Spherical Functions of Mathematical Geosciences: A Scalar, Vectorial, and Tensorial Setup (Advances in Geophysical and Environmental Mechanics and Mathematics)

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. 1 Introduction -- 2 Basic Settings And Spherical Nomenclature -- 3 Scalar Spherical Harmonics -- Green Functions And Integral Theorems -- 5 Vector Spherical Harmonics -- 6 Tensor Spherical Harmonics -- 7 Scalar Zonal Kernel Functions -- 8 Vector Zonal Kernel Functions -- 9 Tensorial Zonal Kernel Functions -- 10 Application: Earth’s Gravity Field -- Concluding Remarks -- List Of Symbols -- Bibliography -- Index. Willi Freeden, Michael Schreiner. Includes Bibliographical References (p. 579-596) And Index. 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 Collects the material developed by the Geomathematics Group, TU Kaiserslautern, to set up a theory of spherical functions of mathematical (geo- )physics. This work provides the palette of spherical (trial) functions for modeling and simulating phenomena and processes of the Earth system In recent years, the Geomathematics Group, TU Kaiserslautern, has worked to set up a theory of spherical functions of mathematical physics. This book is a collection of all the material that group generated during the process.