynamic Ocean Topography	p. 482
Elastic Field	p. 496
Density Distribution	p. 515
Vector Outer Harmonics and the Gravitational Gradient	p. 542
Tensor Outer Harmonics and the Gravitational Tensor	p. 551
Gravity Quantities in Spherical Nomenclature	p. 560
Pseudodifferential Operators and Geomathematics	p. 564
Bibliographical Notes	p. 568
Concluding Remarks	p. 571
List of Symbols	p. 573
Bibliography	p. 579
Index	p. 597
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