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## Über dieses Buch

This book presents, in a consistent and unified overview, results and developments in the field of today´s spherical sampling, particularly arising in mathematical geosciences. Although the book often refers to original contributions, the authors made them accessible to (graduate) students and scientists not only from mathematics but also from geosciences and geoengineering. Building a library of topics in spherical sampling theory it shows how advances in this theory lead to new discoveries in mathematical, geodetic, geophysical as well as other scientific branches like neuro-medicine. A must-to-read for everybody working in the area of spherical sampling.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
An essential objective of mathematics is to create settings and concepts to understand better our world. Mathematics is present in everyday life. Even more, almost all sciences undergo a process of “mathematization” due to increasing technological progress. An example is geomathematics. It is a key discipline for observing, understanding, and forecasting the complexity of the system Earth. We are living in a world of rapid climate, environmental, and societal change. Emphasis must be also put on the interactions with the ecosystem. For all these interactions to be managed there is a strong need for geomathematical research that becomes increasingly apparent. Understanding phenomena requires the handling and analyzing of data. Usually, data sets are discrete manifestations of continuous processes of the system Earth. As an urgent consequence, geosciences have to take care in maintaining and improving the links between the Earth’s system and its simplifications by theories, models, and simulations based on discrete data sets. The building bridge between the real world and its virtual counterpart based on mappings in form of measurements and observations consequently is one of the fundamental roles of geomathematics.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 2. Basics and Settings

Abstract
Keeping the geoscientifically motivated background in mind this book aims at specifying, developing, and classifying spherical sampling structures and methods.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 3. Spherical Harmonics

Abstract
Spherical harmonics are the analogues of trigonometric functions for orthogonal (Fourier) expansion theory on the sphere. Spherical harmonics constitute an ideal frequency-limited polynomial basis. This property makes spherical harmonics attractive for global modeling. Whenever trend representations of a signal are required on a global scale, spherical harmonics are good candidates.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 4. Zonal Functions

Abstract
The “inverse Fourier transform” providing a function F from known Fourier coefficients F(n, j) allows the (geo)scientist to think of the function F as a superposition of “wave functions” Y n,k of different frequencies.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 5. Slepian Functions: Basics and Settings

Abstract
Functions cannot have a finite support in spatial (or temporal) as well as in spectral domain. Representing signals that are optimally concentrated in both is a fundamental problem in signal processing and information theory. One-dimensional resolutions go back to Slepian and others in the early 1960s ([238], [239], [400], [397], [399]).
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 6. Stereographic Shannon-Type Sampling

Abstract
In its customary one-dimensional formulation, known in communication and electrical engineering, the Shannon sampling theorem is usually related to time-dependent signals, for which a condition between a bandwidth and sample rate has to be established (see [229], [384], [441]).
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 7. Plane Based Scaling and Wavelet Functions

Abstract
In this chapter, we present wavelet transforms on the plane which have important properties that carry over to the sphere including the ability of harmonic continuation (as, e.g., needed in physical geodesy). Due to its base on plane integration techniques and two-dimensional wavelet transforms the method of transfer to the sphere can be organized very economically and efficiently without relying on specific integration grids on the sphere.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 8. Sampling Based on Bivariate Fourier Coefficient Integration

Abstract
A standard approach on the sphere is modeling a function (signal) by its truncated Fourier expansion in terms of spherical harmonics. The calculations amount to the approximate integration of the Fourier coefficients. In doing so, economical computations can be conveniently related to a plane rectangular domain. The resulting latitude-longitude integration rules are useful whenever data sets are available in all nodes on the sphere that are originated by the rectangular grid points in accordance with the coordinate transform.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 9. Orthogonal Zonal, Tesseral, and Sectorial Wavelet Reconstruction

Abstract
Functions describing geophysical quantities, such as the Earth’s gravitational or magnetic potential, the air pressure and wind field, the deformation field of the Earth’s crust, ocean circulation, etc., are significant sources of information in geosciences. For more than two centuries such quantities have been analyzed globally in spherical approximation by orthogonal (Fourier) expansions in terms of spherical harmonics. However, this approach is not efficiently and economically applicable to data sets of substantial local variation (an example is the modeling of the Earth’s gravitational potential for coastal areas of the Pacific ocean with the Andes). Furthermore, local changes and undulations of geodata as, e.g., caused by tectonic movements, seismic activities, ocean topography, climate changes, etc., unavoidably require the application of space localizing structures.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 10. Biorthogonal Finite-Cap-Element Multiscale Tree Sampling

