Seibert, Katrin: Spin-weighted spherical harmonics and their application for the construction of tensor slepian functions on the spherical capSpin-gewichtete Kugelflächenfunktionen und ihre Anwendung zur Konstruktion von tensoriellen Slepian-Funktionen auf der sphärischen Kappe. 2018
Content
- Title page
- Acknowledgements
- Zusammenfassung
- Abstract
- Contents
- Introduction
- Fundamentals
- Basics and Notations
- Tensor Nomenclature
- Spherical Nomenclature
- Scalar Spherical Harmonics
- Vector Spherical Hamonics
- Tensor Spherical Harmonics
- Spin-Weighted Spherical Harmonics
- Spin Weight and Function of Spin Weight
- Definition of the Spin-Weighted Spherical Harmonics
- Properties of the Spin-Weighted Spherical Harmonics
- The Addition Theorem for the Spin-Weighted Spherical Harmonics
- Properties of the Operator ð
- The Uniqueness of the Eigenfunctions of Δ^(∗,N)
- Spin-Weighted Legendre Polynomials
- Additional Properties of the Spin-Weighted Spherical Harmonics
- Relation to the Scalar, Vector, and Tensor Spherical Harmonics
- Scalar Slepian Functions on the Sphere
- Vector Slepian Functions on the Sphere
- Tensor Slepian Functions on the Sphere
- Derivation
- The Normal Part
- The Left Normal/Right Tangential Part
- The Left Tangential/Right Normal Part
- The Tangential Part
- Complete Tensor Solution
- Properties
- Shannon Number
- Spectral Concentration of Spacelimited Functions
- Slepian Functions on the Spherical Cap
- Calculation of the Kernel Matrix
- Commuting Operator on the Spherical Cap
- Computation of the Slepian Functions on the Spherical Cap
- The Shannon Number of the Slepian Functions on the Spherical Cap
- Implementation of Tensor Slepian Functions on the Spherical Cap
- CMB Polarization on the Spherical Cap
- Definition of the CMB Polarization
- Tensor Slepian Functions for CMB Polarization on Spherical Caps
- Application and Numerical Experiments
- Conclusions
- Appendix Integration on the Sphere
- Gauß Quadrature
- Gauß-Tschebyscheff Quadrature
- Gauß-Legendre Quadrature
- Gauß Quadrature with Spin-Weighted Legendre Polynomials
- Integration over the Spherical Cap
- Appendix Point Grids on the Sphere
- Appendix List of Symbols
- Bibliography