Spectral methods in geodesy and geophysics 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.