Spectral methods in geodesy and geophysics: Fid-Bau Portal

Spectral methods in geodesy and geophysics

Jekeli, Christopher

Cover -- Half Title -- Title -- Copyrights -- Dedication -- Preface -- Contents -- Chapter 1. Introduction -- 1.1 Definitions and Notations -- 1.2 Geophysical Motivation -- 1.3 Mathematical Preliminaries -- 1.4 Summary -- Chapter 2. Fourier Transforms of Functions on the Continuous Domain -- 2.1 Introduction -- 2.2 Fourier Series -- 2.2.1 Properties of the Fourier Series Transform -- 2.3 The Fourier Integral -- 2.3.1 Properties of the Fourier Integral Transform -- 2.3.2 Rectangle Function -- 2.3.3 Gaussian Function -- 2.3.4 Dirac Delta Function -- 2.3.5 Fourier Transforms Using the Delta Function -- 2.4 Two-Dimensional Transforms in Cartesian Space -- 2.4.1 Two-Dimensional Fourier Series Transform -- 2.4.2 Two-Dimensional Fourier Integral Transform -- 2.4.3 Special Functions in Two Dimensions -- 2.5 The Hankel Transform -- 2.6 Legendre Transforms -- 2.6.1 One-Dimensional Legendre Transform -- 2.6.2 Two-Dimensional Fourier-Legendre Transform -- 2.6.3 Properties of Fourier-Legendre Transforms -- 2.6.4 Vector Spherical Harmonics -- 2.6.5 Cap Function -- 2.6.6 Gaussian Function on the Sphere -- 2.6.7 Spherical Dirac Delta Function -- 2.7 From Sphere to Plane -- 2.8 Examples of Fourier Transform Pairs -- Exercises -- Chapter 3. Convolutions and Windows on the Continuous Domain -- 3.1 Introduction -- 3.2 Convolutions of Non-Periodic Functions -- 3.2.1 Properties of the Convolution -- 3.2.2 Convolutions in Higher Dimensions -- 3.3 Convolutions of Periodic Functions -- 3.4 Convolutions on the Sphere -- 3.5 Filters on the Line, Plane, and Sphere -- 3.5.1 Filter Examples for Profiles and Spherical Signals -- 3.5.2 Bandwidth and Resolution -- 3.6 Window Functions -- 3.6.1 Classical Tapers -- 3.6.2 The Concentration Problem -- 3.6.3 Truncated Convolutions on the Sphere -- Exercises -- Chapter 4. Transforms, Convolutions, and Windows on the Discrete Domain.