Abstract
In this chapter we are interested in a compromise for multiscale sampling, thereby following closely [151]. The scaling and wavelet functions are intended to be rotation-invariant, i.e., zonal functions, but they should also reflect structured latitude-longitude grids in order to obtain a fast algorithm that is easy to implement.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 11. Spherical Harmonics Interpolatory Sampling

Abstract
If the data are localized, sampling problems on the sphere can be solved through application of methods designed for the two-dimensional Euclidean space. However, problems like the determination of an Earth’s Gravitational Model (EGM) involve essentially the entire surface of the sphere so that modeling of the data as arising in the two-space is no longer appropriate. Since there is no singularity-free mapping of the entire sphere to a bounded planar region, there is a need to develop coordinate-free approximation methods over the sphere itself.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 12. Bandlimited Multiscale Tree Sampling

Abstract
Over the last decades multiscale methods have found important applications in numerous areas of mathematics, physics, engineering, and computer science. Scaledependent functions form versatile tools for representing general functions, decomposing data sets, or decorrelating signal ingredients. They especially become more and more important in Earth sciences, since most recent satellite missions deliver millions of data distributed around the globe.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 13. RKHS Framework and Spline Sampling

Abstract
Spline theory is canonically based on a variational approach (cf. [94, 104, 117, 155, 430]) that minimizes a weighted Sobolev norm of the interpolant, with a large class of spline manifestations provided by pseudodifferential operators being at the disposal of the user.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 14. Orthogonal/Non-Orthogonal Wavelet Approximations and Tree Sampling

Abstract
The concept of spherical wavelets as presented in [147] is different from earlier approaches: Rather than beginning with an understanding of dilation defined continuously over a scale interval, we restrict dilation here to discrete values.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 15. Non-Orthogonal Finite-Cap-Element Multiscale Sampling

Abstract
The idea of the low discrepancy method (see [62], [203], [325], [403] and many others) leading to spherical “finite-cap-element sampling” is simple: Approximate the integral of a function $$F:\mathbb{S}^2 \to \mathbb{R}$$ by a finite mean of functional values at prescribed points. This procedure is a reasonable approach only if the data set is somehow “equidistributed” over the sphere.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 16. Non-Orthogonal Up Function Multiscale Tree Sampling

Abstract
Spacelimited wavelets usually suffer from their physical interpretability. In addition, most of the locally supported zonal kernel functions on the sphere possess a non-monotonically decreasing L2($$\mathbb{S}^2$$)-symbol, which prevents their applicability for building up a multiresolution analysis (in the sense that the space L2($$\mathbb{S}^2$$) of square-integrable functions on the sphere can be decomposed by a nested sequence of subspaces including a closure property).
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 17. Sampling Solutions of Inverse Pseudodifferential Equations

Abstract
The main dilemma of modeling inverse problems is that most of them are illposed. The characteristic of such problems is that the closer the mathematical model describes the ill-posed problem, the worse is the “condition” of the associated computational problem (i.e., the more sensitive to errors). Therefore, the indispensable problem is to bring additional information about the desired solution, compromises, or new outlooks as aids to the resolution of ill-posed problems.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 18. Sampling of Potential and Stream Functions

Abstract
Usually, spacelimited wavelets do not reflect appropriately physical constraints. They mainly are of importance from numerical point of view because of their economy. In this chapter, however, we construct spacelimited wavelets which are close to the physical reality. The essential idea is to find multiscale regularizations of Green’s functions to spherical operators so that spacelimitation is a result of forming two certain differences of mollifications (regularizations). The concept was first realized in local geoid determination from deflections of the vertical (cf. [149]). It can be applied to potential methods of exploration (cf. e.g., [30]) and in seismic reflection as well (see, e.g., [110]). Our aim here is to explain exemplarily multiscale regularizations to potential and stream functions on the sphere (in canonical extension to [149]).
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Chapter 19. Applicabilities and Applications

Abstract
In accordance with J. R. Higgins [199] sampling a signal is considered in two variants within this book:
• the first stating the fact that a bandlimited function is completely determined by its samples,
• the second describing how to decompose or reconstruct a function using its samples in an appropriate way.
Willi Freeden, M. Zuhair Nashed, Michael Schreiner

### Backmatter

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